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# Help A to make a strategy to win the Game?

Topic: Example of writing a position paper
June 17, 2019 / By Utai
Question: A and B, are two infinitely (equally) intelligent people. They play a game. The game is described as: - There are N numbers written on a paper (1 to N). When a player's move comes, he can strike off any one number of his choice which are available on the paper. After striking off a number, he also has to strike off the number's factors. So, striking off 10 would result in striking off 5, 2, and 1. If a factor is already striked off, no need to strike it off again. All striked off numbers are out of the game. The game proceeds with rest of the numbers. The person striking off the the last number wins. If A begins the play and both A and B play their best possible moves, who will win for N = 1000?

## Best Answers: Help A to make a strategy to win the Game?

Roderick | 7 days ago
'A' will win. In fact, 'A' will win for any number N. I can give a nonconstructive proof; that is, I can show that a winning strategy for A must exist, but my proof won't tell you what it is. I am still investigating a description of an actual strategy (although I am not yet sure if a simple one exists). For k > 1, Let S(k) denote the state of the game after the first move if k was crossed off. For example, S(2) means that all numbers but 1 and 2 are left; S(3) means that all numbers but 1 and 3 are left; S(4) means that all numbers but 1, 2, and 4 are left; S(12) means that all numbers but 1, 2, 3, 4, 6, and 12 are left; and so on. Case I. Suppose that there exists some k > 2 such that, if A crosses it off, then he has a winning strategy thereafter. Then A just crosses off this number k, and then has a winning strategy thereafter. Case II. Suppose that no matter what number A crosses of that's bigger than 1, then B has a winning strategy. This means that if B encounters any of the states S(2), S(3), S(4), ..., S(1000), then he has a winning strategy. I'll show that if A crosses off 1, then he can win. Say A crosses off 1. Now B is left in the position of being forced to pick some other number to cross off. What if B crosses off 2? Then just 2 and 1 are gone, so the state of the game is now S(2) with A to play, so A can follow the winning strategy for S(2). What if B crosses off 3? Then just 3 and 1 are gone, so the state of the game is now S(3) with A to play, so A can follow the winning strategy for S(3). What if B crosses off 12? Then 1, 2, 3, 4, 6, and 12 are gone, so the state of the game is now S(12) with A to play, so A can follow the winning strategy for S(12). No matter what number k is crossed off by B, it leaves the game in the state S(k), which means A can then proceed to win. In summary: -If A has a winning strategy which starts with crossing off some number greater than 1, he uses that. -If crossing off ANY number greater than 1 allows B to win, then A crosses off 1, which forces B to return the game to a state in which A can win.
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Mickey
We can "waste moves" by choosing small #'s Find all the prime #'s less than 1000 2 has 500 #'s that have 2 as its factor 3 has 333 5 has 200 and so on... all the way to 31 because 31^2 is the greatest perfect square under 1000 we see that we can cross out 9 #'s that face 2 as a factor with 512(2^9) so we know that for every prime # we can eliminate all its roots with one turn. There are also #'s with two or more different multiples eg 2*3 for example 2^8*3 is unique so is 2^6*3^2. (sorry this part is hard to explain) Find all unique numbers for all different sets of possible numbers. The problem should then be solvable from there. Note: my solution is only a guideline as to how I think the problem can be solved. Check my work and make sure what I am saying is correct before you actually find out all the different possible combinations etc...
👍 90 | 👎 5

Jude
This is the next day. Thinking about the hole I mentioned when I outlined my strategy yesterday, now I believe the hole is quite significant. However, it can be shown the first player wins by force, as The Mathemagician demonstrates a few posts down. It's one of those arguments where you can prove something exists, but you can't say what it is. If the first player has a winning move involving a number larger than 1, he plays it. This move also takes out the number 1. If the first player does not have a winning move involving a number larger than 1, he just takes out the number 1. That passes the situation to the second player, except the second player can't duplicate the first player's strategy because the number 1 is gone. As for what the actual winning strategy is, it must be very complex.
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Originally Answered: How would I make a new video game?
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