Math Olympiad Question

Probability Level 3

The following question is from a Math Olympiad competition.

You and Adam were playing a game. There are a pile of stones. On every turn, each player may take 1, 2, or 4 stones from the pile. The person who takes the last stone loses.

If Adam will be playing first, which of the following options for the number of stones in the pile will guarantee you a win?

15 12 11 13 14

Winston Choo by Winston Choo , 46, Singapore

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2 solutions

Jesse Li
Nov 11, 2018

If Adam takes 1 stone on the first turn, you can take 2. If he takes 2 on the first turn, you can take 1. If he takes 4 on the first turn, you can take 2. You can repeat this pattern for every move. The amount of stones taken after every round (that is, after Adam goes and you go) will be a multiple of 3. Eventually, 12 stones will be taken and Adam has to take the thirteenth one.

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Jeremy Galvagni
Nov 11, 2018
1 L
2 W
3 W
4 L
5 W
6 W
7 L
8 W
9 W
10 L
11 W
12 W
13 L
14 W
15 W

The table shows W=winning position for the player whose turn it is. These positions can be changed to L=losing position by a fair turn. The counter moves can be seen on the table.

The losing positions can only be changed to winning positions.

Since 13 \boxed{\text{13}} is the losing position among the choices, Adam will lose and I will win.

1 Helpful 0 Interesting 2 Brilliant 0 Confused

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