Consider the following game. There is a list of distinct numbers. At any round,

Question

Consider the following game. There is a list of distinct numbers. At any round, a player arbitrarily chooses two numbers a,b from the list and generates a new number c by subtracting the smaller number from the larger one. The numbers a and b are put back in the list. If the number c is non-zero and is not yet in the list, c is added to the list. The player is allowed to play as many rounds as the player wants. The score of a player at the end is the size of the final list.

Suppose at the beginning of the game the list contains the following numbers: 47,49,72,127,165 and 173. What is the score of the best player for this game?

Explanation / Answer

Let us start by taking two numbers 47 and 49. After subtracting 47 from 49 we get 2.Now 2 is non-zero and not present in the list so we add 2.

Now we take 2 and 173 we get 171. Now 2 and 191 we get 169 and so on. Hence we get all the odd numbers from 1 to 173.

Now let us take 173 and 1. We get 172. This will give us all the numbers from 1 to 173

Hence the highest score of a player is 173

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