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    ASSORTED

    BRAIN-TEASERS

    Kundan Pangtey

    ISBN: 978 - 81 - 905947-0 - 7

    Published by:

    K. S. Pangtey

    B-5/11

    Green Tower (ONGC)

    Golf Green

    Kolkata-700095

    West Bengal, India

    Phone: 033 24296972

    e-mail: [email protected]

    2

    CONTENTS

    01. BEAUTY CONTEST

    07

    02. MAP ENLARGEMTN

    10

    03. COVER ON COVER

    15

    04. THE WALL

    18

    05. POINT OR PERIOD

    23

    06. TOUCHING BALLS

    28

    07. CLOSING-IN V/S ZOOMING-IN 30

    08. PLOTS & SUB-PLOTS

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    34

    09. PLANTS HATE GREEN

    40

    10. KILO CASE

    41

    11. HOUSE CATEGORY

    43

    12. DEGREE OF DISTANCE

    46

    13. HEAD MASTER

    51

    14. LAT-X

    54

    15. TELESCOPIC EFFECT

    58

    16. NEW DELHI OR NEWDELHI 61

    17. WHOSE SIDE IS IT ANYWAY 64

    18. LOOSER CAN BE WINNER 68

    19. SERVING FIRST

    75

    20. CURSE OF CURVE

    81

    21. WHITE ALWAYS WINS

    91

    22. HONEY COMB

    94

    3

    4

    TEASER - 01

    BEAUTY CONTEST

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    Imagine a contest of beauty pageant. Participants come onto

    stage one by one in front of Judges for each event. The

    participants and Judges do not know each other. The Judges

    are asked to award points on a scale from 1 to 10 (no fractions)

    to each candidate.

    If you are made one of the judge and is asked to choose one

    out of following choices for the first participant in first event

    which one would you choose ?

    (a) 01

    (b) 05

    (c) 02

    (d) 10

    (e) 09

    THE EXPLANATION:

    The contests are generally conducted to select a winner from among a group of contestants. In a contest, it is not important how much marks or points (absolute)an individual gets, rather the whole process is to make a proper sorting from among a chaotic group through a comparative analysis. Sorting cannot be done in isolation, it can only be done if there are more than one candidates put

    5

    together and compared together. If the winner candidate gets 5 point on a scale

    of 1 to 10, it does not mean that she or he is not worth 10, it merely indicatesthat all other candidates have points less than 5.

    When 1st candidate comes onto stage, judges may find it difficult to award a point as they would not know whether she is best or worst among all candidates. Judges would normally rely on their own past experience to make an imaginary benchmark or standard and attempt a comparison with that. Therefore the whole processwould be highly influenced by an individuals experience which would amount to biased judgment, especially during initial stage.

    If 1st candidate is awarded 1 point, for example, it would mean that an assumption has been made that all remaining candidates are better than her and would geteither one or more than 1 point. Now if all remaining participants turn out to

    be worst than 1st candidate then there would be a big dilemma as no numbers areavailable less than one hence all shall end up having 1 point each. Similarly if1st candidate is given 10 points same situation would arise in reverse order

    Let us compare beauty contest (crude comparison) to a ball game

    where participants are replaced by balls of various sizes chaotically placed ina cluster. These balls are required to be sorted out based on their sizes.

    CHAOTIC CLUSTER OF BALLS

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    OF DIFFERENT SIZES

    The balls are ten in numbers and named as A, B, C, D, E, F, G, H, J

    & K and judges award points on the basis of their sizes on a scale of

    6

    1 to 10 (no fractions). The smallest one gets 1 and largest one gets 10 points.The sizes of balls are not known to judges prior to contest.

    BALLS SORTED AS PER SIZES

    The name and diameter of balls are shown in a table below:

    SN

    ITEMS

    BALL CHARACTARISTICS

    1 Ball Name D B I

    G E H A J C F

    Ball Size

    2

    1

    2

    3

    4

    5

    6 7 8

    9 10

    (Dia) (cm)

    Let us imagine that D is the first one paraded in front of judges.

    If D is awarded 1 or 2 (minimum marks), the result would be correct up to 99%.

    However, if D is awarded 10 or 9 marks (maximum) the result will be 99% wrong.

    In case D is awarded 5 marks (mid point of scale), at least half of the remaining candidates would get correct markings.

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    Suppose E is the first candidate paraded in front of judges and is awarded 1 or2 marks (minimum), the result of contest would be 50% correct and if awarded 10or 9 marks (maximum) still result would be 50% correct, however, if E gets 5 point (middle), the result would be 100% correct.

    7

    Awarding of marks at extreme end is a total gamble and there is every danger ofgoing wrong by 99% and candidates may end up getting incorrect marks.

    To be on safe side the correct way would be to choose the midpoint, i.e.5 so that a fair amount of points are available on either side and a dead end is avoided.

    Following conclusion can be drawn from the analysis;

    (a) If points are picked up from extreme ends of a scale (1, 2 or 9, 10) there is a danger of going 100% wrong or right by 0%. So the probability of choosing points correctly would always vary from 0% to 100%.

    (b) However, if points are chosen from middle of the scale (5) the answer wouldalways be 50% to 100% right and will never be 100% wrong. The probability of choosing points correctly would always vary from 50% to 100% and will never be less

    than 50%.

    (c) Therefore the correct way would be to choose the mid-point for initial participants so that the danger of going wrong by 100% is avoided and a fair amount of points are made available on either side for awarding later participants and adead end is averted.

    ********

    TEASER - 02

    MAP ENLARGEMENT

    If a map (A), having a size equal to A4 paper and scale of 1:100

    is enlarged by 200% (2 times) (B), what will be the scale of

    enlarged map?

    (a) 1:50

    (b) 1:71 (approx)

    (c) 1:75

    8

    A: MAP OF AN AREA WITH SCALE

    B: MAP SHOWN AT A ABOVE IS ENLARGED

    THE EXPLANATION:

    We know that a scale or ratio of 1:100 means 1m on map equal to 100 m on ground.

    Distance on map

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    1

    1 : 100 = ____________________________ = ____________

    Distance on ground 100

    When a map is enlarged or compressed, two things change, (i) scale of map (ii) size or dimension (length & width) of map. Both are linked in such a way that size is inversely proportional to scale (ratio). Larger the map narrower the ratioof a unit distance on map to unit distance on ground.

    9

    A map is a kind of a scaled down model of an area (2D) where an actual surface area is represented on a paper (media) suitable for human use. Three types of similar relationship (between actual and model) can be envisaged;

    1. Line (length - one dimensional)

    2. Area (length & width - two dimensional)

    3. Volume (length, width& depth - three dimensional)

    1) If a scaled down model of an actual line (one dimensional), having a ratio of1:100, is enlarged by 200% (2 times) in same way as a map is enlarged, the ratio of model (enlarged) to that of actual one would become 1:50 (exactly half).

    ENLARGEMENT OF A LINE (1D)

    2) If a map (scale down model) of an area (earth surface), having a scale of 1:100 is enlarged by 200% (2 times) the scale of enlarged map would become 1:71 (approx) and not half as in case of a line.

    ENLARGEMENT OF AN AREA (2D)

    10

    3) Now if we want to change the scale of same map from 1:100 to 1:50 (half), themap shall have to be enlarged by 400% (4 times) and not 200% (2 times) as wouldbe in case of one dimensional objects such as a line. The area being a 2 dimensional entity, enlargement will have to be done twice along X and Y direction (length and width).

    Area

    =

    X x Y

    =

    2(X) x 2(Y)

    =

    4 (X x Y)

    or

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    =

    200% x 200%

    =

    400%

    CHANGING SCALE OF AN AREA (2D)

    4) On same analogy, if a scale down model (similar to map) of a 3D solid objector volume (having a ratio / scale of 1:100) is enlarged 200% the ratio would be1:79 and not 1:50 as in case of a line and neither 1:71 as in the case of an area.

    ENLARGEMENT OF A SCALE MODEL (SOLID: 3D)

    11

    5) Now if scale of same 3D model is changed from 1:100 to 1:50, it would be enlarged by 800% (8 times) and not 400% (4 times) as would be in case of a map. Thesolid model being 3 dimensional, the enlargement will have to be done volume wise, along X, Y and Z axis.

    Volume

    =

    X x Y x Z

    =

    2(X) x 2(Y) x 2(Z)

    =

    8 (X x Y x Z)

    Or

    =

    200% x 200% x 200%

    =

    800%

    CHANGING SCALE OF 3D MODEL (VOLUME: 3D)

    The solid objects have two elements which can be compared for map to ground typerelationship. One is the surface of the object, similar to earths surface (2D)& second is volume contained within its body (3D). Therefore the ratio or scalewould depend upon which aspect is being considered while doing enlargement. Forexample the ratio of surface area would be governed by 2D conversion like that of map to ground, where as ratio for volume would be governed by 3D conversion formula.

    ********

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    12

    TEASER - 03

    COVER ON COVER

    Consider that there is a book with a specific cover design which

    is named here as K. Now a stamp size (medium size)

    photograph (named as M) of cover of K is to be printed on the

    left bottom corner of same cover (as is done for the cover of

    this book). Whether it is possible to do so?

    (a) Yes

    (b) No

    THE EXPLANATION:

    1) We have a book cover design named as K and its exact photograph M of smallersize (stamp size). The smaller one M is to be placed or printed on bigger one K.

    Name of Book cover design

    =

    K

    Name of Stamp size photograph of K

    =

    M

    Barring size difference, M is identical to K in all respect.

    13

    A BOOK (M) AND STAMP SIZE

    IMAGE OF ITS COVER (M)

    When M is printed on K (bottom left corner), it is an action of

    adding or introducing new things to the original cover K.

    Any alternation or changes made on the K would amount to changing the original d

    esign therefore K would become KM.

    STAMP SIZE IMAGE OF BOOK COVER (M)

    IS PUT ON ITS COVER (K)

    2) Now as K has become KM, the M (the stamp size exact image of K) will get automatically cancelled and invalid and a new stamp

    14

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    size image, with new name, shall have to be made all over again and put on the new book cover KM.

    Name of new book cover design

    = KM

    Name of Stamp size photograph of KM

    = S

    Now as soon the stamp size image S is placed / printed on the new book cover (KM), the process, as indicated at (1) above will get repeated once again and KM will become KMS.

    NEW STAMP SIZE IMAGE (S) IS

    PUT ON BOOK COVER (KM)

    3) Now KM has become KMS, the S (the stamp size exact image of KM) will get automatically cancelled and a new stamp size image shall have to be put on the new book cover KMS

    Name of new book cover design

    = KMS

    Name of Stamp size photograph of KMS

    = Y

    Now as soon Y is placed on KMS, the cover gets altered and the whole process isrepeated again.

    15

    NEW STAMP SIZE IMAGE (Y) IS

    PUT ON BOOK COVER (KMS)

    The process continues till infinity. Therefore every time a stamp size image orphotograph of a book cover is printed (placed) on itself it gets altered which in turn invalidates its stamp size image. So the process of placing or printing stamp size image on itself triggers another action and creates new cover.

    ********

    TEASER - 04

    THE WALL

    The dimension of a wall is measured by its length (Y), width (X)

    and thickness or depth (Z). The width (X) of a wall can also be

    equated to its Height (H).

    Now, a wall of exact 5 m height (H), not more & not less, is to be

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    erected across an undulating earths surface, similar to that of

    China wall. Indicate whether the width (X) of the wall, shall

    remain constant throughout its length over an undulating earth

    surface?

    (a) Yes

    (b) No

    16

    GREAT WALL OF CHINA CRISS-CROSSING MOUNTAINS

    THE EXPLANATION:

    When a wall is standing on a flat ground its dimension is defined by three elements, they are - width (X), length (Y) and thickness (Z). However, when same wallis build over an inclined ground surface, a fourth element or fourth dimension- height (H) comes into play. This fourth element is same as width (X) when on flat ground.

    GEOMETRICAL ELEMENTS OF A WALL

    The width of a wall (X) is measured along a line normal to ground surface whichmay not be parallel to equi-potential gravity field. On the other hand the height of a wall (H) is measured along a vertical line (plumb line) which is perpendicular to the equi-potential surface of earths gravity field. Normally, sea surface can be considered to

    17

    be parallel to equi-potential gravity field and plumb line is always perpendicular (normal) to sea surface. However, ground surface is not always parallel to equi-potential surface and is highly undulating. On an average, trees are always p

    erpendicular to equi-potential surface, irrespective of ground slope on which itgrows, therefore can be considered to be parallel to Plumb Line.

    TREES GROWING VERTICAL (PARALLEL TO PLUMB LINE) (OBLIQUE TO STEEP SLOPE)

    If height (H) of a wall (measured along plumb line) is kept constant throughoutits length over an undulating ground, the width (X) will not remain same every where. Width (X) would be maximum on a flat ground and would decrease proportionately along sloppy ground.

    HEIGHT OF A WALL IS INDICATED BY ARROWS

    18

    Alternatively if width (X) is kept constant, the height (H) of wall would not remain same everywhere. Wall would have maximum height along slopes and minimum onflat surface.

    WIDTH OF A WALL IS INDICATED BY ARROWS

    There can be varying degree of ground slopes. For example a stair case type slope can be likened to combination of vertical and flat surfaces. Over flat part ofstaircase both plumb-line and width of wall would be parallel, therefore H and

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    X would be same. However, over vertical part, the width (X) would be zero whileheight (H) would be 100%. Geometrically it would amount to putting a new staircase over the ground surface.

    VARIOUS SHAPES OF WALL RESULTING DUE TO

    COMBINATION OF WIDTH & HEIGHT OF THE WALL

    19

    Over a slopping ground, the width (X) of wall can vary from 0% to 100%. The relationship between H and X can be roughly established as follows:

    Q

    X

    = H x ________

    90

    Where:

    X

    = Wall Width

    H

    = Wall Height

    Q = Angle between Plumb-line

    and ground slop

    ANGLE (Q) BETWEEN GROUND

    SLOPE AND PLUMB LINE

    If Q is 90 degree (in case of flat ground) X would be equal to H. The X would progressively decrease with decreasing Q and eventually become zero if Q is zero (in case of vertical ground).

    ********

    20

    TEASER - 05

    POINT OR PERIOD

    If, Year 2001 to year 2005 is equal to 5 years, January (01) to

    May (05) is equal to 5 months and Monday to Friday is equal to

    5 days, then, 10 AM (10 Hrs) to 02 PM (14 Hrs) is equal to how

    many hours?

    (a) 4 hrs

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    (b) 5 hrs

    (c) Both

    THE EXPLANATION:

    Time is a continuum phenomenon however humans have segmented them for their convenience and daily uses. Time segments are designated by various names such as hours, days, months, years etc which represent a specific time interval / period.

    When we say 01 year, we actually mean a period of 12 months starting from January to December. Similarly January month means a time period having 31 days starting from day 1 and ending at 31st day of the month. Same thing applies to each week (Monday to Sunday) and each day which encompasses a time period of 24 hrs, starting from 00 hours to 24 hours.

    Generally, when one says 2000 to 2001, it means 2 years comprising whole of year2000 (12 months) and whole of year 2001 (12 months) and no fractions. On same line January to February implies whole of January and whole of February (a totalof 59 days).

    21

    Any segment of a time period (say a day) has a starting point and ending point,therefore a day can not be called a day unless entire period of 24 hours is completed, similarly a year should have 12 months, else it may not be called a year.

    Therefore each segment of time will qualify to be called by its name only at theend of its specified period. A decade has 10 full years, (1st to 10th year), amonth has full 30 days (1st to 30th days) and a week has seven full days (1st to7th day).

    LARGE TIME INTERVAL / DIVISIONS (PERIOD)

    A watch, used for day to day work, has full 12 hours (00 to 12 hrs).

    DIVISIONS OF TIME INTERVAL (PERIOD) OF A DAY

    (01 PM to 05 PM)

    22

    When we say a match or a game will start at 01 PM or 01 hours and finish at 05 PM or 05 hours (or will last from 01 to 05 hours) it can have following meaning;

    1. 01 PM (hrs) to 05 PM (hrs) = 4 hours

    (00 - 01 hour not included)

    SHORT TIME PERIOD (01 to 05 HRS)

    2. 01 PM (hrs) to 05 PM (hrs) = 5 hours

    (00 - 01 hour included)

    SHORT TIME PERIOD (01 to 05 HRS)

    23

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    As per prevailing convention, 01 hours or 01 0 Clock meant end of 01 hour (or beginning of 02 hours). It never means beginning of 01 hrs hence does not include 00to 01 hour. However, same convention is not applied when we refer in terms of years or months. When we say that so and so was a number one player from 01 year to 05 year it invariably means 5 years and includes year from 00 to 01 also.

    Similar to time, space is also a continuum phenomenon and space has also been segmented for convenience and daily uses. On human scale, space is generally segmented into kilometers, meters, centimeters, feet, inches etc, horizontally as well as vertically.

    For example, on a horizontal scale, whenever we say 02 inch to 06 inch it can have two meanings;

    1. 02 inch to 06 inch = 04 inches (2nd inch not included)

    SHORT DISTANCE (SPACE) INTERVAL (02 to 06 INCH)

    2. 02 inch to 06 inch = 05 inches (2nd inch included)

    SHORT DISTANCE (SPACE) INTERVAL ( 01 to 06 INCH)

    24

    On a vertical scale, measurement of a depth interval can also create similar anomaly. For example, in petroleum exploration industry, oil, which occur in rock layers, is extracted by drilling a well (8 to 10 inch diameter) deep into earth.The properties of rock layer containing oil and its depth from surface is recorded and measured by wire line logs (as shown below).

    DEPTH INTERVAL OF AN OIL WELL

    (00 to 1000 m & 390 to 420 m)

    Now, imagine that a rock layer containing oil has been found to occur between 400 m to 410 m in a well. Its thickness would be measured as 10 m (as shown above).

    The depth interval of oil layer can be written in two options:

    i. 400 m to 410 m

    = 10 m (400th m not included)

    ii. 400 m to 410 m

    = 11 m (400th m is included)

    25

    However, in petroleum industry practice, the depth interval is generally demarcated from a point on top of layer to a point at bottom of layer. The well path ismeasured and marked continuously so that a point will always fall on some numeral. So if top of layer falls exactly at 400 m, its depth interval would start from a point at 400 m and downward to 410 m, hence thickness of interval, as per industry practice would be taken as 10 m and not 11 m.

    Depending upon the convention and practice, one of the options can be followed provided it is qualified with a definition or explanation.

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    ********

    TEASER - 06

    TOUCHING BALLS

    How many balls of same size can remain in touch with each

    other at the same time?

    (a) Four

    (b) Three

    (c) Six

    (d) Nine

    26

    THE EXPLANATION:

    On a two dimensional plane (2D), a maximum of three balls can remain in touch with each other at a time.

    THREE BALLS OF EQUAL SIZE

    ON A FLAT SURFACE (2D)

    In three dimension (3D), a maximum number of four balls can remain in touch witheach other at a time and form a tetrahedron. Three balls together forming eachface of a tetrahedron.

    FOUR BALLS OF EQUAL SIZE IN A BOX (3D)

    27

    On a plane (2D), a single central ball can remain in touch with a maximum of sixballs of same size at a time, making an hexagonal shape.

    SIX BALLS OF EQUAL SIZE ON

    A FLAT SURFACE (2D)

    In three dimension (3D) a single central ball can remain in touch with 12 ballsof same size at a time.

    TWELVE BALLS OF EQUAL

    SIZE IN A BOX (3D)

    ********

    28

    TEASER - 07

    CLOSING-IN V/S ZOOMING-IN

    Zooming-in is defined here as a simulation of an act of moving

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    closer to an object-in-view with the help of a magnifier such as

    telescope or binoculars. Closing-in is defined here as an act of

    physically moving closer to an object-in-view.

    One aspect which is common in both processes is that things

    get magnified and look bigger to the viewer. The process of

    magnification is identical in both cases?

    (a) True

    (b) Falls

    THE EXPLANATION:

    Zooming-in

    Zooming-in is a simulation of an act of movement toward or away from an object and not the actual act of movement. This is performed with the help of a magnifier such as zoom lens. In motion picture or television, zooming-in is a process of

    changing rapidly from a long shot to a close-up while maintaining the subject or scene in focus or vice-versa.

    The observer and object do not move physically towards or away and remain fixedat their places and the relationship with regard to

    29

    angle and depth between observer and object remain unchanged. The process of zooming-in can be equated simply to the process of enlargement of a 2D picture where only X and Y (length and width) is stretched and 3rd axis Z (depth) does not play any role.

    Geometrically following things happen in zooming-in process;

    All objects seen within view-field progressively appear bigger.

    All objects, near or farther within view-field magnify at same rate, hence the viewers relation with object, with respect to depth and angle does not change while zooming-in or zooming-out. The process can be likened to enlargement or stretching of a photograph on a paper (2D).

    IMAGE OF AN OBJECT SEEN BY NAKED EYE

    ZOOMIN-IN: IMAGE OF AN OBJECT SEEN

    WITH THE HELP OF A MAGNIFIER

    (OBJECT IS ENLARGED)

    30

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    Closing-in:

    The closing-in process involves both viewer and object and they come closer progressively to each other and eventually come into physical contact.

    In real world the three dimensional (3D) objects have three axis, X, Y and Z (length, width and depth) and each point on an object is at a different depth and at different angle at a given time and space with respect to viewer. When viewermoves toward an object, the depth and angle of object keep changing and become pronounced at close range.

    The image on our retina forms in 2D format (length & width) however as viewer moves toward an object, 3rd dimension (Z-axis) comes into play as a result of movement of the viewer.

    The image of each objects seen within the view-field magnify at different rate depending on its depth and angularity. The nearer one magnifies at much higher rate than the farther one.

    Geometrically following things happen in closing-in process:

    All objects within a view-field progressively appear bigger.

    All objects, seen within view-field magnify at different rate.

    The nearer objects magnify at much higher rate then the farther objects, hence the viewers relation with object, with respect to depth and angle constantly change while closingin or closing-out.

    IMAGE OF AN OBJECT SEEN BY NAKED EYE

    31

    CLOSING-IN: IMAGE OF AN OBJECT SEEN

    AT CLOSE RANGE BY NAKED EYE

    (OBJECT IS ENLARGED)

    Conclusion:

    In both processes objects within view-field magnify and appear progressively bigger, however similarity ends here. The major difference between two, with regardto magnification, is that in case of closing-in nearer object magnify at much h

    igher rate than the farther one where as in case of zooming-in all objects within viewfield magnify at same rate.

    ********

    TEASER - 08

    PLOTS & SUB-PLOTS

    A square piece of plot / land (green) with open space and road

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    (brown) on all side is to be divided into many sub-plots (parts).

    With following conditions,

    1) All sub-plots should be equal in size

    2) All sub-plots should be equal in shape

    3) All sub-plots should have equal accessibility to road and

    open space present on all sides

    Under aforementioned conditions, this piece of land can be

    divided into how many maximum sub-plots (parts)

    32

    (a) 4

    (b) 8

    (c) 12

    (d) 2

    SQUARE PLOT WITH ROADS ON ALL SIDES

    THE EXPLANATION:

    Division of a square plot of land:

    The land being an areal quantity, it can have only two dimensions (2D), therefore individual sides or individual corners can be divided 2 times only to make them symmetrical on either side of a line. The square plot of a land has 4 corners(4C) and 4 sides (4S) and each side has fixed length and each corner has fixed a

    ngle, hence the sum total of length of sides and angle of corners when divided should remain constant (4C + 4S).

    Corners (C) = 4

    Sides (S)

    = 4

    SQUARE PLOT WITH 4 SIDES & 4 CORN

    33

    (1) When divided into 2 sub-plots:

    2 Plots = 2 x (2C+1S+S+S)

    = (4C + 4S)

    Or = 2 x (2S+1C+C+C)

    = (4S + 4C)

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    SQUARE PLOT DIVIDED INTO 2 EQUAL PARTS

    (2) When divided into 4 sub-plots:

    4 Plots = 4 x (1C+S+S)

    = (4C + 4S)

    Or = 4 x (1S+C+C)

    = (4S + 4C)

    SQUARE PLOT DIVIDED INTO 4 EQUAL PARTS

    (3) When divided into 8 sub-plots:

    8 Plots = 8 x (C +S)

    = (4C + 4S)

    34

    SQUARE PLOT DIVIDED

    INTO 8 EQUAL PARTS

    Keeping in focus the conditions laid down above, it is observed that a square land can be divided into 2, 4 and 8 equitable sub-plots only and cannot be dividedinto 3, 5, 6, 7, 9 or more number of sub-plots.

    The shape of a piece of land can be trigonal, pentagonal, hexagonal, circular and so on.

    The sub-divisions or sub-plots are in a way symmetrical divisions and their number is dependant on the shape of a plot. The corners and sides of a square or a hexagonal shaped land have fixed number of corners and sides and are linked together in a definite pattern. To make identical sub-divisions (sub-plots) corners a

    nd sides would have to be equally shared without changing their original positions.

    The relationship between the number of corners or sides and the maximum possiblesub-divisions or sub-plots equal in all respect can be established in followingmanner:

    P

    =

    C x 2

    Where;

    P

    =

    Maximum number of sub-plots

    C

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    =

    number of corners or sides

    Square plot (4 sided)

    35

    C

    =

    4

    P

    =

    C x 2

    =

    4 x 2

    =

    8

    8 EQUAL DIVISIONS

    Trigonal plot (3 sided)

    C

    =

    3

    P

    =

    C x 2

    =

    3 x 2

    =

    6

    6 EQUAL DIVISIONS

    Pentagonal plot (5 sided)

    C

    =

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    5

    P

    =

    C x 2

    =

    5 x 2

    =

    10

    36

    10 EQUAL DIVISIONS

    Hexagonal plot (6 sided)

    C

    =

    6

    P

    =

    C x 2

    =

    6 x 2

    =

    12

    12 EQUAL DIVISIONS

    Round plot (infinite sided)

    C

    =

    infinite

    P

    =

    C x 2

    =

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    infinite x 2

    =

    infinite

    37

    INFINITE EQUAL DIVISIONS

    An infinite number of radial rays can be drawn from the center of a circle (round plot) outward. A circle can have any number of sides and corners hence it canbe divided into an infinite number of equal parts or sub-plots.

    ********

    TEASER - 09

    PLANTS HATE GREEN

    We talk of greening our earth, but plants hate green color?

    (a) True

    (b) Falls

    THE EXPLANATION:

    The color of any object is due to its inherent quality of dislike for that color. In other words all colors are consumed by the object except the one that is reflected and reaches to the eyes of viewer.

    We love green color and want to green whole earth but what we do not realize isthat had plants liked it we would not have seen green as we see it today. Therefore green color is the enemy of plants. But

    38

    why plants hate it and we love it. May be its a deliberate ploy used by plants sothat animals are befriended for their own survival.

    This principal is used universally by all kinds of creature, like colors of flowers.

    Therefore when we say let us green earth, its a misnomer, its a selfish way of seeing through the eyes of humans and not plants.

    If sun suddenly starts beaming only green color it would be soothing to our eyesbut it will be an end to the majority of plant life.

    ********

    TEASER - 10

    KILO CASE

    If Y2K stands for year 2000, then indicate which of the

    following is correct ?

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    (a) 1 KM2

    = 1000 M2

    (b) 1 KM2

    = 1000000 M2

    (c) Both

    THE EXPLANATION:

    In day to day life, the long words are frequently squeezed or compressed and arerepresented by a single alphabet, especially in written form, may be perhaps tosave time, space and energy. Such short forms, when used in combination with other similar short forms or numerals, for deriving equations or formula, has thepotential of creating confusion, especially in the minds of students not conversant with such usage.

    The K is one of the most used single alphabet in modern times which is a short form for Kilo. The K symbol is used extensively as independent entity or in combination with other alphabets or numerals. For example in metric system K is commonly used in

    39

    combination with other alphabets such as KG or Kg (Kilo-gram), KM

    or Km (Kilo-meter), KL or Kl (Kilo-liter), Y2K (Year 2000) and so on.

    When K is used in combinations:

    i

    The equations, where K, used in combination with other alphabets such as M as asingle dimensional entity, the confusion level is zero because K and M, whetherallowed independently or together, the end result will remain same.

    1 KM

    =

    1 x K x M

    =

    1 x 1000 x M

    ii Problems may start cropping up when KM is squared (multi dimensional) as thisinvolves multiplication of itself. Since K

    and M also represent independent entity confusion may be created that whether both are part of same unit or not? There is a possibility that K would be perceived as an independent alphabet separate from M and only M is squared (M2) and notK. This scenario is very much possible on the assumption by an individual that Kbeing a dominant alphabet is separate even when combined with alphabet like M.

    1KM2

    =

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    1 x K x M x M

    =

    1 x 1000 x M x M

    =

    1000 x M x M

    =

    1000 M2

    iii However since general convention provides that both K and M must be writtentogether as KM or Km for Kilometer, hence must be squared together as KM2 or asK2 M2. This is similar to a numerical quantity squared, such as 1232 which wouldbe red as 123 x 123 and not as 1, 2, 32 or 1 x 2 x 32.

    40

    1KM2

    =

    1 x KM x KM

    =

    1 x K x K x M x M

    =

    1 x 1000 x 1000 x M x M

    =

    1000000 x M x M

    =

    1000000 M2

    To avoid confusion or any other anomalous perception once for all, Kilometer orany similar quantity or entity should better be either denoted by one alphabet,such as T or R or if two alphabets are essential then it could be placed withina bracket (KM)2 then squared.

    ********

    TEASER - 11

    HOUSE CATEGORY

    In a particular locality, large number of residential houses were

    to be constructed in different sizes. Due to certain problems

    only few houses of different sizes, scattered haphazardly, could

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    be constructed initially. While making allotment, the completed

    houses were graded and grouped in A, B, C and D category as

    per their sizes in ascending order, where A is smallest and D is

    largest house. Whether the categorization of houses in A, B, C

    & D pattern was a full proof method?

    (a) Yes

    (b) No

    41

    THE EXPLANATION:

    The grading of houses are generally done either in ascending order, where A is at bottom or in descending order where A is at top. But there is an anomaly in this system, especially if more houses of different sizes are yet to be constructed.

    Unidirectional (one way expansion) growth pattern:

    In A, B, C and D grading system, A is a starting point or dead end. As alphabetsare not available prior to A, growth is not assumed prior to A. Alphabets are available only beyond D hence growth is assumed beyond D only. Therefore this type of grading or categorization is unidirectional or one way expandable.

    If houses are required to be constructed in sizes smaller than A (in case of ascending order) or bigger than A (in case of descending order) it will not be possible to assign category in same series, therefore unidirectional pattern has thedanger of running out of alphabets for new houses and bidirectional pattern would be preferred.

    HOUSES NAMED (NUMBERED) FROM BEGINNING OF A

    SERIES LEAVING NO NUMBERS AT BEGINNING

    42

    Bi-directional (two way expansion) growth pattern:

    In a bidirectional grading nomenclature (categories), the grading is done with no dead end. For example house categories can be named as J, K, L and M so that plenty of alphabets are made available at either end for future expansion of houses at both end.

    However, if houses are required to be constructed in the middle (inbetween) of the series, say between K and L we will again run out of alphabets, as no gap isleft in-between, therefore a multi-directional pattern would be most suitable toaccommodate all possibility of future expansion.

    HOUSES NAMED (NUMBERED) FROM MIDDLE OF

    SEIRES LEAVING NUMBERS AT BOTH ENDS

    Multi-expansion growth pattern:

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    In uni-directional or bi-directional pattern, middle (in-between) expansion is not possible. What would happen if houses are required to be constructed in the middle of series say between J and K? Two things can be done to accommodate middle expansion:

    (i)

    Sufficient gaps can be created in bi-directional series such as D, F, K and P or

    (ii)

    A second series can be added to the first series. Like J(1), J(2), K(1), K(2) and so on.

    43

    HOUSES NAMED (NUMBERED) WITH GAPS LEAVING

    NUMBERS BETWEEN TWO HOUSES

    Since more houses are yet to be constructed and also because original plan of construction of new houses may change in terms of size depending on requirement, the nomenclature of the categories of houses (already completed) shall have to be

    in such a way to accommodate future houses without disturbing the existing series of categories, otherwise we will end up having multiple series with confusingand chaotic house addresses within same locality.

    ********

    TEASER - 12

    DEGREE OF DISTANCE

    A contour map, by definition, is a map of a surface showing

    undulations with the help of lines joining equal elevation, also

    called topographic map.

    On a regional scale, the ratio of a unit distance on a contour

    map at any point to that on ground is always same.

    (a)

    True

    (b)

    Falls

    44

    THE EXPLANATION:

    A map or any drawing made on a flat sheet of paper are 2D transformation of 3D objects. Even the image formed on our retina (eye) is in 2D format. The 3D perception in our brain is caused due to 2D image received from two different eyes setat an angle with respect to center of an object.

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    During earlier days relief on a map used to be shown with the help of variationsof light and shade to give effect of 3D objects in 2D images. As the use of shading became systematized the contour lines, visually representing different elevations of land came into general use. Contour map is a graphical conventional method for showing changes in height or depth on a 2D map with the help of lines that connect points of equal elevation called contour lines.

    CONTOUR MAP WITH VARIABLE SHADING

    The pattern of contour lines can depict landforms. Areas of tightly spaced contour lines describe steep changes in elevation and sparse contours depict flat areas. The Mean Sea Level (MSL) is generally taken as base or datum surface, on whichcontours are

    45

    referenced or plotted. With each contour line representing a standard distance above or below the base line, one can accurately calculate height or depth simplyby counting the lines from the base line.

    Theoretically, the distance measured on a contour map, in accordance with its scale should match to that on ground. However earth, being a spheroid, making a regional contour map is a tricky one.

    Two things can happen while making contour maps (especially on a regional scale)of earth surface;

    i.

    The globe surface is curved along two directions X & Y

    axis. When flattened or stretched it cracks vertically (when one axis is curved)and also horizontally when both axis are curved.

    RIGID CAKE CURVED AND FLATTENED

    RIGID HEMI-SPHEROID (EARTH)

    CURVED AND FLATTENED

    46

    ii. The tall mountain (vertical component: height or depth) is squashed onto a 2D flat base or datum surface.

    TALL OBJECT SQUASHED ON A FLAT SURFACE

    While making contour maps, the plumb lines (or longitude lines) that converge downward towards center of earth are made parallel with constant distance between

    longitudes, irrespective of undulations.

    LONGITUDE LINES BECOME PARALLEL ON FLATTENING

    The earth surface can be likened to a curved rubber sheet over which cones (similar to mountains) are placed. When rubber sheet is flattened the distance between summits (S1 S2 S3) of cones progressively decreases where as distance betweenbase (root) of cones (R1 R2 R3) does not change.

    47

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    RUBBER SHEET WITH CONES (CURVED & FLATTENED)

    CONTOURES OF CONES PLOTTED ON RUBBER SHEET

    Where as the distance between summits of cone (S1, S2, S3) in a contour map, does not change, irrespective of bending or flattening of rubber sheet. The contourmap assumes that the distances between S1 to S2 to S3 is same as that between R1 to R2 to R3.

    In a regional contour map, the bigger area of a mountain, such as Himalaya or Tibetan Plateau, are squeezed on to a smaller area on a datum plane, such as MeanSea Level (MSL). Conversely, the depths of oceans are brought-up and stretched.Therefore one (single) scale may not be applicable to measure a ratio between aunit distance on map to that on the ground at all points in a regional map.

    CONTOUR MAP OF AN UNDULATING SURFACE (2D)

    48

    Although a single scale has been provided for above contour map, the ratio (mapto ground) may be different for depths (dark region) and different for high ground (light region).

    RELIEF MAP OF TIBETAN PLATEAU AND INDIAN PLAIN

    (ONE SCALE MAY NOT BE APPLICABLE)

    Similarly a single scale may not be applicable for a map or a photograph of large area depicting Himalayas, Tibetan plateau and Indian Planes together. There may be huge elevation difference between them.

    ********

    TEASER - 13

    HEAD MASTER

    Ramayan is one of the greatest epic of Hindu (Indian)

    Mythology. It is a saga of victory of god Ram, with the help of

    god Hanuman over the evil Ravan. The saga is entwined with

    multiple kings and warriors. One out of following kings had an

    asymmetrical body, which one was that?

    (a) Ravan, the demon king

    (b) Sugreev, the monkey king

    (c) Jatayu, the vulture king

    49

    THE EXPLANATION:

    Symmetry is the preservation of form and configuration across a point, a line, or a plane where shape on one side match exactly to the shape on another side.

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    There can be several different types of symmetry and in each type, characteristics such as angles, side lengths, distances, shapes, and sizes are maintained. The main types of symmetry include: a) rotation, b) translation, c) glide-translation and d) reflection etc.

    The rotational symmetry results from the transformation called rotation, which is the turning of a shape around a center point or the center of rotation. The distance to the center of rotation is kept constant and the amount of turning called the angle, is measured in degrees.

    The translational symmetry results from the transformation called translation, which is just another term for "movement" with specific direction and magnitude.Translation is making a copy and then moving it.

    The glide reflective symmetry results from the transformation called glide reflection and is actually a combination of a reflection and a translation.

    The Reflection symmetry is the most familiar type of symmetry which is also known as line or bilateral symmetry. This result from the transformation called reflection occurring across a line or the axis. The corresponding point is at same distance from the axis as is the original point. Reflection may happen first or second but the figure that results after a reflection and translation is called glide reflection of the original figure.

    50

    SYMMETRICAL IMAGE ON EITHER SIDE OF A VERTICAL LINE

    (Reflection symmetry)

    Human and animal bodies are bilateral symmetries because one half (right side) is exact correspondence of other half (left side) on opposite side of a dividingplane passing through the center of the body.

    In the Hindu epic Ramayan, Ravan was the demon king and he had ten heads on hissolder. The Ravan had tormented god Ram and had abducted his wife Sita. God Ram

    killed the demon king Ravan over a long battle with the help of Hanuman, the monkey god, the Sugreev, the monkey king, Jatayu, the vulture king and Jamvant, thebear king among others.

    Ten heads of Ravan was arranged in T shaped pattern. The central head, attached toa single neck, was flanked by four heads on one side and five heads on other side making total ten.

    TEN HEADED RAVAN WITH SINGLE NECK

    51

    Therefore the unequal number of heads on either side of central head created an

    imbalance and assymetricity in the body of Ravan, the demon king.

    In terms of assymetricity, Ravan had inherited a defective body. It is possiblethat he was constantly troubled by the assymetricity of his body manifesting inhis violent behavior and hot headedness.

    To create a perfect symmetrical body for Ravan with ten heads, the heads must have to be arranged in a Y shaped pattern with two central heads attached to a Y shapd neck. The neck would be branching so that five (5) heads are equally divided on either side and are balanced.

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    TEN HEADED RAVAN WITH TWO NECKS

    ********

    TEASER - 14

    LAT-X

    52

    The latitude reading / value (Degrees) on the summit of a

    mountain, (say Everest, Chomolungma or Sagarmatha) and at

    its root is always same:

    (a) True

    (b) Falls

    THE EXPLANATION:

    The earth s surface is divided by an imaginary set of grid lines, the longitude

    and the parallels of latitude, which are referred as geographic coordinate system. These coordinates appear on all standard maps and on some they may be the only way of locating and referencing a specific point.

    (a) Latitude lines

    Latitude lines on the earth s surface are drawn connecting all points equidistant (parallel of latitude) from the equator and form a set of rings around the globe. The latitude value is measured from the equator, with positive values goingnorth and negative values going south. The equator is 0 latitude and the North Pole and South Pole are 90N and 90S latitudes, respectively. The length of one degree of latitude averages about 69 mile (110 km), it increases slightly from the equator to the poles as a result of the earth s polar flattening.

    53

    (b) Longitude lines:

    Longitude lines are a set of north-south rings crossing the equator at right angle and converging at poles. These lines form great circles and always divide earth into two. By international agreement, the meridian passing through the original site of the Royal Greenwich Observatory at Greenwich, England, is designatedthe prime meridian, and all points along it are at 0 longitude. All other pointson the earth have longitudes ranging from 0 to 180E or from 0 to 180 W.

    The Angle LAT-X

    The height of an elevation or depression on earth is measured along the plumb line. A plumb line is defined, as a line from which if a weight is suspended it isdirected exactly toward the earths center of gravity.

    54

    The longitude plane and plumb line pass through the center of earth, and fall onsame plane and do not cut each other. On the other hand, barring equator, the latitude plane does not pass through earths center hence both plumb line and latit

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    ude line do not fall on same plane and cut each other therefore are oblique to each other. The crux of the anomaly lies in the phenomenon of obliqueness of latitude plane with respect to plumb line.

    The plumb line cuts latitude at an angle which increases from zero degree at equator, to 90 degrees at poles. This angle is referred to as LAT-X here.

    NORTHERN HEMISPHERE SHOWING ANGLE LAT-X

    As we know, contour maps are generally made on a base or reference plane, so that height or depth are calculated by counting contours up or down from the reference or base plane. While making a contour map, the location of a point on earthsurface, such as a mountain top, is transferred straight down onto a base planealong plumb line and not along latitude line. Except along equator, the Latitudelines and plumb lines (S - R line) are always oblique to each other, hence thelocation of S and R would not be at same point with respect to latitude values.However, on a contour map both S and R

    would be located exactly at same place, irrespective of height hence would havesame latitude value.

    55

    CROSS SECTION (X-Y) SHOWING S-R LINE (PLUMB LINE)

    OBLIQUELY CUTTING LATITUDE LINES

    The summit (S) and root (R) of a mountain can be likened to two floors of a building with S being first floor and R being ground floor. The floors are linked bya stair case which is parallel to latitude plane. Walking along the staircase from ground floor to first floor is like walking along a latitude plane. The staircase is always diagonal cutting ground and 1st floors at different points. Theslope of staircase would depend on angle X and gap between two floors.

    LATITUDE LINES CUTTING A BUILDING DIAGONALLY

    Therefore, the summit (S), like first floor, shall not have same latitude (stair

    case) as that at root (R) or ground floor, contrary to a contour map where bothS and R are placed at same point.

    ********

    56

    TEASER - 15

    TELESCOPIC EFFECT

    On a local scale, compare an aerial photograph (seen from top)

    of a mountain with contour map of same mountain or an aerial

    photograph of a tall building with contour map of same

    building.

    The shape of the mountain, or the tall building would be exactly

    same in both aerial photograph and contour map:

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    (a) Falls

    (b) True

    THE EXPLANATION:

    The aerial photograph is a method of creating 2D image of relief of an undulating surface by means of variations of light and shade. The contour maps are systematized form of showing same variations of light and shade by means of lines connecting equal elevation or depth.

    The most important thing in determining shape and size of 3D objects in contourmap or aerial photograph is whether relative position of each point, with respect to each other, have been maintained taking into account the element of height,length and width.

    The contour map (especially on a local scale) maintains the relative position ofeach points with respect to each other as seen in ground

    57

    at all level. However, even though aerial photographs do not hide anything, it deforms the feature and distorts the relative position of points seen in ground.The nearer objects appear bigger and farther one becomes smaller telescopically.

    The objects seen within the

    view finder have to be adjusted for depth element because photographs or image in2D formats lack depth.

    One scale would not be applicable for all the objects seen within same photo-frame and shall vary according to depth. Therefore shape of object get distorted proportionately with depth in an aerial photograph.

    Take for example a pyramid shaped structure with staircase type slopes as shownin figure below (A).

    In the map view (contour map) of the structure (B), the relationship between each step with respect to their size and position remain constant irrespective of relative distance of each step from top to bottom. The ratio of an area between top step and bottom step would remain same as would be in reality.

    Single map scale would be applicable for each step, irrespective of their depthor height from reference level (datum level) on which map is made.

    In an aerial photograph (C), taken from very close range from top, the step closer to the viewer would appear bigger than the step away from viewer (towards bottom). The ratio between top and bottom steps with respect to size and position would not be same as would be in reality. The ratio would depend on how close thephotograph is taken from.

    Single scale would not be applicable for all steps in the aerial photograph (C).Each step shall have to have its own scale depending upon its depth from the le

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    vel on which photograph is referenced.

    58

    SHADED CONTOURS (B) & PHOTOGRPH (C) OF A PYRAMID (A) SEEN FROM TOP

    In an extreme case (D), only the top stair would be visible in the aerial photograph and all below would be hidden behind. It may appear like an inverted pyramid, like a tall building seen from above (E).

    D: A PYRAMID SEEN FROM TOP AT CLOSE RANGE

    E; TALL BUILDING SEEN FROM ABOVE

    ROOF IS MUCH BIGGER THAN BASE

    ********

    59

    TEASER - 16

    NEW DELHI OR NEWDELHI

    New Delhi is the capital city of India. In 1912, during British Raj

    the Delhi (which was the capital city of India during Moghul

    Raj), was expanded and the expanded part of city was named

    as New Delhi, probably to reflect the new development and to

    distinguish it from old city of Delhi at that point of time.

    Let us assume that another expansion of New-Delhi on same

    style is again undertaken today and expanded part of city is

    proposed to be renamed in same manner as was done earlier

    so that it reflects new development. If that happens New Delhi

    would be called as?

    (a) New New Delhi

    (b) New Newdelhi

    (c) Brand New New Delhi

    THE EXPLANATION:

    During Moghul raj, Delhi was the capital city of India. In 1912, British Raj moved the capital of India from Calcutta to metropolis of Delhi. They constructed anew city adjacent to Delhi which was completed in 1931 and was named as New Delhi, because this side of city was new at that time. The New, prefixed to the name of Delhi was adopted, perhaps to reflect the new development and also keepingin line the new trend prevailing during those days.

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    60

    Literally, new means recent but how recent cannot be known, it can only be sensed relatively. Like new year becomes old within few weeks and new baby becomes old within few months. Time does not stop for any one, todays new is tomorrows old,therefore New added by British raj to Delhi may not convey same meaning today, as it would have conveyed at the time of its naming.

    Since its formation, New Delhi has expanded in all direction, virtually adding several new city to it. If new regime decides to name the expanded part of the city in same way as was done earlier why it cant be re-named as New New Delhi (or New Newdelhi) again? With the passage of time, New and Delhi may not remain separate words in same sense as it was thought initially but both may merge and become a single word Newdelhi in same way as Newton or Newman are single words, having no relation with the literal meaning of new

    Imagine that if an old city, named after Newton, is to be renamed in same way asthat of Delhi to reflect new expansion, it would be called New Newton and old part of the city would always be called as old Newton.

    NEW CITY AND OLD CITY SIDE BY SIDE

    61

    If New Delhi becomes Newdelhi, (a new word) with no relation with its literal meaning, the expanded and new part of the city can be easily renamed again as NewNewdelhi and old part can continued to be called Old Newdelhi.

    CLUSTER OF NEW & OLD CITY SIDE BY SIDE

    This process can continue till big cities stop growing and adding new cities.

    ********

    TEASER - 17

    WHOSE SIDE IS IT ANY WAY

    Every object such as, cars, houses, dogs, humans, planes,

    books, watches or TVs, have their own right, left, front and back

    sides.

    62

    When we refer left or right side of a book or a watch or a TV or a

    road, we refer left or right side in same way as it would be in

    case of a car or a plane or a dog or a human?

    Similarly when we say clockwise or anticlockwise we refer to it

    from watchs own side?

    (a) Falls

    (b) True

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    THE EXPLANATION:

    When humans are driving cars, both car and driver face in same direction, henceleft hand side of driver is also left hand side of the car.

    CARS AND HUMANS FACE SAME DIRECTION WHEN DRIVING

    HENCE BOTH HAVE SAME SENSE OF RIGHT AND LEFT

    63

    When a woman looks into a mirror everything gets reversed. Her left hand becomesright hand of her image in mirror because both of them are facing each other and the image behaves as if she is an another person looking at her.

    RIGHT IS LEFT (AND VICE VERSA) WHEN

    LOOKING INTO MIRROR

    However, when a man is reading a book, even though both book and man are facingeach other in same way as he would be facing his image in the mirror, the left and right side of book does not get reversed.

    The left side of the man (reader) is also the left side of book and his right side is the right side of the book. Book itself does not have its own left or right side. Had it been so our right would have been left side of the book.

    RIGHT IS RIGHT (NOT LEFT) WHEN REFERRING TO BOOKS

    64

    Similarly when we are facing a watch and referring to a direction called clockwise or anticlockwise movement, we are explicitly referring to the direction from viewer / user point of view.

    The circularity of clockwise movement would be from top to right, (in same way a

    s cars steering is moved when turning to right). The opposite sense of rotation would be counterclockwise or

    anticlockwise.

    CLOCKWISE MOVEMENT (GENERALLY LEFT TO RIGHT)

    Assume that a watch has grown its own hands and legs and is behaving like a human being. What would be the clockwise

    movement ?

    WHAT IF CLOCKS HAD HANDS & LEGS AS WE HAVE

    (CLOCKWISE MOVEMENT WOULD BE REVERESED)

    65

    Clockwise movement would be, from TOP to LEFT (in same way as cars steering is moved when turning to left) and not TOP to RIGHT

    as is normally considered. (Anticlockwise movement would be opposite to that):

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    In strictest sense, the clockwise movement is a misnomer because we never considerwatch having left / right side like us, but we refer it from our own point of view. Hence it can be called viewer-wise clock-movement. The left or right side ofan object is dependant on whether they face in same direction or opposite direction as that of humans (user) while in use.

    User centric world view:

    1. As the reader and the book or the TV or the photograph or the watch always face each other when in use, there was perhaps no need to assign left or side to them but was always referred from users point of view. On the other hand, cars, buildings, trains, animals etc face in same direction as humans while in use, theyhave left and right side similar to humans.

    2. All man made materials or even natural ones are referenced from user (humans)point of view because we are the ones who use them and we are the ones who require communication in such languages which fundamentally is based on naming of each and every possible aspect perceived by humans.

    ********

    TEASER - 18

    LOOSER CAN BE WINNER

    66

    The scoring system in Lawn Tennis goes through a tortuous

    four course steps. A player has to win first points, then games,

    then set and only then he wins a match.

    The winner always wins more games than the looser,

    (a) True

    (b) Fall

    THE EXPLANATION:

    Lawn Tennis is played with four tiers / steps of scoring system.

    A player has to win five (5) points first with a gap of two to win one game, then he has to win six (6) games first with a gap of two to win one set and win maximum set out of five or three to win a match.

    In this system each set is like an independent match and have to be won separately. No weightage is given to the games won in previous set and it does not matter whether previous set is won by 6 - 0 or 76, both have same value.

    In case of a score line of 0-6, 7-6, 7-6, 0-6, 7-6, the winner gets away with 9fewer games than the looser. Out of a total possible 51 games in the match, thelooser have won 30 (58%) and winner only 21 (42%) thanks to rules framed to favor one with no breaks, barring tie breakers.

    67

    TENNIS MATCH: MOST TORTUOUS SCORING SYSTEM

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    Badminton is also played on same scoring system but with one tier fewer than Tennis;

    Here, a player has to win points first, then sets and only then a match is won.Each set is like an independent match and have to be won separately and no weightage is given to points won in previous set whatever may be the margin.

    BADMINTON: LITTLE LESS TORTUOUS SCORING THAN TENNIS

    The scoring system in basket ball, on the other hand is the most simplest one. No segments, no layering, no tier, you win more points you win match. Entire match is one entity. First part is not forgotten

    68

    and points won from the start get full weightage and is carried forward till theend of the match.

    A great tempo and excitement is generated from the word go and never a dull moment creeps throughout the match.

    BASKET BALL: SIMPLE SCORING SYSTEM

    Some Examples of Close Tennis Matches:

    Womens Events:

    1. US Open Womens Final 1985

    Hana Mandlikova def. M. Navratilova: 7-6, 1-6, 7-6.

    Or

    Martina Navratilova def. Hana Mandlikova : 18 - 15

    Two-time US Open champion Martina Navratilova won a total of 18 games (6+6+6), where as Hana Mandlikova won 15 games (7+1+7), 3 games less than the Martina, in

    the US Open final of 1985 but still Martina lost the match.

    The result would have been just opposite, had basket ball type scoring system was followed. Hana took her chance more

    69

    carefully and reserved energy for the crucial set where it mattered most. Had itbeen 5 setter Martina would have definitely won the match as she showed consistency.

    2. US Open Womens Final 1994

    Arantxa Sanchez def. Steffi Graf: 1-6, 7-6, 6 - 4.

    Or

    Steffi Graf def Arantxa Sanchez : 16 - 14

    Steffi Graf won a total of 16 games (6+6+4), 2 games more than the total of 14 games (1+7+6) won by Arantxa Sanchez Vicario in US Open 1994 but still she lost the match. Had the points / games scored in previous set been carried forward till the end of the match Steffi would have been a clear winner (16 - 14).

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    3. Wimbledon Womens Final, 1993:

    Steffi Graf def. Jana Novotna: 7-6, 1-6, 6 - 4.

    Or

    Jana Novotna def Steffi Graf: 16 - 14

    Steffi Graf won a total of 14 ( 7+1+6) games, 2 games less than Jana Novotna, who won a total of 16 (6+6+4) games in the entire match but still she lost match?The cumulative system would have produced just opposite result.

    Mens Events:

    1. Wimbledon Mens Final, 2009:

    Roger Federer def. Andy Roddick: 5-7, 7-6, 7-6, 3-6, 16-14

    Or

    Andy Roddick def. Roger Federer : 39 - 38

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    One of the longest and keenly contested match ever played at the Wimbledon. Inspite of winning more games (39) than Roger (38), Andy lost the match, why? At least I am hugely disappointed. I thought Andy had upper hand and looked more sharpthan the aging war horse Roger throughout the game.

    2. Montreal Masters Mens Final, August 2007

    Novak Djokovic def. Roger Federer: 7-6, 2-6, 7-6

    Or

    Roger Federer def. Novak Djokovic 18 - 16

    The serb, Djokovic beat Roddick, Nadal and Federer, the top 3, 2

    & 1 of tennis at that time in a row, a big feat indeed. However he was very lucky in the final as he won title with a deficit of two games. The king Federer could have easily won had it been five setter, I am sure, as he had already collected more breaks and more games than Novak. Tennis, curiously does not recognise previous set. New set starts as a new match. Federer could not do any thing but look to next event.

    3. ATP/WTA Miami Masters Event, 4th Round, April 2007

    Guillermo Canas def. Roger Federer: 7-6, 2-6, 7-6.Or

    Roger Federer def. Guillermo Canas: 18 - 16

    Roger Federer faced same situation here too. Guillermo Canas of Argentina, had stunned him, 7-6, 6-2, 7-6 in the forth round of ATP/WTA Miami Masters Series event in April 2007. Federer had said then that it was one of those matches where he should have never lost. But did he loose? In fact he won more games, but thanks to weird rules that did him in.

    4. US Open Men s Singles Fourth Round, 1987

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    71

    J. Connors def. M. Pernfors: 1-6, 1-6, 7-5, 6 - 4, 6 - 2 Or

    M. Pernfors def. J. Connors: 23 - 21

    This is considered to be one of the greatest come back match of the US Open tennis where Connors made it to quarterfinals in spite of 2 set down. On a cumulative basis Pernfors won a total of 23 games (6+6+5+4+2), 2 games more than Connorswho won a total of 21 games (1+1+7+6+6). Continuous scoring system, without anybreak between sets would have made Pernfor winner.

    5. Wimbledon Mens Final, 2007:

    Roger Federer def. Rafael Nadal: 7-6, 4-6, 7-6, 2-6, 6 -2Or

    Rafael Nadal drew. Roger Federer: 26 26.

    Locked in the toughest test, Roger Federer finally overcame Rafael Nadal in a five-set epic final to win his fifth consecutive championship at the All England Club. However, the match could have gone either way. Both won equal number of games (26 each) but still Nadal was denied further chance.

    At this stage the match was equally balanced and should have ended in a draw. Nadal had more breaks and showed more consistency and should have been given morechances to prove his prowess, however, laws of the game, made by wise men had other things up the sleeves.

    6. Wimbledon Mens Final 1992

    A. Agassi def. G. Ivanisevic: 6 - 7, 6 - 4, 6 - 4, 1-6, 6-4.

    Or

    G. Ivanisevic drew. A. Agassi: 25 - 25.

    Goran blasted 37 aces and won a total of 25 games (7+4+4+6+4) equal to that wonby Agassi (25 games) and still lost. The dual was a photo finish and needed an extraneous and crooked process of scoring system to decide the winner.

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    Why this discrepancy?

    In all above matches the looser has won more or equal number of games and showedbetter consistency but they still lost the match. The rules seem to favor the opportunistic, cunning and to those who are conserving energy deliberately and ruthlessly pouncing on few chances coming their way. Unlike basket ball match, where tempo is maintained throughout the match, tennis is played in hiccups. Each s

    et is like a separate match and at times players deliberately slowdown to come back in next set and dullness creeps in. Had points been carried forward in a seam less manner the boring moments would have been minimized and consistency wouldhave been rewarded. So it is not necessary that winner always wins more games,it is the looser, may win more games but still shown the door. It is a shame !

    We see a great ambiguity here. On one hand a lot of weightage and importance isattached to the distant past performance of a player while seeding them (so to give them unfair advantage), but the performance in the immediate past (previousset) is totally ignored and a new set begins afresh from scratch.

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    ********

    TEASER - 19

    SERVING FIRST

    Andy (A) and Sandy (S) are facing each other in a tennis match

    where Andy (A) is serving first in first set;

    i

    What will be the score if Andy retains his service and

    breaks 1st service of Sandy (S) in the 1st set ?

    (a)

    6 4

    (b)

    6 3

    ii What will be the score if Sandy retains his service and

    breaks 1st service of Andy in the 1st set ?

    (a)

    4 6

    (b)

    3 6

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    THE EXPLANATION:

    Andy (A) is serving first in 1st set.

    (a) Andy (A) retains his service ( ) and breaks (X) the 1st service of Sandy (S)and wins first set,

    Table a

    Service

    Games

    Result

    Score

    A

    S

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    1

    1 0

    2

    X

    2 0

    3

    3 0

    4

    3 1

    5

    4 1

    6

    4 2

    7

    5 2

    8

    5 3

    9

    6 3

    TOTAL

    5

    4

    9

    9 (5+4)

    (b) Sandy (S) retains his service ( ) and breaks (X) the 1st service of Andy (A)and wins first set,

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    Table b

    Service

    Games

    Result

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    Score

    A

    S

    1

    X

    0 1

    2

    0 2

    3

    1 2

    4

    1 3

    5

    2 3

    6

    2 4

    7

    3 4

    8

    3 5

    9

    4 5

    10

    4 6

    Total

    5

    5

    10

    10 ( 5+5 )

    When A breaks 1st service of S (Table - a), the maximum gap created between A an

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    d S is of the order of 3 games where as when S breaks 1st service of A (Table-b)the maximum gap is of the order 2 games only. Here A and S do exactly same thing but A achieves a gap of 3 games to 2 of S without doing any thing extra.

    It can have a psychological impact on the performance of S. When A breaks 1st service of S, he wins set at 6 3 and plays a total of 9 games (Table-a) where as when S breaks 1st service of A, he wins set at 4 6 and plays a total of 10 games,one game more than A (Table-b). By breaking only once in the whole set A gets away by playing only 9 games (serves five time to four of S), where as for exactly same number of break S is required to play 10 games for none of his fault, whythis discrimination ?

    (c) Now, what happens if score line is:

    6 1 in favor of A (Table-c) or

    1 6 in favor of S (Table-d)

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    Table - c

    Service

    Games

    Result

    Score

    A

    S

    1

    1 0

    2

    1 1

    3

    2 1

    4

    X

    3 1

    5

    4 1

    6

    X

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    5 1

    7

    6 1

    Total

    4

    3

    7

    7 (4+3)

    Table d

    Service

    Games

    Result

    Score

    A

    S

    1

    1 0

    2

    1 1

    3

    X

    1 2

    4

    1 3

    5

    X

    1 4

    6

    1 5

    7

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    X

    1 6

    Total

    4

    3

    7

    7 (4+3)

    The total number of games played (in both cases) (Table - a

    & b) are seven (1 to 7) with 4 odd numbers (1st, 3rd & 5th) and 3 even numbers (2nd, 4th & 6th). By virtue of serving first, A shall get an opportunity to serveodd numbers (4 times) and S shall have opportunity to serve even numbers. So Agets 4 serves and S gets 3 serves. Therefore, for same score line (6

    1) A is required to break serve of S only two times (Table-c) where as S toils to break A three times (Table-d).

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    Why this inequality ?

    In all above cases, it is very obvious that A is enjoying a certain amount of advantage over S only because he is serving first. In todays hyper sports and intense competitive era even a hair line gap advantage may have impact on the final result, especially if winner and loser are of same caliber.

    What is the solution: Equality ? :

    Each player must be given equal number of services in each set and in each match, so that equal opportunity level is achieved.

    Each set / match can be played in two stages.

    1st stage to be of 4 serves in total (2 each continuously). Whosoever win maximum points with a gap of 4 wins the set else set goes to 2nd stage.

    2nd stage would consist of maximum of six Steps (S-1 to S-6). Each player will be given one service alternatively at each step. One who wins maximum points witha difference of four (4) at any step wins the set.

    In case of a tie at the end of 6th step, either the set is terminated, irrespective of points difference and points in each set are added at the end of match and winner is decided on the basis of maximum points won cumulatively in a match.

    Alternatively set can continue till a difference of 4 points is achieved. Set it

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    self can be converted into a match beyond a certain point.

    1) 1st STAGE: Total: 4 services (2 each with no gap).

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    SCORE/ RESULT

    PLAYER

    STAGE SERVICE

    Go to

    WIN

    A

    S

    2nd STAGE

    1

    1st

    2

    4-0 3-1 2-2

    STAGE

    3

    4

    2) 2nd STAGE: 2nd stage consists of maximum 12 services

    SCORE / RESULT

    PLAYER

    STAGE SERVICE

    Go to

    WIN

    A

    S

    2nd STAGE

    1

    1st

    2

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    STAGE

    3

    4-0 3-1

    2-2

    4

    Go to NEXT

    WIN

    SERVE

    5

    S-1

    5-1 4-2

    3-3

    6

    7

    S-2

    6-2 5-3

    4-4

    8

    9

    E S-3

    7-3

    6-4

    5-5

    G

    10

    AT

    11

    Sd S-4

    n

    8-4

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    7-5

    6-6

    2

    12

    13

    S-5

    9-5

    8-6

    7-7

    14

    15

    S-6

    10-6

    9-7

    8-8

    16

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    In each set there will be seven (7) opportunity (one in 1st stage & six in 2nd stage) to win the set early. If each set ends in 1st stage itself, match will beover very quickly. It would indicate that the looser is not of equal caliber and

    match deserves to end fast.

    Alternatively, games or points won in each set can be added together at the endof match and who so ever wins maximum points, in whole match (cumulative) with adifference of 4 shall win the match.

    ********

    TEASER - 20

    CURSE OF CURVE

    Consider there are two humanoid robots K1 and K2. Both have

    exactly identical capabilities, both run at same speed from start

    to finish. Now both are competing with each other for 800 m

    race (cycle) together (pack running) in an indoor stadium with

    circular / oval track. Who will win?

    (a) One of them (K1 / K2)

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    (b) Both

    79

    THE EXPLANATION:

    Game at micro-second scale:

    In modern day sporting events, the electronic gadgets, like 3rd umpire in cricket or stretched photographs in sprinting are used extensively in order to extractthe hair line gap to decide who is in or who is out. This micro-second precision level is rigorously applied in all games and even one hundredth (1/100) of a second or one tenth of an inch is not spared in separating winner from looser.

    CLOSE FINISH: REQUIRE MACHINE SCRUTINY

    But what is startling is that same precision of microscopic level analysis is seldom applied to the game in entirety to see whether factors other than individual efforts are influencing the results.

    There are many events where extraneous factors other than individual effort appear to be influencing the result. One such area is track & field event where tracks are designed in such a way that each lane is different from other and each tr

    ack is curved at every turn. Although variation in lane qualities may appear miniscule at human scale but can get magnified many time at sub-second (microscopic) scale.

    The lanes in 100 m sprint events are all straight and parallel hence are exactlyidentical, however, in 400 m oval (circular) track, the conditions are not exactly same for all lanes.

    80

    FULL CIRCULAR AND SEMI-CIRCULAR TRACKS

    The total length of an individual lane (circumference) in case of round track (a

    n extreme case) can be shown as follows:

    L

    = PR2

    Where:

    L

    = Length of individual lane (say Lane-1)

    P

    = 22 / 7 (Constant)

    R

    = Radius of individual lane

    Or in case of an oval track (semi circular track) the formula for total track length would be as follows;

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    L

    = Z + PR2

    Where:

    L

    = Length of individual lane (Lane-1)

    P

    = 22 / 7 (Constant)

    R

    = Radius of individual lane

    Z

    = Straight part of the track

    Now, take for example lane one (L-1) and lane two (L-2) are having radius R1 & R2 of 50 feet & 54 feet respectively, (assuming a

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    minimum lane width of 5 feet) the length of individual lanes would be as follows, (Z being constant for both lane):

    L1

    = P(R1)2

    = 22/7 x 502

    = 7857 feet

    L2

    = P(R2)2

    = 22/7 x 542

    = 9165 feet

    The ratio:

    L1 / L2

    = 7857 / 9165

    = 0.86

    Or

    (R1)2 / (R2)2 = 54 / 52

    = 0.86

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    Therefore Lane-2 is dissimilar to Lane-1 (1: 0.86) in terms of total distance (end to end). In case of 400 m or longer race, any discrepancy due to this would get multiplied.

    The degree of curvature or circularity for each lane can be determined by the ratio between total length of lane (circumference) and total length of maximum number of straight dashes one can make within the limits of an individual lane.

    C1 = D1 / L1

    Where

    C1 = Degree of curvature of Lane-1

    D1 = Total length of straight dash in Lane-1

    L1 = Total length / circumference of Lane-1 (PR12 )

    82

    THE OUTER-MOST CIRCULAR TRACK

    HAS THE LONGEST STRAIGHT DASHES

    Lane-3 has 5 straight dashes (SD) of equal length (5SD)

    Lane-2 has 4 straight dash of equal length and one is 1/3 (4SD+1/3SD).

    Lane-1 has 3 straight dashes of equal length and one is

    (3SD+1/2SD).

    There are three areas of concern in case of oval / circular tracks; 1. The lanesare not 100% equal in terms of total lengths.

    2. The lanes are not 100% equal in terms of curvature.

    3. The ratio between curvature and straight portions of track are not 100% equalbetween the lanes.

    If two lanes are equal in terms of all characteristics, the ratio shouldbe always 1:1 (100%) between lanes as happens in case of 100 msprint event. Therefore anything less or more than 100% wouldmean un-equality among lanes, which when subjected tomicroscopic level would magnify many times.

    Who would win K1 or K2

    The races are generally run in packs or individually along single lane. Pack orbunch running involves body touching and constant

    83

    fight for inner circle where as independent running does not involve body touching.

    PACK RUNNING AND SINGLE LANE RUNNING

    In case of pack or bunch running, as long both, K1 and K2 run on straight portion of the track, they will be running side-by-side without body touching, howeverthe moment they enter into curved portion, the one on inside track (say K2), be

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    ing nearer would occupy the inner circle and would forge ahead of K1 simply because inner side of the curve has less distance. K1, being towards outside, wouldimmediately fall back, as he has to cover longer distance and would remain behind, thereafter till end and finish second. Therefore only one (K1) would win therace.

    RUNNING AROUND BENT PART OF A TRACK

    On the other hand, independent running, although does not involve body touchingand fight for inner circle, one has to tackle the curvature of the track to maintain the speed and body tilt.

    84

    The impact on races: the impact of circular track on results can be summarized as follows:

    (i) Body Tilt Effect:

    The inner most lane has maximum curvature and outer most the least. To overcomelane curvature athletes tilt their body inside proportionate to their speed andcurvature so as to avoid falling out of lane due to centrifugal force. Since curvature is variable for each lane the body tilt varies from lane to lane, maximumin inside lane and vice versa. The variable tilting affects the speed and momen

    tum and ultimately the result.

    BODY TILTING WHILE NEGOTIATING A BEND

    (ii) Cumulative Effect:

    The tracks are generally laid in an oval shape where the ratio of straight portion to curved portion of track varies from stadium to stadium. The speed is maximum in straight portion and minimum in curved portion. In case of small indoor stadium, if one is allotted inner lane, he may have to pass through curved portionmore frequently than the one allotted outer most lane.

    Any amount of variations among lanes, however negligible it may be, can turn int

    o a snow ball on a cumulative basis, especially in events where participants are required to run multiple times through same lane.

    85

    (iii) Scale Effect:

    The impact of curvature may appear miniscule at human scale but can get magnified many times at micro-second level. The impact can be seen more easily in the events such as motorcycle / formula car race where participants need to maintain high speed over a long and tortuous track.

    BODY TILT AT BENDS INCREASES

    WHEN HIGH SPEED IS MAINTAINED

    It is evident wide from above that the participants are not provided equal running conditions at same standard of precision as their results are subjected to. Athletes have every right to demand identical conditions on a microscopic scale.

    The glaring examples:

    The result of 400 m hurdle in 1984 Los Angeles Olympics for bronze medal was sub

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    jected to a very high degree of magnification and nothing was spared in denyingPT Usha of India a bronze. It was amazing that she was made to loose by one hundredth of a second (1/100 second).

    She literally lost by a whisker. The scale of stretching of time and space was amazing and unbelievable. The time difference is so small (1/100 second) that itstill amazes even today and creates

    86

    doubt whether it was her deficiency or track design deficiency that cost her bronze. There can be no doubt that the result would have been definitely different(plus or minus) if she had been allotted another lane than she was running in.

    SPRINT QUEEN P.T. USHA OF INDIA

    Similarly, Milkha Singh of India, was denied bronze medal in 400 m race in 1960Rome Olympics by a wafer thin margin of 1/10 seconds, although he had broken theOlympic records.

    FLYING SIKH MILKHA SINGH OF INDIA

    87

    There may be hundreds of such close cases where doubts would always remain in the minds whether failure was due to extraneous causes. These cases should be putto modern tests involving computer software analysis to see if results were influenced by factors other than individual efforts, such as lane variation, especially over long races.

    What is the solution:

    1. The dissimilarity among the lanes in a circular track is very obvious, in terms of variable curvature, variable total distance and variable straight to curved ratio.

    2. The first and foremost requirement is to thoroughly analyse the existing trac

    k & field design at microscopic level with respect to its effect on the outcomeof results so as to eliminate any doubt in the minds of participants, especiallythose who loose by whiskers.

    3. In order to achieve identical lane properties, track designs as indicated below can be tried:

    a. ASHTAMIUM (8): 8 Shaped Track Design.

    In an ancient Hindu Sanskrit language eight (8) is referred to as Ashtami, hencename Ashtamium.

    Each individual lane is curved at two ends and the product of sum total impact of degree of curvature, elevation, undulation or any other element is always constant for each lane.

    Each lane is of same length of 200 m.

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    The lanes do not cross on same level but cross one over the other like an over bridge so as to avoid clashing of athletes.

    The races are always run in 200 m or in multiple of 200 m.

    88

    There is no need to run in packs as several tracks can be accommodated in a single stadium.

    ASHTAMIUM: NEW TRACK DESIGN

    b. SWIMMING POOL TYPE RACE TRACK

    Another variation could be similar to that of swimming pool type where all tracksare parallel and straight. Each individual runs in his own track to and fro (bytouching the end of each lane).

    SWIMMING POOL TYPE TRACK DESIGN

    TRACKS ARE ALL STARIGHT AND PARALLEL

    *******

    89

    TEASER - 21

    WHITE ALWAYS WINS

    Two computers C1 & C2, with identical capabilities are pitted

    against each other in a game of chess. If C1 starts with white

    pieces first and both make no mistakes, who will win match in a

    best of 5 (five) games ?

    (a) C1

    (b) C2

    (c) End in a draw

    THE EXPLANATION:

    Theoretically, the game should end in a draw as both have equal capabilities andboth make no mistakes. However, simple commonsense logic proves it otherwise.

    The game of chess in its present form is highly complicated. There is multiplicity of pieces and there are tens of moves and whole game thrives on mistakes committed by opponents.

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    It is a well known fact that in the game of chess, the balance is tilted in favor of white. This aspect is very difficult to see in complicated and long windingmoves. Therefore in order to analyse game of a chess in a transparent and convincing manner, the chess board and pieces can be scaled down to: a) 36 squares (6x 6) with 24 pieces or to b) 16 squares (4 x 4) with 8 pieces.

    90

    36 SQUARES 16 SQUARES

    24 PIECES 8 PIECES

    The simplified version will have less complicated moves and will be easily understandable even to a layman. This is required to see and magnify the inherent flaws in a game of chess. In the strip down version the positions and movements ofeach piece remain same.

    As the number of moves become very less and short it becomes easy to foresee future best moves. Any body can easily understand the move and see himself whethergame ends in a draw or in favor of either of one. Let us imagine that C1 with white and C2 with black pieces play the game:

    (a) First move by white: White Queen to B3

    (b) First move by black: Black Queen to B2.

    91

    (c) Second move by white: White rook to D1 (Game is over)

    In this simplified game, it is very evident that C1 (white) attacks from the word go and is always one step ahead of C2 (black).

    It is evident here that C1 (white), getting the first opportunity to move alwaysgoes for best move and inflicts maximum damage and if no mistake is done, as would happen in case of competing machines, C2 (black) would never win. C1 (white)would en-cash best move and will go for a kill. C2 (black) is left with no opti

    on but to be content with second best move and is always busy defending C1(white) and never gets to attack

    The game of chess is unique in the way that it is like a battle field where every move kills pieces (soldiers) and reduces opponents ability to fight from the word go. Unlike many other sports where both opponents are allowed to fight at same time, in chess opponents are allowed to hit alternatively and white getting first opportunity it unleashes best move. Black is left with second best move andalways remains behind white. Black receives the first lethal blow and never recovers thereafter.

    92

    For that matter any one who starts (attacks) first in any game has some advantage, but in chess white starting first, has 100%

    advantage.

    ********

    TEASER - 22

    HONEY COMB

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    Honey bees make hexagonal cells / combs, because?

    (a) Round combs are out of fashion

    (b) Bees know no other comb making

    (c) For economy and geometric reasons

    THE EXPLANATION:

    The honeybee, one of man s oldest friend is a social insect living in large colonies. The central feature of the bee hive is its nest the honeycomb and this marvel of engineering consists of pan