it's a simple fact that the first player wins for some numbers and the second for others. So the question is not well-specified as it relies on the meaning of a random natural number.
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The question was, "given 'n', show which player (the player going first or second) has a strategy to guarantee a victory".
What's not well specified about that?
E.g. if I'm given 346 it's me, if I'm given 240 it's my opponent.
This is getting complicated, and we need to use general deduction instead of playing with examples!
I think my solution (when leading zeros aren't allowed) works. The rationale being that no palindrome ends in zero, therefore if I give you a number that ends in zero, you can't do the same back to me... so as long as I hand you a multiple of 10 I win (I think).
For example n=903,421 I might subtract 111,111 to make it end in a zero, and I'm not sure how you can avoid that strategy.
There is a hole in that logic. He can't return a number to you ending in a zero, but there could be other winning numbers. Eg you give him 20, he gives you 11 or 13 (loss in 3 as discovered in my analysis).
Eg you give him 20, he gives you 11 or 13 (loss in 3 as discovered in my analysis).
He gives 13 (or 11), then you give him 10, which loses. What am I missing?
This is getting complicated, and we need to use general deduction instead of playing with examples!
I think my solution (when leading zeros aren't allowed) works. The rationale being that no palindrome ends in zero, therefore if I give you a number that ends in zero, you can't do the same back to me... so as long as I hand you a multiple of 10 I win (I think).
For example n=903,421 I might subtract 111,111 to make it end in a zero, and I'm not sure how you can avoid that strategy.
There is a hole in that logic. He can't return a number to you ending in a zero, but there could be other winning numbers. Eg you give him 20, he gives you 11 or 13 (loss in 3 as discovered in my analysis).
He gives you 11, you give him 11-1=10. He gives you 13, you give him 11-3=10. Either way he's stuffed.
If you get a number not ending in 0 you can leave a smaller number ending in 0 and you'll always get back another number not ending in 0 because no palindromes without leading 0s end in 0. So by descent you're going to leave him with 0. But if you get a number that ends in 0 he's going to do the same to you instead.
If you give him 10 he subtracts 9 and leaves you 1. You have to play the last move, which is a win for him according to my understanding of the rules.
The Nim rule is that the player who leaves zero loses. I assumed that was what was meant. i.e. if G(N) is the game where you start with number N, G(0) is won for the first player.
@MEGACHE3SE - please clarify this!
My first idea was that if I chose 12, then the first player can choose 1, giving the second player the win, 2, giving the first player the win, any other number, giving the second player the win. But that any random number under 10 is an immediate win for the first player. Any number which can be reduced to ten by any player is a win for that player.
It was pretty obvious that Mega gave the question wrongly because for instance, the random number ten is a loss for the first player. So there's no consistency and the question implies consistency. Your problem is that you don't know what a random numbr is. You don't get to choose it. Lack of basic IQ.
question never implied that the same player would win each time. reading comprehension issue.
you are deliberaely trying to misinterpret the problem and drawing (still fallaciious) conclusions off of your misreading.
The Nim rule is that the player who leaves zero loses. I assumed that was what was meant. i.e. if G(N) is the game where you start with number N, G(0) is won for the first player.
@MEGACHE3SE - please clarify this!
the player who leaves zero after their subtraction is the winner
I suppose I have to treat you both as really thick. The point is that if it doesn't work for 8, it doesn't work.
the solution works for 8. your "point" is moot. the solution works for any positive integer chosen.
@llama_l, yes, I got both volumes of "Winning ways for your mathematical plays" as presents long ago. Too much for someone not working in the area, to be frank! This book contains a huge amount of development of the theory of combinatorial games which provide examples of surreal numbers. There is a much richer structure than one would guess.
If you give him 10 he subtracts 9 and leaves you 1. You have to play the last move, which is a win for him according to my understanding of the rules.
" until one player reaches 0, of which then that player wins", seems clear enough to me.
@MEGACHE3SE, you haven't answered my clarifying question. It's crucial to the discussion!
[EDIT: oops. I missed it]
@MEGACHE3SE, you haven't answered my clarifying question. It's crucial to the discussion!
i did it just got lost in the discusison. player who leaves zero after their subtraction is the winner.
If you give him 10 he subtracts 9 and leaves you 1. You have to play the last move, which is a win for him according to my understanding of the rules.
" until one player reaches 0, of which then that player wins", seems clear enough to me.
It could mean the position before they move or after they move.
Thanks, @MEGACHE3SE, I must have missed it in the fold.
That invalidates my analysis, which was for a different game. Perhaps we can move on to that, since the answer for your game is clear and quite simple.
If you give him 10 he subtracts 9 and leaves you 1. You have to play the last move, which is a win for him according to my understanding of the rules.
" until one player reaches 0, of which then that player wins", seems clear enough to me.
It could mean the position before they move or after they move.
Dubious. If it's before he moves he can hardly be said to be reaching it.
In chess you would generally say you reach a position where you have the move, but nothing more needs to be said on a minor semantic misunderstanding.
Would anyone like to generalize my analysis of the modified game where the Nim rule applies?
In chess you would generally say you reach a position where you have the move.
You could say a position is reached where you have the move, but you'd generally say White reaches the position where Black has the move. I found it pretty clear anyway.
So P2 wins if n is a multiple of 10, otherwise P1 wins? Thats what I got, maybe I'm wrong though