Bob Jewett

Combinations and ThrowSome surprising insights into the world of "throw."

In my column last April, I covered someof the details of throw. Here are two relatedand surprising experiments for you to try.As a reminder, a ball is said to be "thrown"when its path is not directly away from thespot where another ball contacts it. Thisdeviation can be due to spin on the cue ball,or simply from the motion of the strikingball across the struck ball on a cut shot.

Most beginners will shoot the shot inDiagram 1 wrong. The two object ballsare frozen together and pointed aboutsix inches away from the pocket. Anovice will attempt to "cut" the secondball by playing to side A of the balls,perhaps expecting the first ball to moveto the right before pushing the secondball towards the pocket. Of course, weall know that you have to hit the shoton side B, and let the friction betweenthe balls drag the second ball towardsthe pocket. But how does the shotchange if the balls aren't touching? Withsome separation, as in Diagram 2, therewill be two effects, the throw from the sur-face friction, and the cut because the firstball does move to the side before it hits thesecond ball. Which effect will dominate?

If the balls are separated by a hair'sbreadth, the shot hasn't changed much andyou would expect nearly the same result asfor frozen balls. With an inch of separation,the cut effect will probably dominate. Ourgoal in the experiment is to find the separa-tion at which the cut exactly cancels thethrow, and the second ball goes straight upthe table. In Diagram 2, a ball is placed onthe far cushion along the line of the twoballs, so we can easily see the amount ofcut or throw.

For repeatability of ball placement, getsome self-adhesive donut-shaped paperreinforcements. There is a new style avail-able made of thin, tough plastic that can belifted and moved several times for reposi-tioning. A trick I use for minor tweaks is toplace a fingernail as a marker at the edge,and then lift and replace the donut therequired distance from the nail.

As a first guess, place die two balls one-quarter inch apart. That's probably close tohalf the diameter of your ferrule, in caseyou don't carry calipers. Place the rail ballon another donut, and shoot straight alongthe combination line to be sure that all threeballs are in line. Once you have the target in

the right place, shoot the shot as an anglecombo as shown, with full contact on thefirst ball, which in turn has a half-ball con-tact on the ball that's driven up the table.

Is there more throw or cut on the shot?That is, where does the ball land on the farcushion relative to the target ball? Is it thesame from the other side? If it's not, some-thing is screwy with your setup or table.Next, try a fuller hit; place the cue ball clos-er to the line of the shot, so the first ball isdriven about 3/4 full into the second ball.What happens for a more extreme angle?Does speed change the shot? If you havethe donuts on the table to replace the balls,you can try all of these changes in a fewminutes.

Of course, the results will depend on howsticky the balls are. When I tried this shot ata local pool hall, I got cancellation for ahalf-ball hit when the separation was closeto a quarter-inch. This is exactly the rule ofthumb that I've used for over 30 years, butI had never measured it before. Commonrule of thumb: "If the balls are a quarter-inch apart, there is no throw or cut, no mat-ter how you hit them." Is it a quarter-inchwith your equipment?

When I tried a fuller hit, I got a surprise.The throw dominated. With a thinner con-tact, such as 45 degrees or 1/4-ball full, thecut dominates. The result of this experi-ment is that the simple rule of thumb isn'taccurate, and you need to do some testing

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Bob Jewett

under your own conditions.I've seen ball-to-ball frictionvary by a factor of two, andthis will surely change the"zero-throw" spacing. If youtry the experiment, pleasesend me your results in careof Billiards Digest.

The second experiment inthrow was suggested byHugh Hilden, who is a pro-fessor of mathematics at theUniversity of Hawaii, andwho sometimes uses poolproblems in his calculuscourses. The setup is shownin Diagram 3, and is similarto an ancient trick shot. Tenballs are lined up straight to apocket. The critical factor in the experimentis the separation of the balls. In theory, ifthe balls are separated by one-ball diame-ter, any small aiming error you have on thefirst pair — one degree — will be copiedexactly to each subsequent collision.

Suppose you space the balls by two balldiameters instead. Geometry says that anyinitial error will be doubled in each colli-sion, and with a one-degree error at thestart, the sixth ball won't even hit the sev-enth. There is geometrical growth of the

error, and the shot is theoretically nearlyimpossible.

Where does throw enter? It turns out thatthrow between successive balls, which iscaused by the cut angle of the error, tendsto correct the error. This is partly due to theballs being thrown back in line, and partlydue to spin that the thrown ball picks up, inthe direction which tends to cancel theerror.

Professor Hilden reports that when theballs are one ball and a half apart, the shot

could be made consistentlyeven when the last ball wastwo diamonds from thepocket. For this case, theerror multiplication factorwithout correcting throw isabout 58, and the permittederror at the end is only fivedegrees, so the simple theorywould require a tenth of adegree accuracy in the aim-ing. This is a sixth of an inchin the length of the table.With the balls set two ballsapart, the percentage wasdown, but the shot was stillpossible. With a one-ballseparation, the shot becomesunmissable, even with some

intentional aiming error. This shot is fun totry, and it sounds like a machine gun whenyou shoot it. Try it with all fifteen balls.

These experiments will improve your feelfor combinations, and the next time you runinto either multi-ball or not-quite-frozencombos, you'll be ready. And if you're ajunior player deciding about college, con-sider math at the University of Hawaii. Youmay have already done the homework.

Bob Jewett is an Advanced-level BCACertified Instructor.

20 BD-JANUARY 2001

Bob Jewett

Newton on the BallThe flaws of the 90-degree rule.

Guest columnist (and fellow billiard-physics fanatic) Dr. George McBane is aprofessor of chemistry at Ohio StateUniversity, where he and his graduate stu-dents study collisions between molecules.Tools used there can be applied to billiardballs, as you will see.

Anyone w h o has played pool for morethan ten minutes has figured out that thethinner the cut, the slower the object ballgoes, and the faster the cue ball goes afterthey collide. And the first thing most playersare told when they start to learn position playis "After the collision, the cue ball leavesat right angles to the object ball's path."The first of those statements is true, thesecond is only sometimes true.

People who study collisions—of plan-ets, of subatomic particles, of balls—usea simple diagram, called a Newton dia-gram (after Sir Isaac) or velocity vectordiagram, to help figure out what laws ofconservation of energy and momentumrequire of a collision.

The diagram is easy to draw. Cueistscan use it to show how fast the cue andobject balls will go in a cut shot, what theangle will be between the cue and objectballs' paths after a collision and how thebehavior of the balls will change if thecue ball is lighter or heavier than theobject ball. It can also occasionally dis-prove plausible-but-incorrect statementsabout how balls behave, including the"right-angle rule."

The simplest version is shown in Diagram1, along with the shot it corresponds to onthe table. To draw it, start with a line alongthe direction you will shoot the cue ball. Theline's length represents the cue ball's speedjust before the collision.

Mark the beginning of the line with S (for"stationary"), and the end with I ("initial").At the midpoint of that line, put a dot (C).Draw a circle with its center at C that passesthrough S. Now, starting at S, draw anotherline, parallel to the path the object ball musttake to the pocket; extend it until it touchesthe circle. Mark the point of its intersectionwith the circle O ("object"). Draw a linefrom O through C to the circle on the otherside; mark that intersection F ("final"), andfinally draw a line from S to F.

The line from S to O gives the directionand speed of the object ball after the colli-

sion. The line from S to F gives the directionand speed of the cue ball. It's easy to see thatas the cut angle (the angle from I to S to O)gets bigger, the object ball speed will getsmaller and the cue ball speed will get big-ger, until finally for a perfect 90-degree cutthe object ball will not move at all and thecue ball will go straight forward withoutslowing down. If you remember your geom-etry, you might also be able to show that,with this diagram, the angle between thefinal cue and object ball directions is exactly90 degrees, no matter what the cut angle is;the "right-angle rule" is correct in this case.

In carom games, one-pocket and safeties atpool, it is often important to control thespeeds of both object ball and cue ball, andthis diagram can show you how those speedsvary with the cut angle (The diagrams tellyou only about the collision between theballs, so they apply directly to stun shots.Follow or draw will affect the speed anddirection of the cue ball; those effects mustbe "added on" to these).

What assumptions lie behind this picture?First, these diagrams assume that all theaction takes place in a single plane. If thecue ball is airborne, or is different in sizefrom the object ball, the slate enters the pic-ture in an intimate way and the diagrams arenot as useful. Diagram 1 also assumes thatthe cue and object balls have the same mass,and that the total translational energy (ener-gy of motion along the table) is the samebefore and after the collision. These latter

assumptions are often violated, and slightlydifferent diagrams must be used.

It's rare for there to be no change in thetranslational energy. Usually there is somechange in the spins of the two balls duringtheir collision, so that translational energy isconverted to rotational energy or vice versa;to prevent that, you have to either hit astraight — in stop shot, or you have to hit a stunshot with just enough outside english thatthe surfaces of the two balls do not rubtogether when they collide. When the ballsdo rub together, some energy changes frommoving the balls along the cloth to making

them spin, or vice versa. Some also goesto producing the sound of the hit, andsome warms up the balls; both of thoseamounts are usually too small to worryabout.

In the Newton diagram, changes intranslational energy change the size ofthe circle. In the majority of shots, trans-lational energy is lost, and the circle getssmaller. This is true for shots where thethrow tends to decrease the cut angle:inside English, center ball, orlittle/enough outside English that the cueball still "slips forward" on the objectball as it hits. If you use enough outsideenglish that the cut angle is increased(which is easiest for nearly straight-onhits), then some of the initial spin of thecue ball ends up as translational energy,and the circle can get bigger.

Diagram 2 shows how this works. Thetwo diagrams drawn there are both for 30-degree cut shots to the right. On the left, theshot was hit with center-ball. At the momentof contact, the balls rubbed together, and thefriction from that rubbing threw the objectball to the left and imparted some clockwisespin to each ball (See Tech Talk, April 2000).The throw has no effect on the diagram,since the line from S to O was drawn in thedirection the object ball actually traveled. Ittakes energy to makes the ball spin, though,and that energy comes from the initial trans-lational motion of the cue ball; that makesthe circle smaller, so that the line from O toF is shorter than the line from S to I. Now theangle from O to S to F is less than 90 degrees.

On the right, the same shot was hit withheavy outside (left) English (The playeraimed differently than on the left, so that theobject ball would still leave in the samedirection—toward the pocket). In this shot,

14 | B D - FEBRUARY 2001

Bob Jewett

when the two balls came together, the rub-bing between them was in the other direc-tion; the object ball was thrown to the rightand picked up a little counter-clockwisespin, and the cue ball lost spin. The trans-lations! energy increased overall, so theline from O to F is longer than that from Sto I. Now, the angle from O to S to F islarger than 90 degrees; the separationangle widens in this case.

How big are these changes in separationangle? It's reasonable to think of them thisway: the cue ball always leaves at rightangles to the line between the two ballcenters at contact, while the object ballwill be thrown to the right or left of theline between centers. So, the changes in sep-aration angle are the same size as the throwangles, and with clean balls those are rarelybigger than five degrees. That argument isnot exactly right; some energy is lost to heatand sound, and the balls do move slightlyduring the time they are in contact so the"line between centers" is not preciselydefined, but it gives a good estimate of themaximum change from right angles.

You can occasionally use these changes inseparation angle to maneuver around anobstructing ball in your path to the next shot,or even to take a free carom shot at the nineball while still pocketing your main objectball. The diagrams also show that it is notpossible to make a cut shot without havingthe cue ball move in the direction opposite

the cut, as is sometimes claimed.If the cue ball and object balls do not

weigh the same, there is a dramatic effect on

the cue-ball path. This situation is most com-mon on coin-operated tables, but can alsoappear on other tables if the cue ball is mis-matched or worn. To draw this diagram,instead of placing point C at the midpoint ofthe first line, you put it at the "balancepoint": the point where a light, stiff rod withits ends at S and I would balance if the cueball was put at I and the object ball at S. Inother words, the distance CI times the cueball mass must equal the distance CS timesthe object-ball mass. Then, draw two cir-cles. One should have its center at C andpass through I; that is the "cue-ball circle."The other, the "object-ball circle," has itscenter at C as well, but passes through S.Then draw the line from S in the object balltravel direction as before, and label its inter-

section with the object ball circle O. Draw aline from O through C and on to the cue ballcircle; its intersection with the cue ball circle

is F. Finally, draw the line from S to F thatshows the final direction and speed of thecue ball. Diagram 3 shows a heavy cueball (left) and a light cue ball (right).(Translational energy changes wouldmake both circles smaller or bigger;Diagram 3 shows the case where there isno change.)

If the cue ball is heavier, the separationangle varies smoothly from zero for astraight — in hit to 90 degrees for a very thincut. A heavy cue ball produces "instantfollow"; the cue ball will start out moving

forward of the right-angle line, even if itarrives with back spin, and before frictionwith the cloth has had any effect.

If the cue ball is lighter than the object ball,then you get "instant draw." For a straight — inshot, the cue ball will back up after contacteven if it did not have any spin (Think ofthrowing a soccer ball at a bowling ball). Asthe cut gets thinner, the separation angle willdecrease from 180 degrees, and finally forvery thin cuts it will approach 90 degrees.

For both heavy and light cue balls, themost dramatic effects appear for shots nearstraight in. Because the separation anglechanges dramatically with the cut angle,carom shots are much more difficult withmismatched balls.

—George C. McBane

Bob Jewett

Weird TechniquesGet a better grip on hard-to-reach shots.

With perfect play, every shot is goingto look the same: comfortable stance with-out stretching; smooth, straight stroke; easyposition on the next ball using little or nospin; repeat. This can be dull. When playersget into trouble, you get tosee some interesting tech-niques. You should learnsome of these — at leastthe legal ones — so you'llbe ready on those rareoccasions when you getout of line.

One common question iswhat to do when you can'treach a shot normally. Thestandard techniques are touse a mechanical bridgeor shoot opposite handed,and both of these shouldbe part of your practice.Suppose, though, thatyou're faced with the shotin Diagram 1, and yourname isn't Shaquille. Youneed to hit the ball on thecushion, and most of therack is in the way.

The standard method forthis situation is to placeone mechanical bridge ontop of another to get added height — thetop bridge's shaft goes in the slot whereyou would normally put your cue stick. Itcan also help to turn one or both bridges ontheir sides to get more height. If lots ofballs are between you and the cue ball, thetop bridge can be pushed forward to can-tilever out over the obstructions. Be sure togrip both bridge handles firmly together forstability. As with most bridge shots, don'tbe ambitious about what you are going todo with the cue ball; easy does it. If youhaven't practiced this one before you needit, good luck. It is illegal to stack more thantwo bridges. It's also illegal to rest yourhand on top of the rack, even when playingby "cue-ball fouls only."

There are several bridges on the marketthat help with this kind of shot. At snooker,a bridge (snooker players call it a "rest")with a long "swan's neck" that arches outover obstructions is available. Anothersnooker bridge called the "spider" has feetwide enough and long enough to straddle asingle ball — think bow-legged. On this side

of the pond, bridges are available that locktogether for additional stability.

A more creative use of the bridge on thisshot is to place the head on rail A and thebutt at rail B. The bridge handle will be

high enough to clear the obstructing balls.Now place your hand on the bridge handleand form a more or less normal bridge forthe shot. This technique seems to be legal.

Another technique is useful when youneed to hit the ball pretty hard, and youdon't trust the mechanical bridge for thatmuch power. An example would be at bil-liards where you have to take the cue balloff the left side of the object ball inDiagram 1 and go six cushions for thescore. The shot can be reached from theside of the table, but you can't get yourhead over the cue stick to sight. RaymondCeulemans has been known to place thecue stick on the correct line while standingat the other end of the table, walk around tothe side, carefully pick the stick up withoutmoving it off line, and then stroke "blind-ly." Under Billiard Congress of Americarules, this is legal provided that you main-tain contact with the cue stick.

Jump shots are so common now that theyhardly deserve mention, except for thewrong ways to do them. It is not legal to

make the cue ball clear an obstructing ballby miscuing (scooping the ball). It is notlegal to shoot jump shots with just yourshaft. Shaft jumping is effective becausethe resulting very light stick stops on con-

tact with the cue ball andlets it rebound freelyfrom the table. It is notlegal to use a very hardmaterial such as phenolicfor a tip. Such "tips"seem to help the cue stickstop faster, but can behard on the equipment.

Speaking of miscues,Shot A in Diagram 2shows where some play-ers are tempted to use anintentional miscue. Thecue ball and object ballare pointed straight at apocket, but are only aquarter of an inch apart.The problem is to avoidhitting the cue ball a sec-ond time as the cue stickfollows through. If youaim to hit the cloth andthe ball at the same time,a miscue will result, andthe cue ball will hit the

object ball and jump straight up in the air.Players who try this in tournaments protestthe resulting foul call because they say theywere not playing a jump shot, but the shotis ruled a foul nonetheless.

A legal technique for 2A is to move yourgrip hand forward so that it will be at therail just before the tip hits the ball. If youstroke with the stick rubbing the rail, yourhand will smash into the rail and stop thestick before it hits the cue ball a secondtime — correct placement of the grip handis critical to getting power without a foul.This technique was said to be a favoriteproposition of Luther Lassiter. With prac-tice, it doesn't hurt much.

Diagram 2B shows a straight — in shot thatrequires draw. Being right-handed, youcan't reach it, and the mechanical bridge ismissing in action. Jerry Briesath has thecute solution: lay your stick flat on the tablewith the tip nearly at the cue ball. With yourleft hand, pinch under the stick just a littleat the joint so the tip is the right height onthe cue ball. Now jam the heel of your right

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Bob Jewett

hand into the rubber bumper onyour stick. Jerry can draw the cueball the length of the table. Canyou? I suppose a pencil under thestick would also work to raise thetip to the right height, but thatwould be Illegal Use of a Device.

One technique for getting a lotof side spin is to aim as if for noEnglish and then swerve to theside on the final stroke. Someplayers claim this works but Idoubt that it gets more spin thatcoming straight through, and Ican't think of a better way todestroy consistency. SemihSayginer and Mike Massey seemto get enough spin without thistechnique. The only benefit ofthis method is that it can compen-sate for squirt under some condi-tions.

A related technique some professionalplayers use is to always set up at the bottomof the cue ball and then hit high, low, left,right, center or anywhere. The rumor is thatold-time players used this to baffle theiropponents. There is one thing it will helpwith: If you have a problem seeing wherethe center of the cue ball is, starting at thebottom is the best place to see if you are

centered.Masse shots give plenty of room for

"interesting" techniques. Grips include theregular, the "dart" grip, and "The Claw."For this last grip, get into full masse posi-tion, and make a "V for victory" sign withyour grip hand, palm down. Put the stickinto the V and then curl your two fingers tograb the stick between them. You can getsurprising power with this grip. If the

owner of the table asks, you don't knowme.

Have you seen something really strangeon the pool table? A technique, that is. If so,let me know about it, in care of this maga-zine, and I'll describe it in a future article.In the meantime, perfect the above tricks.You won't need them often, but they'reinteresting to try, and it's good to be pre-pared.

14 BD-MARCH 2001

Bob Jewett

Three Draw DrillsSomething old and something new.

How's your draw? I recently wasworking on mine to improve my finesseposition for one-pocket and 14.1. The fol-lowing drills have something new and chal-lenging even for the old hands.

I saw the first drill nearly 40 years ago inWillie Mosconi's Winning Pocket Billiards.Place a number of balls in a semi-circlearound the side pock-et and shoot them inorder. Where theballs start is up toyou; I find it's best toput the balls as closeto the pocket as pos-sible without block-ing a shot. You arenot allowed to touchany cushion with thecue ball or to bank anobject ball. Reset allthe balls as soon asyou have missed ashot.

Diagram 1A showshow the start of a perfect run looks. The cueball follows a zig-zag path; after each shotit comes to rest about eight inches from thenext ball, and lined up so that a shot to thecenter of the pocket gives just enough angleto get to the next ball. You will quicklylearn what that angle is, but a good rule-of-thumb is to leave the cue ball so that shoot-ing full at the object ball will barely put theball in on the right side of the pocket.

Now for a new wrinkle on this old drill: ifyou make all the balls without a miss, addone ball to the semi-circle for the next run,but if you fail, subtract one ball from thenext set. This is now a form of "progressivepractice" and you can keep track of howyou are doing by simply noting how manyballs are in the present semi-circle. If youneed a grade for motivation, three is a C,six is a B, and nine is an A.

Diagram IB shows a problem you willlikely encounter. The cue ball stopped toosoon for a good shot on the 3 ball as shownby the dashed line, which points down therail several inches past the pocket. If youhave perfect soft draw and cheat the pocketto the verge of missing, you may be able tokeep the cue ball from going past positionfor the 4 ball. In this situation, try using aslevel a cue as possible and right Englishwith the draw. The idea is that the side spin

will throw the object ball into the pocket sothat not as much cut angle is needed. It willtake you a while to get the feel for this.Accurate aim is mandatory, since you willstill have to cheat the pocket.

Diagram 1C shows the opposite prob-lem; the cue ball has wandered too far andis on the wrong side of the 3 ball. You need

to hit the object ball on the left side and getthe cue ball to come to the right.Amazingly, the answer is once again rightside-spin. The trick is to elevate to about 45degrees and play a half — masse shot. The cueball will curve before and/or after hittingthe 3 ball, and with the right touch will getback in line for the 4 ball. This shot callsfor a solid raised V — bridge and finesse.Don't shoot it like you're killing snakes;instead, it should be more like kissing yourGrandma.

In Diagram 2 is a chance for you to testthe idea that follow is better than draw forposition. You have ball in hand on the 1 ballat a game of 9-ball, and need to get to theshaded area for a shot on the 2 ball. Should

you place the cue ball on the A side fordraw or the B side for follow? Try each way10 times to see which works best for you. Ifyou have trouble with the fallow shot, themost likely problem is failure to hit highenough on the cue ball; the higher you hit,the less speed is needed for the position.Next try placing the 1 ball at A. Is draw bet-

ter than follow now?Finally, place the oneball at B. There is arule-of-thumb thatsays with ball inhand, you shouldnever place the cueball for a draw shot.Is any of these shotsan exception for you?

Finally, Diagram 3is an interesting drawdrill that can be usedas a challenge. Thegoal is to pocket the 9ball in Pocket X witha draw-carom after

pocketing the 3 ball. You don't have to getthe 9 all the way there in one shot, though;leave it where it rolls to and try again. The3 ball comes back up to the same position,and the cue ball is in hand for each try. Thepositions A, B and C show possible succes-sive positions of the 9 if you shoot softly.And it is good to land softly on the 9, sincethis will teach you precise finesse position.Softer shots also tend to keep the 9 near thecushion where it is easy to get to. If you failto touch the 9, start the drill over.

While you work on this drill, try to get afeel for the angle the cue ball draws to for aparticular cut angle on the 3 ball. With a lit-tle practice, you'll be able to pull the cueball right along the end rail even with astarting angle over 30 degrees. For thisdrill, you will probably want the tip as lowas possible on all shots to minimize thecue-ball speed. In the first drill in Diagram1, it is often useful to take a little draw offthe ball to keep from pulling back to far.

One variation is to go for as many con-tacts as possible. For this, soft shots aremandatory, and hitting the cushion with leftside spin seems to keep better control of thecue ball. My record is about 18 shots beforepocketing the stripe. Another variation is togo for the fewest shots, but you'll see thatmuch better accuracy is required.

14 | BD- APRIL 2001

Bob Jewett

One last word on this drill. If youover-do it and run the 9 ball to Y, not allis lost. Very carefully line the cue ballup between the 3 and the upper edge ofthe 9, and then do your best to draw thecue ball perfectly straight back to hit the9 on the upper edge. This shot is wortha couple of tries even if you miss thefirst time, just to get a feel for this veryprecise skill.

For all of these drills, remember tochange sides. The shot may look a lot

different to you when it's reversed. Ofcourse, the shots in Diagram 2 may needa bridge on one side for players who don'tswitch hands. Draw is always a good skillto have ready for position play. Finesse-draw is even better.

Bob Jewett is an Advanced-level BilliardCongress of America-certified instructorand partner in the San Francisco BilliardAcademy, which is BCA-certified forinstructor training.

BD-APRIL 2001 1 5

Bob Jewett

Charting the CourseConverting spin to speed.

This month w e ' r e going to look intodetails of the physics of straight draw andfollow shots. Don't worry about equationsand algebra — most of the work is going tobe done by a simple graphical tool. Nextmonth the study will be extended to cutshots with draw and follow, and the toolwill show us immediately the angle thecue ball will take for any cut angle andany amount of draw or follow.

The basic notion of the tool is that anyball has a speed and a roll, and these canbe shown on a diagram with two arrows.A simple rule will tell us how the speedand spin change if they don't "match."Consider a cue ball rolling smoothly onthe cloth. From its center, we draw anarrow in the direction of its movementwith a length that shows its speed.Physicists call such an arrow a vector,but we'll stick with "arrow."

We will also represent the follow ordraw on the ball with another arrow.Let's call this the spin arrow, while thefirst one will be the speed arrow. Thespeed arrow is shown as a solid arrow,while the spin arrow is shown with adashed line. Since rolling smoothly onthe cloth is the natural state of the cueball, let's make the two arrows equal inthat case.

The arrow diagram for our rolling cueball is in Diagram 1. The two arrows arethe same length and in the same direc-tion. How fast is the ball going? Thatdepends on the scale of the arrows. As we'llsee below, most of the results give ratios ofspeeds, so each diagram will apply to allballs in a particular situation without regardto the actual speed or spin. For example, allsmoothly rolling balls have an arrow dia-gram like Diagram 1, regardless of speed.

It's pretty clear that a smoothly rollingcue ball will remain like that until it hitssomething, so the two arrows will remainmatched. Suppose this cue ball hits anobject ball full. What is the arrow diagramfor each ball right after the collision? InDiagram 2, the object ball is shown on theright and the cue ball on the left. At first,the object ball has a speed equal to the orig-inal speed of the cue ball and no spin; it'ssliding on the cloth. The cue ball is theopposite; it has no speed but retains all ofthe follow it had just before the collision.

How do these diagrams change the sec-

ond or two afterwards? We all know fromexperience that the object ball will pick upsmooth forward roll, while the stopped cueball will accelerate forward with its excesstop spin until it too is again rolling smooth-ly on the cloth. The two arrows for eachball will match when each ball reaches that

state, but what happens in the interim?The amazingly simple rule that describes

how spin and speed change to match is this:the tips of the two arrows move towardseach other at a constant rate, and the spinarrow moves two and a half times fasterthan the speed arrow. Diagram 3 showshow the arrows change with time — per-haps in each tenth of a second. On the top,the object-ball speed decreases while thespin (forward roll) increases. Similarly, thecue-ball spin partly turns into speed. Notethat the spin changes more than the speedon each ball, by that factor of two and ahalf.

The actual rate at which spin and speedbalance is determined by the frictionbetween the cloth and the balls. On stickycloth, the transition period will be shorter;on slippery cloth, or with waxed balls, theequilibrium will take longer to occur. The

first surprising result that we can see fromthese diagrams is that for our full-follow-shot case, the cue ball and the object ballwill reach smooth rolling at the sameinstant, because the two sets of arrow headsbegin with the same separation.

Also note in the diagram that the objectball ends up with more speed than thecue ball. This is because of the 2.5:1ratio of how quickly the arrow headschange. This ratio is determined by howefficiently a solid sphere (like a poolball) stores energy in rotation comparedto simple forward motion. The ration offinal speeds is also 2.5:1, and if wesquare this we get the ratio of how farthe balls will travel, or 6.25:1. This fac-tor was discussed here in December oflast year and is useful to know whenplaying soft-follow shape; the cue ballwill go forward about 1/6 as far as theobject ball is driven.

Of course, if the cue ball had no followor draw when it hit the object ball, itwould have neither speed nor spin afterthe collision, and it would have no rea-son to move. Suppose the cue ball had"perfect" draw. Then its spin arrowwould be back away from the objectball, and would be just as long as itsspeed arrow at impact. The spin-to-speed transformation would take placejust as before, but in the opposite direc-tion. (Remember that "perfect" draw isdefined as just as much spin as a rolling

ball but back spin rather than follow. It isabout the limit of what you put on the cueball with a level stick.)

What would happen if the cue ball hadonly partial natural roll when it struck theobject ball? If you knew how much followit had, you could draw the spin/speedarrows for it and find the final result. Try anexample with "half-follow" on the cue ball,which you can get by striking the cue ballat about 6 mm above its center. For bonuscredit: how far will the cue ball move for-ward compared to the object ball for thishalf-follow case?

Where is side spin in all of this? Hiding.Until the cue ball hits a cushion, side spinhas almost no effect. The dynamics of drawand follow shots are unchanged by thepresence of side spin on the cue ball.

As a last example, consider a cue ballthat's struck with perfect draw, but without

14 BD-MAY 2001

Bob Jewett

an object ball close by. Its arrow diagramis shown in Diagram 4, as is the timedevelopment of the spin and speed. If yougo through the ratios, you'll discover thatthe cue ball ends up rolling at only 3/7 ofits initial speed, and it will go only about20% as far as a ball that's been struck for"perfect" follow at the same stick speed.This explains the working of the "drag"shot, in which you shoot a long shot hard-er but with draw so roll-off can't hurt youso badly but you can still land softly on thedistant shot.

In next month's column, we'll extend thisidea to cover draw and follow with cutangles included. This will let you plan anycarom with any amount of draw or follow— theoretically, at least. The diagrams willhelp explain the working reasons for sever-al draw and follows systems that have beencovered in previous columns.

Bob Jewett is a partner in the SanFrancisco Billiard Academy, which offersclasses at all levels from beginning playersto advanced instructors.

BD-MAY 2001 15

Bob Jewett

Drawing DrawPart 2 of charting the cue ball's path.

1

2

Last month I described a graphic tech-nique for understanding how draw and fol-low interact with the forward speeds of thecue ball and object balls, and how speed isconverted to spin and vice versa. The basicidea is shown in Diagram 1, where the cueball is struck with draw. Ifthere is no object ball in thecue ball's path, the speed andspin arrows evolve as shown,with the spin (draw) arrow,getting shorter and finallyturning into follow as thespeed arrow decreases fromthe initial value due to thedrag from the draw. Animportant point is that thespin arrow changes five unitsfor each two units that thespeed changes. Another isthat the two arrows move at aconstant rate towards eachother. That constant rate isdetermined by how slipperythe cloth is.

While this analysis is bothinteresting and useful, thereal action starts when thecue ball hits an object ball atan angle. At that instant, thespeed of the cue ball, whichwas in line with the draw orfollow arrow, is knocked bythe collision to a differentline with a different speed. InDiagram 2 the cue ball thatwe loaded up with draw isseen from above as it hits anobject ball for a half-ball hit.The initial speed and spin areequal, and in opposite direc-tions. Remember that wecalled this amount of draw"perfect," as there was just asmuch draw as a smoothly rolling ballwould have follow.

What does the speed of the cue ballbecome? To find it, draw a rectangle asshown in the diagram. One corner is at thecenter of the cue ball, and one side is alongthe line joining the centers of the cue balland object ball. The final detail that com-pletely sets the rectangle is that the speedarrow of the cue ball is a diagonal of therectangle. Believe it or not, there is onlyone rectangle that satisfies these three

requirements.The new speed arrow of the cue ball is

exactly the side CA, and the speed arrow ofthe object ball is side CB. This is a strangesituation for the cue ball. The speed and thespin are no longer in the same direction (or

Spin (draw) Speed

Spin (follow)

Draw

Speed

along the same line) — what happens toeach, and where does the cue ball go?

The two arrows still move according tothe rule given earlier. They move towardseach other (along a straight line) with thespin arrow changing 2 1/2 times as fast asthe speed arrow. This is shown in Diagram3, where the successive arrows are shownfor five different times in sequence. At 5th-time sequence, the arrows are the samelength and in the same direction, so thechange stops at that point.

What sort of path does the cue ball followduring this time? It is easy to get a closeapproximation of the path by joining thespeed arrows head-to-tail in order.Diagram 4 shows what this looks like; thepath shown is roughly a curve. If we used

twice as many times, with thetimes more closely spaced,the ten arrows would form aneven smoother curve.Technically, the ideal curve isa parabola, which is also thepath a ball follows whenthrown. At the end of thecurving part of the path, thecue ball will continue to rollalong the direction of Arrow5, since by then the speed andspin have reached their happycommon equilibrium.

This draw shot repays timespent in practice. Notice thatthe final path of the cue ballis a little past the perpendicu-lar to the initial path. (Thecalculated angle is about fivedegrees.) Previous columns,including Dr. GeorgeOnoda's column in May1989, suggest that it is hardto pull the cue ball backbehind the perpendicular.How does the shot work foryou? (It's no fair if you cheatby hitting more than half theobject ball.) If the perpendic-ular is your limit, what aresome reasons you might notbe getting the angle predictedfor "perfect" draw?

Usually the final directionof the cue ball is far moreimportant than the exactcurve it takes before it settles

into that path. A simple case is shown inDiagram 5. This is the same shot as inDiagram 2, but with follow rather thandraw. The speed is the same, but the spin isin the opposite direction. The final direc-tion can be found by joining the speed andfollow arrows, and then finding the pointalong that line that divides it in the ratio of2:5.

The angle between the initial and finalpaths in this case is a very important one toknow; it is the natural angle for a half-ball

1 4 BD-JUNE 2001

Bob Jewett

hit. Position play that involvesanything close to half-ball (a30-degree cut) and a rollingcue ball will produce a deflec-tion angle very close to this(about 34 degrees).

The geometry of the rectan-gle and the arrows can be usedto develop various systems forcue-ball control. For example,you can show that if you playa follow shot with a small cut

angle to the left, the cue ball will be deflect-ed to the right by 2 1/2 times the angle. Ina previous column, I suggested 3 as theratio. See what happens for you with realballs on real cloth.

Where does this factor of 2:5 come from?Roughly stated, it says how much moreeffective the simple movement of the massof the ball is than the movement you getfrom the spin rubbing on the cloth.Physicists call this the "moment of inertia."The usual assumption is that the ball is uni-form, but this is not always the case. Somecue balls have heavier centers, and theeffectiveness of spin on such balls is small-er; they would be more lively if they werehollow with a heavy shell. Think about itthis way: if you had a bicycle tire made outof lead, it would be hard to stop its spin by

grabbing the rim, but if all the lead were inthe hub, the spin would be much easier tostop with that long lever arm from thespokes. How large is the effect for cueballs? It seems to be no larger than theeffect of the cue ball's being small and lightfrom wear.

In theory, this graphic system for figuringout where the cue ball will go can give youprecise results. Since it depends on know-ing how much spin the cue ball has, the eas-iest case is when the cue ball is rollingsmoothly on the cloth. Fortunately, playingwith a smoothly rolling cue ball is the eas-iest way to play position, next object ballwilling. This system may also be useful forletting you know which shape shots areimpossible — the speed arrowhead can bepulled over only so much by the spin arrow.This system can even be applied to masseshots, and for a given stick elevation andoffset, you can draw out the curved part ofthe path as in Diagram 4. That extensionwill have to wait for a future column.

Bob Jewett is an advanced-level BilliardCongress of America Certified Instructorand a partner in the San Francisco BilliardAcademy, which offers classes at all levelsfrom beginning players to advancedinstructors.

BD-JUNE 2001 15

Bob Jewett

The RackIt's more than a torture device.

One w a y or another, a major change iscoming to pool. The advent of the SardoTight Rack is forcing players and officialsto re-examine what a rack should be andhow to deal with a nearly perfect framing ofthe balls. At least one rule has alreadychanged, although you might not havenoticed.

Let's begin by asking a question: Shouldthe rack be tight? The norm for many lazyand/or cheating rackers is to leave a sloppy,loose grouping of balls on the table, result-ing in a break that is unpredictable andoften ineffective. The rules require — and Ithink it's only fair — that the balls beracked as tightly as possible, but how tightis that?

It is not theoretically possible to freeze allthe balls in a rack. Here, we're talkingabout real pool balls that are all slightly dif-ferent sizes. A new set of good balls will bethe same diameter (and round) within onethousandth of an inch. The rules permit fivetimes that deviation in each ball. With realballs, it is usually impossible to get every-thing frozen. Start with the 9 ball in a 9-ballrack. Assemble the rack by putting the 9 inposition, and then freezing the 2 ball to itsside. Freeze the 3 ball to those two, and soon around the 9, forming a ring of six ballswith the 9 in the middle, as shown inDiagram 1. With the addition of each ball,there is no choice about where to put it,since it must be frozen to the two otherballs. Suppose the 7 ball is a little small, butall the other balls are perfect. The result isas shown. The 7 can be frozen to the 9 and2, or the 9 and 6, but it cannot be frozen toall three of its neighbors.

Suppose the 7 was larger than the otherballs. Then the hole the 7 is supposed to fitin would be too small, and while you couldfreeze it to both the 6 and 2 balls, it couldnot also touch the 9 without forcing someball away from the 9.

You complete the rack by adding the 1ball and the 8 ball at top and bottom, andboth of those can be frozen easily to the 6-7 and the 3-4, respectively. This means thatfor any normal set of pool balls, it is possi-ble to rack nine balls with only one gap,and because balls are never exactly thesame size, it is almost certain that there willbe one gap among the balls around the 9. Ifthere are two gaps, the rack could be better,and it is reasonable to ask the racker to try

again.In other games, the number of gaps

varies. At 6-ball, you can obviously alwaysdo a perfect rack: start with a triangle ofthree balls, and then put the "corners" on —everything can be frozen. At the 15-ball

games, you can work from our 9-ball rackby putting three-ball "corners" against the8-4-5 and the 8-3-2 sides. With a littlethought, it's clear that each set of three ballsmight require one additional gap, givingthree required gaps in any 15-ball rack.

Pat Fleming has pointed out a way to getrid of these "necessary" gaps that uses thefact that most pool balls aren't round. Onmany sets of old balls, the "eyes" of theballs — where the numbers are — bulgeout. In our example above, you might makethe 7 ball "wider" by rotating it until itseyes are against the 6 and 2 balls. If thatdoesn't quite close the gap, you couldrotate the eyes on the other balls into ser-vice. This is all a little far-fetched, but it'ssomething to keep in mind if you just can'tget a normal rack tight.

Until the recent introduction of the Sardo

rack, the chance of getting a rack with min-imum gaps by using a wooden triangle wasnil. On new cloth and with a good set ofballs, you can get close, and players oftenremark on how well the balls break undersuch conditions. As the cloth wears andgets small craters in the rack area, and espe-cially as the rackers move the rack aroundand spread out the craters, the chances getslim, and soon Slim leaves town.

Several things help the Sardo rack pro-duce nearly perfect racks. In tournaments,the cloth is marked to show exactly whereto put the edges of the rack, so the chanceof "crater spreading" is reduced. Comparedto the typical wood triangle, it is far moreaccurate mechanically — just think abouthow often you have to turn an standard tri-angle to find the one good corner. Finally,the top part of the rack that comes down toposition the balls pushes them gentlytogether, rather than forcing them into spe-cific positions, so any mismatched ball isaccommodated as well as possible.

What happens differently with a near-per-fect rack? At 14.1, the ideal break is for thetwo corner balls (the 1 and 5 in Diagram 2)to go to cushions and return to the rack withno other ball moving — a perfect rerack.With a good rack, this shot is impossible.As shown, the cue ball will hit the 1 ballinto the 4. The 4 will move to the right andit will contact both the 9 and 3. The energytransmitted to the 9 will eventually emergewith the 10, 14 and 11 balls leaving therack. If you want to do the perfect 14.1break, just leave gaps between the 4-9, 3-15and 2-12. If those balls are frozen, extraballs are guaranteed to move.

What happens differently at 9-ball? Thefirst thing that a lot of players noticed withthe new Tight Racks is that if you find thecorrect spot for the cue ball, one of thewing balls (5 or 2 in Diagram 1) is almostguaranteed to go in, even with a moderatespeed (and well-controlled) break shot. Forat least a year, the solution in tournamentshas been to move the rack position so thatthe nine ball rather than the one is on thespot. This makes the wing balls much hard-er to make, as they must be driven morethrough the balls behind them.

I noticed a second difference in one ofAllison Fisher's matches at the Hopkins'Super Billiards Expo this year. When shebroke, the 1 ball reliably found the side

14 BD-JULY 2001

Bob Jewett

pocket. Try this yourself: break from theside cushion and hit the one ball nearly full.Test two cases, one with the front six balls

all frozen — it's always possible to get this— and one with the one ball slightly sepa-rated from the balls behind it. I think you'llfind that a millimeter can make all the dif-ference in the world to this shot. What wasremarkable was not that Allison played the1, since that is the most predictable ball tomake with the "forward-spotting" rule inforce. What caught my eye is that sheplayed the shot at moderate speed andseemed to be playing position on the 2 ball.The routes of the other balls in the rackstart to be predictable if the rack is the sameevery time.

A final difference at 9-ball is that the 9almost never moves. Although the two ballsin front of it will push on it, the two ballsbehind will take up the energy, just as forthe middle ball in a three-ball combination.If the nine isn't kissed — and on manybreaks it isn't — it will still be in placewhen the commotion dies down. This canlead to some interesting shots. At the recentBilliard Congress of America Open 9-BallChampionship at the Riviera Hotel &Casino in Las Vegas, Oliver Ortmann wasplaying Ernesto Dominguez, and he neededtwo more games for the match. On hisbreak, the 9 stayed at home, as expected,and the back ball, shown as the 6 inDiagram 3, came four cushions directly at

the 9. In a minor miracle, the 6 hit the 9 justright to send it into the corner pocket.Tough luck for Dominguez; if any ball hadtouched either the 6 or 9, Ortmann would-n't be "on the hill." Ortmann broke again,and as if it were on tracks, the 6 camearound four cushions, hit the waiting 9 ball,and the match was over.

If we are going to accept tight racks at 9-ball, how should the rules change? Here'sone possibility for your consideration:After the break, all balls that were madewill spot, and the breaker gets the next shot.A 50-mph break will no longer be useful,but the knowledge and skill to play specif-ic position on the 1 ball will be essential.The 1 ball could go back on the spot, sinceit would actually be a disadvantage to makea ball on the break. I think it would be abetter sort of 9-ball.

If you would like to see what a very tightrack at 9 ball plays like, send me email(jewett@netcom.com) and I will send backa "PRN" attachment that will let you makeyour own racking template. You will need acomputer printer and a paper-hole punch.

Bob Jewett is an Advanced-level BilliardCongress of America Certified instructorand was the National Collegiate Championin 1975.

BD-JULY 2001 15

Bob Jewett

When Spheres CollideMore here than meets the eye.

In last month's column, I mentioned inpassing that pool balls compress duringcollisions. This seems to be contrary towhat you see on the table: the balls appearto be hard, incompressible spheres. In fact,there must be some "give" to the surface, orthey would not behave nearly as perfectlyas they do.

In July 1998, I explained that to study acue stick, you should think of it as smalllumps of mass joined by short, stiff springs.Since the naked eye can't see the stick com-pressing along its length during a shot, youmight conclude that the stick was perfectlystiff, did not compress, and delivered itsenergy through the tip to the ball nearlyinstantaneously. In fact, what happens isthat ball pushes on the tip and compressesit, the tip pushes on and compresses the fer-rule, which pushes on the shaft, which thenpushes through the joint, into the butt andfinally to the back end of the cue. As theball comes off the tip, all this compressionis relaxed, and the energy stored in thecompression of both the tip and the stickitself is mostly released into the cue ball. Isay "mostly" because both the stick and thetip are not perfectly springy. Some energyis lost into the stick/tip combination, but ifthere were no springiness at all, the cue ballwould have only about 60% of the speedwe see.

Do you remember from high school whatsound waves are? They are a similar com-pression of the air by something vibrating.Usually the sound energy doesn't comeback, but when it does, it's an echo. You canthink of the compression of the stick asechoing off the back end, and in some sensedoubling the speed of the cue ball. Like anecho, the compression of the stick moveswith the speed of sound. Unlike the echo,the speed in the stick is not the standardspeed of sound — lightning a mile awaywill be heard in five seconds — but is thespeed of sound in the wood of the stick.

By now it should be clear that ball-ballcollisions aren't as simple as they mayseem on the surface. When the cue ball hitsan object ball, a sequence like the one forthe stick-ball begins. This is shown inDiagram 1, with the compression slightlyexaggerated. At the first instant of contact,the cue ball is still moving forward but theobject ball hasn't started to move yet.Something has to give, and what gives is

the spherical shape of the balls. As theymove together, flat spots develop on eachone as the plastic in the contact area com-presses. The cue ball continues to moveforward as the compression starts to movethe object ball. As the object ball picks upspeed, at some point it is moving just as fastas the cue ball.

This "equal-speed" point happens to be atthe time of maximum compression, whenthe flat spot is largest, and when half of theenergy of the shot is stored in the compres-sion of the surfaces of the two balls. Afterthis point, the compression releases like aspring, and that stored energy is put backinto the object ball. If no energy is lost inthe balls — and they are really quite closeto perfect — the push-off of the object ballfrom the compression will stop the cue balldead.

I bet you didn't know that all of that goeson when you shoot a stop.

How big is the flat spot? You can find ityourself by putting a piece of carbon paper(if you can find one) in front of an objectball and then noticing the size of the markthe cue ball leaves. It also works to usefreshly waxed balls or balls with a wet filmfrom your condensed breath. The result isthat for a hard shot, the flat spot is a quar-ter-inch or six millimeters in diameter. Howmuch did the surface of each ball compressduring such a collision? Simple geometrysays about 0.3mm or one hundredth of aninch. That's about the thickness of threesheets of typing paper.

How long does this collision take? It canbe measured just from the size of the flatspot and the speed of the cue ball. It hasalso been measured directly by WaylandMarlow in an experiment described in hisbook, The Physics of Pocket Billiards. Hemeasured the time the balls were in contactby having them make an electrical connec-

tion (that's the hard part) and then measur-ing the time of the contact (that's the easypart). His typical result was 200 millionthsof a second. It is important that this time ismuch longer than the time it takes forsound to travel from one side of the ball tothe other, or just as for the cue stick, the"echo" of energy from the far side of theball couldn't take part in the shot.

The ball-ball collision is hiding a furthersubtlety that can be important in play. Asthe balls compress together, the interactionis not like a simple spring. Instead it is asort of compound spring, because as theflat spot gets larger, more and more surfacearea gets involved. This means that the"spring" gets "harder" the more the ballsare compressed. For a normal cue-ball-to-object-ball collision, this makes little dif-ference, but when another object ball isbehind the first object ball, as when twoballs are spotted on the long string, theexact nature of the collision becomesimportant.

What happens during the shot is illustrat-ed in Diagram 2. This is for balls that areperfectly springy (elastic) and the predic-tion is a little surprising. Theory says thatfor perfect balls, the cue ball is expected tobounce back from the two-ball collisionwith some speed. For the "progressivespring," the speed back is expected to beabout 8% of the incoming speed. If theballs behaved like regular springs, gov-erned by Hooke's Law, the bounce-back isexpected to be twice that large.

Why do three balls behave so much dif-ferently that two? It turns out that for justtwo balls, the Laws of Conservation ofEnergy and Momentum forbid anythingexcept the stop shot from happening. Ifthree balls are involved, there are many out-comes that satisfy the two Laws, and whichoutcome we see can only be predicted if weinclude the exact details of how the ballsinteract. The three balls must be all touch-ing at the same time during the collision.

So, if the cue ball is expected to bounceback, how come we never see it do so?(Draw doesn't count.) The answer seems tobe that enough energy is lost in the colli-sion that the cue ball is fully stopped, butdoesn't get enough push-back from thesprings to get negative motion. You mighttry the experiment yourself with an old setof balls, and then some right out of the box.

28 BD-AUGUST 2001

Bob Jewett

What happens to the middle ball is theuseful part. It is going forward some, whichis contrary to the usual teaching of "stop-shot physics." To make a shot from this forthe double-spot shot, as shown in Diagram3, place the cue ball a little off line, and hitthe front ball full. It will get some speed tothe side, but will also have some speed for-ward, due to the "three-ball effect." If youhave the right small angle, you can makethe front ball in the corner pocket. Thiskind of shot has been described here beforeas the "10-times-fuller" system. The factor

of 10 can be predicted from the physics.Also, since the compression length is sosmall (less than 0.3mm), the shot is greatlychanged if the two balls that are supposedto be frozen are even the thickness of a dol-lar bill apart.

The compound spring law that governsspheres in collision is called Hertz' law, andit was discovered by the same physicistwho discovered radio waves and whosename you see in "megahertz" and suchterms. The details of the law are an activearea of research, and not just on the pool

table. It turns out that industrial processeswhich transport beads or pellets of materialhave lots of ball-ball collisions going on,and Hertz' Law is needed to understandthem. If you have access to the Web, enter"Hertz contact law sphere" in a good searchengine, and you should get plenty of equa-tions.

If you've waded through all of this rathertechnical stuff, you deserve a reward.Here's an old puzzle, slightly reworded. Ifyou get the correct answer and are the entrychosen by our autocratic judge — me —you'll get a one-year subscription to thismagazine. It's better to send e-mail to jew-ett@sfbilliards.com, but real mail sent toBD in my attention is OK.

In 1887, the bright, young Mr. Hertz waswalking down a street in Berlin when heheard an unfamiliar clicking sound comingfrom a tavern. Entering, he saw a teacherand a pupil and three ivory billiard balls.Hertz had heard of this "billiards" but hadnever seen a table before. The teacher shota simple carom — with the click that hadattracted Hertz — and explained the 90-degree initial carom angle. Hertz piped up,"But the angle between the paths of theballs must be less than 90 degrees." Howdid he know? For extra credit, what was hismistake?

BD-AUGUST 2001 29

Bob Jewett

Sliding FrictionHow far will you slide?

It is the various kinds of friction thatmake cue sports interesting. Withoutrolling friction, the balls wouldn't stop untilthey had all found pockets. Without thefriction between tip and ball, which is aidedby chalk, we would have no control of spinand position. Without the friction betweenthe balls, the dimension addedby throw shots would disap-pear. Without ball-cushionfriction, position play wouldbe severely restricted.

Another kind of friction thatis at work on every shot issliding friction between theballs and the cloth. This wasmentioned briefly in the tworecent columns about calcu-lating and plotting the speedand spin of a ball as itachieves normal rolling on thecloth. The natural state of theball is normal rolling, becauseany sliding between the balland the cloth produces a forceon the ball that will tend toeliminate the sliding.

Sliding friction is harder tomeasure than the other kinds.Rolling friction — whichcauses the ball to roll to a stop— can be measured easilywith a stopwatch. Frictionbetween the balls is shown bythe maximum throw angle.You can get at least a feel forfriction on the cushion by theresulting angle for a cue ball with maxi-mum spin going straight into the cushion.

The friction between ball and cloth isdynamic; it always has time as part of itsmeasurement. An example shot of theeffect is shown in Diagram 1. This stan-dard fancy shot demonstrates how to makethe cue ball curve without masse. The ideais to pocket the first object ball, and thencurve around the obstacle ball without hit-ting the side cushion and pocket the hang-er. Shoot slightly fuller than half-ball,which will allow the draw to pull the cueball back enough.

Shown in the diagram are two curves forthe cue ball's path. They represent shootingthe shot at the same speed, and with thesame amount of draw but with differentamounts of sliding friction. If the cloth is

new or the cue ball is waxed — try siliconespray if you want to do the experiment —the cue ball will take the wide course andwill look as if it is moving in slow motion.If the cloth and balls are sticky, the result isthe sharper, faster curve. Under sticky con-ditions, the shot can be fixed just by shoot-

With little friction, the force on the base ofthe ball is less. This leads to less accelera-tion at each instant the rubbing is going on,so the curve happens more slowly. It alsomeans that the spin is not being rubbed offthe ball as quickly, so the curve goes on fora longer time. In some sense these two

effects balance, so that inDiagram 1, the final angle thatthe cue ball takes is the sameregardless of how much fric-tion there is. A sharper curvefor a shorter time gives thesame angle.

The effect is present onstraight shots as well, but it isa little harder to see. The mainresult is that with low friction,the cue ball holds its draw formuch longer. Have some funwith a friend: On a table withold cloth, secretly grease upthe cue ball. Put an object ballin the jaws of a foot pocket,and challenge your friend toshoot a stop shot from behindthe line. If he's got a prettygood stroke, and is calibratedfor the old, sticky cloth, he'slikely to draw the cue ballclear back to the kitchen.Normally, the considerabledraw at the start of the shotwould be worn away over thesix diamonds of travel, but thesilicone reduces the rubbingand holds the draw for longer.

ing harder, but that can get you into mis-cues, jumped balls, and other problems.

Silicone spray has become a standard toolfor fancy-shot artists who want to movefrom amazing to impossible shots. Themain effect is to get much wider, slowercurves with less effort. An example is a shotby Semih Sayginer in his closing exhibitionat the Conlon WorldCup Tournament, July18-23, in Las Vegas. The cue ball was at A,and the target was at about B. The actualshot was a carom shot, but it would alsowork as a pool shot. The cue ball massedaround the obstacle ball over five diamondsaway and came back to the target. With anunwaxed cue ball, the standard shot is tomake just a right-angle turn from C andthen go to the target at B.

Exactly how does "slipperiness" enter in?

How can we measure sliding friction?These shots give you a feel for whether aparticular ball/cloth combination is more orless sticky than you're used to, but it's goodto have an actual number to compare.Physics books suggest measuring slidingfriction between an object and a surfacebelow it by pulling the object sideways, andnoting what fraction of its weight must beapplied to keep it in steady motion. Ballstend to roll when pulled sideways, so that'snot convenient. You could glue three ballstogether, like a mini-rack, and pull thatsideways, but you would need glue, spareballs, and a string, pulley and weights to dothe pulling.

Another way would be to videotape acurve shot like the one in Diagram 1, plotthe result to scale along with the time of

14 BD-SEPTEMBER 2001

Bob Jewett

each location, and do a lot of arithmetic. Asimpler way is shown in Diagram 2. Theidea is that if you shoot the three-ball 1-2-3combo at the 4 ball, the 3 ball will roll for-ward some distance after hitting the 4 ball,and that distance will tell us how much fol-low the 3 ball picked up on the way to the4. If there is more sliding friction, the 3will pick up more follow and will roll far-ther.

Suppose we vary the speed of this shotfrom very slow to fast. For the slowestshots, the 3 will just get to the 4 ball andwill surely be rolling smoothly on thecloth. It will drive the 4 down the tableand roll a little after it. These two dis-tances will follow a simple rule that wascovered here last December: the 4 willroll seven times as far as the three. Thisholds for full shots where the "cue ball" isrolling smoothly on the cloth. As the shotgets faster, there will be a point when the 3just gets to smooth rolling on the cloth as ithits the 4. This should result in the maxi-mum run-through of the 3, since for fastershots, it doesn't have enough time to pickup full follow. In effect, we are shooting astop shot with the 3.

The measured result is shown in Figure1, where the follow distances of the twoballs are shown. For the regular case, thereis a very clear peak which shows the maxi-

mum speed that still achieves smoothrolling on the 3 ball. I also tried polishingthe 3 to reduce the friction, and the lowercurve was the result. It shows only abouthalf the peak run-through of the regularcase, since the slippery 3 ball doesn't pick

up follow as effectively.If we can figure out the speeds of the balls

for these cases, it is a simple matter of alge-bra to figure out the friction of the cloth.The whole calculation goes roughly likethis: A ball is measured to take eight sec-onds to roll 98 inches. This lets us calculatethe speeds of the two balls, given theirrolling distances; We can then calculate thespeed of the balls at impact, and the speedof the 3 ball when it is struck by the 2 ball.Knowing that the distance between the 3

and 4 is 10 inches, we can calculate howmuch the 3 slowed in that distance due topicking up follow from the cloth. This inturn gives the sliding friction on the ball.The final result — details of the calculationavailable on request — is that the cloth-ball

coefficient of friction is 0.25, so that theforce on the ball when it is sliding is 25%of its weight. For the "slippery" case, thecoefficient of friction is reduced to 70%of this value or about 0.18. These valuesagree well with the value that Coriolismeasured over 160 years ago, of 0.20.

How can you use these ideas on thetable? The main thing is to realize that onnew cloth or with waxed balls, somethings will change greatly. Many playerslike to shoot "stun run-throughs" or stopshots that don't quite stop. The success ofsuch shots is critically dependent on how

much sliding friction there is. As the weath-er gets humid, you will see the opposite ofthe slippery condition; the ball-cloth fric-tion goes way up. In this situation, stopshots will turn into follow shots, and drawwill be much harder to achieve.

Bob Jewett is a Billiard Congress ofAmerica Advanced-level instructor, and apartner in the San Francisco BilliardAcademy, which has courses for beginnersto instructors.

BD-SEPTEMBER 2001 15

Bob Jewett

Playing GamesRenew interest in your favorite game by trying a few others on for size.

Has your g a m e reached a plateau? Doyou feel like you're in a rut? What you mayneed is a new game.

Pool halls are largely filled with playerswho know only one game. If they play 8-ball, they will refuse to play 9-ball becausethe shots are too hard and they don'tunderstand the safety play. If 9-ball istheir game, they won't shoot straightpool because they're confused by allthe choices, and of course they don'tunderstand the safety play. If they playpool, they'll never get close to thosetables across the room that are 12 feetlong or don't have any pockets.

Such stick-in-the-muds never stretchtheir minds, never learn new tech-niques, and will be playing the samegame in the same way in 20 years thatthey have for the last 10. I hope youaren't one of them.

When I first started playing, I had anideal situation to try different games.Pool, snooker and carom tables wereall available at the comfortable rate of40 cents per hour. There were fairlygood players on all of those tables —at least they were a lot better than I was —and national- or world-class players couldalways be seen on a trip to "The City."

Among the games that I played during myfirst year or so were straight pool, 6-ball, 9-ball, 8-ball, cribbage, cut-throat, partnersrotation (money ball), kiss pool, call-posi-tion 14.1, and one-pocket on the pool table;snooker, golf, pink ball, and English bil-liards on the snooker table; and straight rail,3-cushion and fancy shots on the carom bil-liard table. Bank pool, bumper pool, bottlepool, cowboy, 21-ball rotation, equaloffense, Fargo, line-up, pea pool, and pinbilliards I played later as the opportunitiescame up.

The first step to adopting a new game is tolearn the rules. The easiest way is to playagainst someone who knows them already,or, if you're shy, to watch a game inprogress. As important as the rules are thetactics and strategy. While watching anaccomplished player, try to predict what hewill do. If a shot really puzzles you, askabout it — most players like to show offtheir knowledge.

Another good source of rules, especiallyif you're striking out on your own, is theBilliard Congress of America rule book. It

contains the rules of over 30 games and isavailable for less than 10 bucks, shippingincluded. Parts of the rules are online at theBCA Web site, or you can use a searchengine such as www.google.com to findother sites.

Billiards.A game rarely seen in the U.S. but defi-

nitely worth learning is English billiards. Itis played with two cue balls and a red balllike carom billiards, but a snooker table isused. Points are scored by pocketing any

Many games will be valuable for master-ing your favorite games because they makeyou polish particular facets that may getneglected in the normal course of play. Forexample, straight-rail billiards (on thepocketless table, just make your cue ball hitboth the other balls to score a point andcontinue shooting) will teach you how tohit the ball softly, because once all threeballs are close, soft shots will tend to keepthem together. Straight rail also teaches youto control all three balls on each shot,which will do wonders for both your preci-sion-banking game and your cue ball con-trol. Allen Hopkins and Dallas West haveboth recommended straight rail to improvepool skills.

If you have trouble locating a caromtable, visit the United States BilliardsAssociation Web site at www.uscarom.orgfor a list of all known rooms in the U.S. thathave tables. The best explanation of ball-to-ball caroms is in Daly's Billiard Book,which was first published over 80 yearsago, but gives far better general cue-wield-ing instruction than many modern books.An excellent modern book that covers a lotof straight rail in a few pages is RobertByrne's Wonderful World of Pool and

ball (three for the red, two for the oppo-nent's cue ball, and three or two for yourcue ball depending on whether you hit redor white first. You also get two points formaking a "cannon," which is what theBritish call a simple carom. It's possible toscore 10 points in one shot by pocketing allthree balls, but this is a bad idea, as youropponent's cue ball stays off the table untilhis turn, and scoring with just two balls ona 12-foot table is quite a challenge. Whenred is pocketed (they say "potted"), itcomes back to the seven or black spot (theysay "billiard spot"). Many turn-of-the-cen-tury British books go into detail about thestrategy and such of English Billiards, andyou can often find these in on-line auctions.The full rules for the game are online atwww.wpbsa.com.

A game similar to English billiards butdesigned for the pool table is cowboy. Thegame uses a shared cue ball and the 1, 3 and5 balls. Points are scored by pocketing theobject balls (score: the number on the ball)or by caroming from one object ball toanother (score: one point for hitting two,two points for hitting all three). The gameis to 101, with some details. You must landexactly on 90 points, or the shot that takes

14 | BD-OCTOBER 2001

Bob Jewett

you over 90 is a foul. On a foul, all pointsof that inning are lost, but there is no otherpenalty. Points 91 through 100 must bescored by caroms only, and it's a foul topocket a ball. Point 101 must be scored bya called scratch off the 1 ball. The 1 spotson the head spot, the 3 on the foot spot, andthe 5 on the center spot. The full rules arein the BCA rule book, and a good searchengine will find several unofficial rulessites on the Web.

In Diagram 1 is an example positionfrom cowboy. Your score is 85. Whatshould you play? Fairly obvious is the 5ball in the side which brings you to therequired first step of 90 points, but youhave to start planning for the carom shotsthat are required after 90. Play the cue balloff the end cushion to end at A, and youshould have an easy carom shot from the 1ball to the 5 when it is spotted on the centerspot (where the shaded ball is). But thatshouldn't be the end of your planning.Since your last 10 points must be scored bycaroms, you want to gather the object ballstogether to make scoring easier. Play the 1ball to hit the cushion near B and to cometo rest near C. If the 5 ball after the caromends up near its starting location, and thecue ball is near the center spot, you can

shoot the 5 towards D and take the cue balltowards the 3. The 5 should bank over tojoin them, if your speed is correct. With allthree object balls together, your run-out isassured.

The games you try should fit your skilllevel. Of course, if a game is fun, stay withit, but I'd recommend that beginners startwith cowboy, cribbage and straight rail. Ifyou've never run out a rack of 9-ball, whynot play 6-ball instead? When I first played,6-ball was the rotation game of choice,while 9-ball was considered too hard formost players in the room.

Advanced players should try more chal-lenging games such as snooker, 3-cushion,bank pool and one-pocket. These will inturn help you work on pocketing accuracy,cue-ball control with spin, cushion reactionand precise speed control.

Good books are readily available for mostgames — does anyone know of a good oneon bank pool? Byrne covers many "alterna-tive" games in Wonderful World as well ashis Advanced Technique book, and hisStandard book is the best available on 3-cushion billiards. George Fels and JackKoehler each have two or three books inprint on strategies and techniques.

If you have a favorite "other" game you

would like to see discussed, please drop mea line in care of this magazine.

In the August issue, I proposed a puzzlerinvolving Herr Hertz and billiard-ball kissangles. A billiard instructor was telling astudent that the angle is a right angle or 90degrees, and Hertz stated correctly that it isless. The first question was: how did Hertzknow that the kiss line is less than 90degrees? The big hint here was that Hertzwas drawn into to the room by the clickingof the balls and didn't even see the colli-sion. That sound is energy being lost in thecollision, and from George McBane's guestcolumn last February you know that lostenergy means the angle between the cueball and the object ball will be less than theideal 90 degrees. The second questionasked what Hertz' error was. The answer tothat is that he should not have corrected theinstructor in front of the student. Of the cor-rect answers, Anthony DeAngelo had themost complete response, so he'll be gettinga year's subscription to Billiards Digest.

Bob Jewett is a Billiard Congress ofAmerica Advanced-level instructor, and apartner in the San Francisco BilliardAcademy, which has courses for beginnersto instructors.

BD-OCTOBER 2001 15

Bob Jewett

Round-Robin FormatsA simple alternative to double elimination.

Do y o u en joy playing in double-elimi-nation format tournaments? If you're likeme, you find them torture. If you lose yourfirst match, you have to win twice as manymatches as the guy who beat you in order tofinish in the same spot. How can that befair?

Round-robin is an alternative for-mat that solves many of the prob-lems with "DE." It was the mostpopular championship format formany years, and was only displacedwhen tournaments started havingmuch larger fields. Even with alarge field, it is possible to have amodified round-robin that lets mostof the players play more games, andis fairer in the selection of whoadvances.

In the basic round-robin tourna-ment, everyone plays everyone elseonce. This is a standard arrange-ment for most league play. A mainproblem is to construct a schedule.There are programs and pamphlets for this,but it is very easy to do by hand. For exam-ple, suppose we have eight players, orteams, numbered 1 through 8. Write thesedown in two rows like this:

1 3 4 52 8 7 6Pair these by columns to find the first-

round matches: 1-2, 3-8, 4-7, 5-6. Ideally,all these matches happen at the same time.Now here's the tricky part. Keeping the " 1 "in its place, rotate all the rest counterclock-wise:

1 4 5 63 2 8 7The numbers on the top row moved to the

left and the numbers on the bottom rowmoved to the right, giving the second-roundmatches: 1-3, 2-4, 5-8, 6-7. Continue thisuntil you have seven rounds. Since eachplayer plays everyone else, the number ofrounds in a round-robin will always be oneless than the number of players.

This method of "construction by rotation"works for any even number of players. Ifyou have an odd number of players, justinsert an extra player named "Bye" andyou're back to the even case. Bye's oppo-nent gets to sit out that round. There will beas many rounds as there are real players inthis case.

If you work through the above example,

you'll see that Player l's opponents in eachround are in numerical order. It may be thatit is better to have a different order. Theusual problem that comes up is that if twofriends are matched up in a late round, theymay decide who should win to have thebest chance to take first place. Suppose

Andy and Bill are buddies, and when theyplay each other, Andy has a 4-0 record andBill is 2-2. There is a temptation for Bill notto shoot his best so that Andy can advanceto 5-0 and have a lock on a major prize. Toavoid this problem, you can invoke the"brother-in-law" rule: matches betweenfriends/cousins/road partners will be in theearly rounds. To accomplish this, simplyrearrange the order of the rounds, or assignthe names to the initial numbers to put allof the buddy-buddy matches in the firstround or two.

The results are usually shown on a specialround-robin chart. Figure 1 shows the par-tial results of a six-player tournament. Theplayers' names are entered on the left sideand across the top (perhaps abbreviated).The entries in the grid show the matchscores for the play so far in this race-to-fourevent. For example, Andy has four wins onhis row from his matches, marked as Wsacross. To find out the scores of his oppo-nents in those matches, just read down the"A" column to see, for example, that Carlwon three games against Andy.

There are three matches left, Andy-Bill,Carl-Dave and Earl-Fiona. At this point inthe tournament, the spectators will be figur-ing out all the possibilities. If Andy wins,he has first place for sure, but if he loseshe'll be tied with either Earl or Fiona,

because one of them has to win and willalso have a 4-1 record.

The tournament director better still havethe paper he read from at the start of thetournament that describes the tie-breakingcriteria. If Andy loses and Earl wins to tiehim with a 4-1 record, there are two com-

mon ways to decide first place. Oneis by a playoff, but often there is notime for that. Alternatively, a two-way tie can be decided by the matchthat the two players played. SinceAndy beat Earl earlier, he would getfirst and Earl would get second.

Other ways to decide ties includetotal points scored, points allowed,and inning average. The actualmethod is not as important as hav-ing it written down and posted.Include all the criteria in order, suchas head-to-head; most points scored;fewest points allowed; one-gameplayoff. Also, the rule for forfeits isimportant to set ahead of time. If a

player abandons the tournament — that is,he fails to play his last several matches —it is reasonable to simply erase his entirerecord. If someone misses one match,assign an F for his score for zero points.

Remember the brother-in-law rule? In thetournament shown, the angles are a littledifferent. Because Andy is guaranteed firstbefore the final round is played, he can loseto Bill without cost. On the other hand,another win for Bill will tie him for thirdplace with the loser of the Earl-Fionamatch, and that might be worth a little moreprize money.

In a league situation where one team trav-els and one plays at home, you also have toassign home/away for each match. Usuallythis can't be done perfectly, so that everyteam gets an equal number of home andaway games, and there are never more thantwo homes or aways in a row for any team.If the league plays a "double round-robin,"so that each match-up occurs twice, a sim-ple rule is to play at home in the secondhalf if you played that team away in thefirst half.

In some situations, it is best to arrange therounds so that the best matches are savedfor the final rounds. It helps if the relativestrengths of the players are known at thestart, and the players are entered into thechart in order of ability. A schedule con-

Bob Jewett

structed to "save the best for last" is shownin Figure 2. The entries in the grid are therounds in which each match takes place.For example, the top-ranked player (1st)plays the worst player (8th) in the firstround. Notice that the nominal "numberone" has progressively more difficultmatches in each round, until he plays thenominal number two in the seventh round.Also note that all the matches among thetop four players will take place in the finalthree rounds (5-6-7).

If a league plays a doubleround robin, the secondhalf can be set up by Seed-ings from the standingsafter the first half, to makethe final weeks of play themost important.

The main problem withround-robins is the verylarge number of matchesfor a large number ofentries. Local pool tourna-ments commonly draw 50players or more, and even ifyou had 25 tables to playon, 49 rounds would bemore than most couldstand. The standard way tohandle this is to play

"round-robins in flights." The players aredivided up into smaller groups, and eachgroup plays its own mini-round-robin. For48 players, eight groups of six would work.Two players could advance from eachgroup, giving 16 in the next round. At thatpoint, everyone would have played fivematches, so to save time you could switchto a single-elimination format, and therewould be only four more rounds. Withplenty of time, perhaps in a two-day tour-nament, you could instead divide the 16

players into four flights of four with oneplayer advancing to a final round of fourplayers. That would leave six rounds on thefinal day.

At the recent Conlon Worldcup 3-Cushion tournament in Las Vegas, theproblem was how to accommodate 132entries on eight tables in three days toselect 12 qualifiers to advance to the maintournament with the seeded players. Oneway would have been to have a single-elim-ination tournament, but it would be brutalfor the many foreign players to travel sucha long way for possibly only one match. Toguarantee at least two matches for every-one, the first round was set up as 48 round-robins in groups of three. Some groups hadonly two players, so they played each othertwice. The single winner from each groupthen went on to two rounds of single-elim-ination, so no one played more than fourmatches in the preliminaries.

Round-robins are especially well suitedto determine the best player among eight orso. They also work well for a league formatwith a long schedule. Even for fairly large,short tournaments, their advantage of giv-ing everyone a reasonable amount of playmakes them worth trying. A free on-lineschedule planner is available at the Web sitehttp://www.playpool.com.

Bob Jewett

Elimination FormatsMore formatting alternatives for running your own tournament.

Last month, I tried to convince you totry a round-robin format for your next tour-nament. Round-robin is the fairest way todetermine the best among a group. For bestresults, especially if the players are dividedinto preliminary flights, the relativestrengths of the playersshould be known sothat the top players canbe seeded into differentgroups. It has theadvantages of moreplay for most entrantsand a chance to recoverfrom a single loss, butit requires more tablesand time than are oftenavailable.

When time and tablesare short, an elimina-tion tournament ofsome kind is better.The standard in theU.S. has been double-elimination tourna-ments, but there areseveral alternatives.This column will goover single-eliminationformats of varioustypes.

The basic idea is simple: win and youplay again, lose and you go home. Figure 1shows an eight-player chart that illustratesseveral important points. The first is that asyou move from where the players start onthe left side to the right, the number ofplayers left is reduced by half in eachround, in the progression 8-4-2-1. The mathwhizzes will recognize these as the powersof the number 2, which have become muchmore popular with the rise of computersand their binary number system. If youneed more room, the next added round tothe left would have 16 spots, then 32, 64,128, and so on.

If you have a number of players that isn'ta power of two, you add enough "byes" toexactly fill the chart. If you happen to playMr. Bye in the first round, you can be pret-ty sure of a win — he rarely makes it to thesecond round.

In the example tournament, 1 beats 8 inthe first round, and continues winningthrough the finals, where 1 beats 2.

The numbers in the diagram have a very

special pattern that is important for bothseeding players and placing byes. Thenumbers are placed starting from the 1 onthe right side. Moving to the left, the 1 runsup the top of the chart. In the "finals"round, the number 2 is added next to the 1

and it runs down to the left. In the semi-finals, a 4 is added next to the one and italso runs to the left. In the quarterfinals, orround of 8, an 8 is added next to the one,and if we had more rounds, they too wouldrun down to the left. The rule is that in eachround, the number next to the 1 is the num-ber of players in that round.

One spot is left in the semifinals is next tothe 2, and the obvious choice is 3, since 4had already been placed next to the 1. Thishints at the other rule for this numberingscheme: in every round, the numbers ineach match add up to the same total. Thus1+4=5 and 2+3=5. Check to see that in thequarterfinals, all the matches add up to 9.This total number is always one more thanthe number of players in the round.

How does seeding work? The purpose ofseeding is to prevent the best players fromplaying each other in the early rounds.Suppose we have the defending championand the defending runner-up in the tourna-ment. If we place them in the number 1 and2 spots on the chart, we have a chance to

see them in the finals again. If we didn'tseed them, and they were paired by chancein the first round, there would be a goodchance that the best player would be quick-ly gone from the tournament.

Of course, you don't always know howstrong the players are,and then a randomdraw for spots mustsuffice. There are alsomiddle-strength play-ers who will argue forrandom draw even ifthe strengths are moreor less known, becausethey would rather havethe possibility of all thechampions in one partof the chart and them-selves in some otherpart. Random drawsoften produce veryunbalanced charts.

This numberingscheme also can beused to place byes.When placing byes, theidea is to spread themas evenly as possiblethrough the chart. Iremember a $100,000

pool tournament in the 1980s in which thebyes were drawn for position just like theplayers, and they happened to clumptogether on the bottom of the chart. In thattournament, Mr. Bye played his brother,and one of them did get through to the sec-ond round. His opponent, who had playedtheir cousin, Mr. Bye, in the first round,was very happy to see him there.

Suppose we have a tournament and sixplayers show up. We need to fill the eight-player chart, so two byes have to be added.Just place them in the highest spots (7 and8) in Figure 1, and have the players drawcards numbered 1 through 6 for their spots.The numbering scheme guarantees thatbyes will be spread out.

Homework exercise: extend the chart oneround to the left for 16 players, and placefive byes.

Sometimes you have to both place byesand seeded players. For this, you have todecide whether the seeded players shouldget a free ride in the first round. If so, justuse the chart as is — the high numbers

Bob Jewett

where the byes go are guaranteed to be nextto the 1, 2, 3, etc. If you decide that theseeded players should not get byes — afterall, they already have seeded spots — putthe byes in the spots starting from half the

number of spotsplus one, and run-ning up. In Figure1, this would be4+1=5 and 6 fortwo byes.

In some single-elimination tourna-ments, a muchstronger form ofseeding is used.The seeded playersdon't even show upfor the first severalrounds; they jointhe tournamentalready in progress.This is illustratedin Figure 2. Thetournament beginswith 16 weakerplayers who playtwo rounds ofelimination to pro-duce four playersfor the quarterfi-

nals. Those are joined by four seeded play-ers to bring the player count up to eight.

Is this a fair format? Of course not. Is it agood format? Maybe. If you want to pro-vide both an opportunity for weaker players

to participate in a top-flight event and guar-antee that all the top players will be in thelater rounds, this format is a good choice.You can think of it as two separate tourna-ments: a qualification event between 16players with four advancing, and the mainevent with eight players, including fourseeds.

A major advantage of single-eliminationevents is the small number of rounds need-ed to determine a champion. With 256 play-ers and an unlimited number of tables, youneed only eight rounds (256, 128, 64, 32,16, 8, 4, 2) to determine a winner. Thenumber of matches is also small, with only255. (Rule: the number of matches is oneless than the number of real players.) Forscheduling, you also have to factor in thenumber of tables available, since you usu-ally won't have 128 tables available. With16 tables, you need 8 rounds of play just toget through the group-of-256. After that,things get easier, as you have only half asmany players in each succeeding round.Software is available on the Internet to helpwith planning and charting.

Next time, I'll go into some other tourna-ment formats. If you have Byrne'sAdvanced Technique in Pool andBilliards, turn to page 77 for some addi-tional ideas.