seanieg89
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A positive integer is written on a blackboard.
Players A and B take turns subtracting a positive square integer from it such that the result is still positive.
The person who plays the last legal move wins.
There are ~180,000 out of the first 40,000,000 positive integers that result in a win for player B (assuming optimal play). Notice this is a very small proportion!
Prove that there are in fact infinitely many starting integers that are winning for player B (quite easy).
Moreover, investigate this game further yourself! Questions like:
-Does the proportion of winning values for Player B less than N tends to zero as N->inf?
-Are there infinitely many "twin pairs" of winning values for B?
-How many winning values for Player B are there less than a given positive integer N, at least asympotically?
-Can we say anything about the divisibility properties of winning values for B? Eg are even numbers more likely to be wins for one of the players than arbitrary numbers?
-What changes is we replace squares with cubes, k-th powers, primes, prime powers etc. Some choices will make the game easier to analyse, some harder.
are examples that I can think of.
Some of these may be intractably hard, some not so much. In any case see what you can guess/prove yourself about this game. It is a lot more fun to think about open-ended problems like this than doing repetitive drills to shave a few seconds off the time it takes you to integrate things.
Players A and B take turns subtracting a positive square integer from it such that the result is still positive.
The person who plays the last legal move wins.
There are ~180,000 out of the first 40,000,000 positive integers that result in a win for player B (assuming optimal play). Notice this is a very small proportion!
Prove that there are in fact infinitely many starting integers that are winning for player B (quite easy).
Moreover, investigate this game further yourself! Questions like:
-Does the proportion of winning values for Player B less than N tends to zero as N->inf?
-Are there infinitely many "twin pairs" of winning values for B?
-How many winning values for Player B are there less than a given positive integer N, at least asympotically?
-Can we say anything about the divisibility properties of winning values for B? Eg are even numbers more likely to be wins for one of the players than arbitrary numbers?
-What changes is we replace squares with cubes, k-th powers, primes, prime powers etc. Some choices will make the game easier to analyse, some harder.
are examples that I can think of.
Some of these may be intractably hard, some not so much. In any case see what you can guess/prove yourself about this game. It is a lot more fun to think about open-ended problems like this than doing repetitive drills to shave a few seconds off the time it takes you to integrate things.
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