July 1st, 2017, 03:40 AM  #1 
Member Joined: May 2017 From: France Posts: 57 Thanks: 1  The game of pebbles
Hi, We consider the game at 2, which consists of a set of N stones, from which one can remove a pebble, or c> 2 pebble, the winner being the one who does not take the last stone. Is there a winning strategy? Cordially. 
July 1st, 2017, 10:47 AM  #2 
Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 872 Thanks: 60 Math Focus: सामान्य गणित 
What are the other rules of the game?

July 1st, 2017, 02:15 PM  #3 
Member Joined: May 2017 From: France Posts: 57 Thanks: 1 
The game of pebbles is a twoplayer game. The game starts with N stones, and a number c>2 is fixed. On a player's turn, he or she must remove either 1 or c stones. The player who does not take the last stone is the winner. Is there a winning strategy? 
July 1st, 2017, 02:32 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,379 Thanks: 542 
If you take c stones you lose, so the only strategy is to take 1 and hope.

July 1st, 2017, 03:10 PM  #5 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
Suppose N = 6 ,c = 3 The player who goes first cannot lose If he takes 3 the other player losses If he takes 1 and the other player takes 1 then he takes 3 , first player wins again If he takes 1 and the other player takes 3 then he takes 1 , first player wins again Suppose N = 5 , c = 3 The player who goes first cannot win If he takes 1 the other player takes 3 If he takes 3 the other player takes 1 Looks like it depends on the choices of N and c so there is no winning strategy for all N and all c Last edited by agentredlum; July 1st, 2017 at 03:22 PM. 
July 1st, 2017, 06:09 PM  #6 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 10,914 Thanks: 716 
This would be a better way to illustrate: 13 pebbles; players take 1,2 or 3 last to pick loses : so winner's strategy is to leave 5 pebbles whoever goes first picks 2: then sure to win 
July 2nd, 2017, 02:19 AM  #7 
Member Joined: May 2017 From: France Posts: 57 Thanks: 1 
There are a loosing position, the solution here : But another difficult, the case, when the choice is 1,2 or 4 stones. 

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