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        riddles >> hard >> take the last chip
        
(Message started by: inexorable on Aug 29^th, 2007, 12:35am)

Title: take the last chip
Post by inexorable on Aug 29^th, 2007, 12:35am

I am not sure if this has been discussed.

Turkey Sandwich was worried about an upcoming test in Discrete Mathematics and was finding it hard to get to sleep. Turkey awoke early in the morning, aroused by devilish laughter, only to see an impish looking homunculus sitting at the bottom of the bed next to a seemingly infinite pile of chips. Hello Turkey it said, would you like to play a little game? This pile contains 43546758343209876 chips and the bottom chip represents your immortal soul. The rules are quite simple. The first player takes some chips, but not all of them. After that we take it in turns to take some chips.

The only rule now is that a player cannot take more in their turn than the previous player took. The winner is the player who takes the last chip. If I win I get to keep your soul and if you win, you get an A in the test. Would you like to go first or second? This seemed a reasonable bet to Turkey. Can you give Turkey a strategy for playing no matter how many chips there are?
What if a player can take up to twice the number of chips that the previous player took. What is the best strategy now?

Title: Re: take the last chip
Post by towr on Aug 29^th, 2007, 1:24am

Searching on Turkey Sandwich, gives two threads on the exact same problem; even phrased identically.

http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_hard;action=display;num=1040828338;start=0#0
http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_hard;action=display;num=1048651023;start=0#0

The extension to taking twice as many is also mentioned further up in the first thread.

Title: Re: take the last chip
Post by mikedagr8 on Aug 29^th, 2007, 1:26am

First question. No strategy for this, I know I should have one, but it seems as though going second is the correct position. It is similar to the round table putting coins on it problem, but I don't know where that is :( .

Title: Re: take the last chip
Post by towr on Aug 29^th, 2007, 1:36am

Hint: work backwards.

Title: Re: take the last chip
Post by mikedagr8 on Aug 29^th, 2007, 1:57am

Just read the threads now.

P.S. Was that aimed at me?

Title: Re: take the last chip
Post by towr on Aug 29^th, 2007, 5:53am

It was a general hint for anyone wanting to solve it themselves rather than look up the discussion in the other threads first.

Title: Re: take the last chip
Post by srn347 on Aug 29^th, 2007, 9:17am

That big number is a multiple of 4, but not 8. Take 4(meaning I go first). If he takes 3, I take 1 and we each alternate with me taking the multiples of 2. If he takes 2, I take 2 with me getting the multiples of 4. If he takes 1, I take 1 and get the multiples of 2. If he takes 4, I take 4 and get the multiples of 4 that aren't multiples of 8. Either way I win.

Title: Re: take the last chip
Post by inexorable on Apr 19^th, 2008, 5:14am

For the extension of taking atmost twice the number of coins, all fibonacci numbers would be losing positions for first player.

There's a nice generalised solution with proofs to this problem here.
http://www.cs.cmu.edu/puzzle/puzzle3_answer.pdf