### Combinatorics + Game Theory Puzzle

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Problem:

Two players A and B play the following game:

Start with the set S of the first 25 natural numbers: S={1,2,…,25}.

Player A first picks an even number x_0 and removes it from S: We have S:=S−{x_0}.

Then they take turns (starting with B) picking a number x_n∈S which is either divisible by x_n-1 or divides x_n-1 and removing it from S.

The player who can not find a number in S which is a multiple or is divisible by the previous number looses.

Which player has the winning strategy and what is it?

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2. B wins. Pair up the numbers in 1...25, except for 1 and primes more than 11, as below:

(2,14) (3,15)
(4,16) (5,25)
(6,12) (7,21)
(8,24) (9,18)
(10,20) (11,22)

When A chooses one of the above numbers, B responds with the other in its pair, which will ensure B always has a number to choose. A will eventually have to pick 1, to which B responds with 23 for the win.

3. A wins. He chooses 16 on his first chance, and on his next he chooses a multiple of 3. After a few moves, B will be forced to choose 1, when A chooses 23

1. Certainly B wins as pointed out by Mike Earnest, but the pairs of number as response for B is not unique.
Certain pairs are uniquely determined. They are:
(22,11); (18,9); (2,14); (10,20); (5,25); (3,15); (21,7)

They have also been pointed out by Mike. But for other the following three sets of possibility are there:

1st : (4,16); (8,24); (6,12) % Same as that pointed by mike
2nd : (8,16); (4,24); (6,12)
3rd : (8,16); (4,12); (6,24)

so say if A chooses 24, then B can respond with 8, 6 or 4.

The substantive question is if n is any natural number, is there always a wining strategy for B?

2. if n=27, then A has a winning strategy,

A will choose 18. then the remaining numbers are paired thus:

(2,14) (3,15)
(4,16) (5,25)
(6,12) (7,21)
(8,24) (9,27)
(10,20)(11,22)
(13,26)

unpaired nos. are {1,17,19,23}

A will follow the strategy of completing pair. ultimately B will choose 1, then A will choose 23.

4. Note that B starts. I claim that B wins.
B chooses 23, A has to choose 1
B chooses 17, A has nothing to choose!

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