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This is a cross-post from MSE: https://math.stackexchange.com/questions/4900281/the-50-game-between-two-players-selecting-numbers-between-1-and-10-inclusive

Let's play a game with two players, with player 1 going first. The players take turns selecting a number between 1 and 10 inclusive. The person who says the number that makes the sum reach or exceed 50 wins. Who wins? Let's go backwards in increments of 11 from 50, we have 50, 39, 28, 17, 6. So player 1 selects 6 on the first turn and then always counters player 2 selecting $k$ with $11 - k$, hitting 50 and so winning the game for player 1. Pretty simple, right?

However, here are two variations on the game I came up which I'm wondering about.

  1. What if players can't say a number so that its sum with the last number said is equal to 11? Which player then has the winning strategy?
  2. What if both players can only use the numbers 1, 2, 4? Which player then has the winning strategy?
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1 Answer 1

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Question 1: The status of a game can be encoded as a pair A-B, where B is the sum including the last number said, and A is the sum excluding it.

For B greater than 40, either B+9 or B+10 is an available winning move.
For B exactly 40, B+10 is available unless the game is 39-40, which is losing. Thus, A-39 is winning unless A is 29.
A-38 is thus losing for all A, since either 38-39-40 or 38-B-B+10 will happen.

For B between 28 and 37 inclusive, A-B is winning unless A is 27 (via A-B-38) and B is not 29 (due to 29-38 and 29-39 both being losing for your opponent). A-27 is winning for every A.
A-26 is thus also losing for all A, since 26-27 and 26-28 are both winning.

Similarly to before, A-B is winning for B between 16 and 25 inclusive, unless A is 15. Dissimilarly to before, there are no extra losing positions to break the pattern. A-15 is winning, A-14 is losing.
In fact, this pattern now repeats every 12 numbers, so 2 is a losing position, and thus a winning move for Player 1!

Question 2:

Player 2 can always ensure that the sum is a multiple of 3, until Player 1 says a number greater than 45, at which point Player 2 wins.

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  • $\begingroup$ Players can only name numbers 1 to 10, so did you mean that A and B are the sums after the previous two moves rather what the players say? $\endgroup$ Commented Apr 17 at 7:10
  • $\begingroup$ Yes, my mistake. $\endgroup$ Commented Apr 17 at 11:49

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