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?
Comments
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Reply DeleteB wins. Pair up the numbers in 1...25, except for 1 and primes more than 11, as below:
Reply Delete(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.
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
Reply DeleteCertainly B wins as pointed out by Mike Earnest, but the pairs of number as response for B is not unique.
DeleteCertain 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?
if n=27, then A has a winning strategy,
DeleteA 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.
Note that B starts. I claim that B wins.
Reply DeleteB chooses 23, A has to choose 1
B chooses 17, A has nothing to choose!
I think that u have to start with a even number
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