The Digits 1 to 9: Hints and SolutionsAuthor(s): Darrell MorganSource: Mathematics in School, Vol. 33, No. 2 (Mar., 2004), pp. 8, 33Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215674 .

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The Digits I to 9 by Darrell Morgan

(1) Write down the digits 1 to 9. Throw two dice and add the scores together. Now you may cross out any digits that add to, or are equal to this total. E.g. dice scores 4 and 5 = 9. I can cross out the 9 or 2 and 7 or 1, 2 and 6, etc. but only one combination. Keep throwing until you cannot cross out any more digits or you have crossed out all the digits. Any numbers remaining are added to make your score. If you manage to cross out all the digits you score zero points. Player two does the same. Play five games each; the player with the lowest total score is the winner.

I I I I

(2) Put the digits 1 to 9 into the squares in this grid so that each row column and diagonal add to the same total.

(3) Place the digits 1 to 9 face up on the table (or write them in your book). Take it in turns to choose a digit. The first player with any digits that add to 15 is the winner. You may need to record the digits chosen by both players.

(4) Put the numbers 1 to 9 into the circles so that each side of the triangle adds to the same total.

(5) Write out the numbers 1 to 9 twice. Player two does the same. Underneath write down a number to play. Your opponent does the same. Simultaneously show your numbers, the player with the highest number scores a point. Cross out the number used by yourself and your opponent. Play the game until both players have used all their numbers. The player with the highest score wins. If both players show the same number no points are scored.

(6) Put the numbers 1 to 9 into the grid to produce a subtraction that works.

Solutions: page 33

Keywords: Digits; Challenge.

Author Darrell Morgan, Cwmtawe Comprehensive School, Pontardawe, Swansea. e-mail: darrell@thejoyofmaths.co.uk

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The Digits I to 9- Hints and Solutions (from page 8)

Pupils may find it useful to make their own number cards from 1 to 9, especially for games 1 and 3. They also prove useful during trial and improvement strategies for the other problems.

(1) Some thought should be given as to which numbers are turned over. E.g. with 1, 2, 4 and 5, remaining, if 6 is thrown then turning over the 5 and 1 may seem the best option leaving 2, 4 or 6 next go (9 chances) but, turning over the 4 and 2 leaves a 5 or 6 (also 9 chances). The best strategy is usually to get rid of the higher numbers first. A probability table of dice scores may prove useful.

(2) After some time pupils may be given the row total (= 15) and then the centre digit (5). Extension to a 4 x 4 grid using the numbers 1-16 may prove challenging.

Solution 4 38 9 5 1 2 7 6

plus rotations or reflections

(3) Pupils need to make a record of numbers chosen by themselves and their opponent if they are to adopt a strategy. After some trials pupils may realize that choosing certain digits guarantees a victory. E.g. choose 8, opponent must choose 7, choose 6, opponent must choose 9 and 1!

Get pupils to answer the following questions:-

Which first choice digits result in a winning strategy?

Can you improve the game to eliminate these winning strategies?

- first choice must be 5 or lower.

- three cards are needed to make 15 (this option is similar to playing noughts and crosses on the solution to question 3).

(4) There are five different possible totals for each side (17, 19, 20, 21 and 23) achievable in a variety of ways. Try using 1, 2 and 3 in the corners for the lowest total and 7, 8 and 9 for the highest.

1 4 4 3 7 48 56 18 24 56

9 6 9 2 9 3 7 8 3 1 3572 1387 6275 9156 8429 total 17 total 19 total 20 total 21 total 23

(5) This very simple and popular game encourages pupils to think about their opponents' strategy and make decisions based on this. Pupils need to record their opponents and their own plays to ensure that they think carefully about future plays. The game can be adapted so that the difference between numbers played is used as the points.

(6) The subtraction cannot be completed without carrying. After some time the pupils may be encouraged to consider the inverse and treat the problem as if it were an addition. This can reinforce awareness of inverse operations. There are 336 different solutions to this problem, although 168 of these simply involve interchanging the bottom two rows. Some solutions include: 783 -659 = 124, 918 -643 = 275, 486 - 157 = 329, 675 - 193 = 483, 819 - 276 = 543, 792 - 134 = 658, 981 - 246 = 735, (182, 218, 273, 318, 327, 546 and 654 are the most common solutions; all can be made in six different ways).

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Mathematics in School, March 2004 The MA web site www.m-a.org.uk 33

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