14 March 2014

# Today’s puzzle

Two students of maths are playing the following game – each takes turn to pick a number from the following set {-4, -3, -2, -1, 0, +1, +2, +3, +4}. Once a number is picked, it is not replaced. And they keep playing till they run out of numbers. And they start all over again.

Whoever can come up with three numbers that add up to 0 first wins the game. It does not have to be the only three numbers she has. For example, she might have picked -3 and then in next chance picked +2. Hoping to pick +1 next. But other player seeing that, would  have taken away the +1 in the next move. So let’s say the original person next picked 4. And then if she got a chance to pick -1 in the next move, she would have -3, +2, +4 and -1. Note that -3, +4 and -1 adds to 0. So she wins.

Now, here is the question: If you are the first player, is there a strategy for you to win? If you are the second player, is there a strategy for you to win?

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Posted March 14, 2014 by Rajib Roy in category "Puzzles

1. By Rajib Roy on

The trick is to realize that what I have given you is a tic tac toe problem. Or one that can be reduced to a tic tac toe problem.
You will quickly note that there are eight combinations of three numbers that gets you 0. Regardless of what else you have, you will need one set of the following:
+1 0 -1
+2 0 -2
+3 0 -3
+4 0 -4
+3 -1 -2
-3 +1 +2
+4 -3 -1
-4 +3 +1

You can arrange the 9 numbers in a 3×3 way such that each row, each column and the two diagonals are one of the above sets. An example: (other ways are possible)

+3 -2 -1
-4 0 +4
+1 +2 -3

So, now, what it means is that you are playing tic tac toe (picking a number means you are putting a X or O there) and you need a straight line to win just like in tic tac toe.

As you know from years of playing tic tac toe, regardless of whether you are the first person or the second person, you can always block the other person from winning (force a draw).

Thus, the answer is, there is no possible strategy that can force a win….

2. By Rajib Roy on

Debatri, your post reminded me. I forgot to mention the people who did crack the puzzle – Chris Kramer, Manojaba Banerjee and Dilesh Bansal. Hope I am not missing anybody….

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