# Nimble game and NIm game

Exercise: Rules of Nimble game: Nimble game is played on a game board consisting of a line of squares labelled: 0,1,2,3,.... A finite number of coins is placed on the squares with possibly more than one coin on a single square. A move consists in taking one of the coins and moving it to any square on the left, possibly moving over some of the coins and possibly onto a square already containing one or more coins. The player alternate moves and the game ends when all coins are on the square labelled 0. The last player to move wins.

Now i have to find winning move in this position with 6 coins: there is 1 coin on the squares 4,9,10,13 and 2 coins on the square 8.

i solve it as normal nim game (with heaps: 4,8,8,9,10,13) i get another solution that answers in ferguson (there is written for example to take coin from square 9 to square 0) and i dont understand why :( And there is also an error because there is also written to take coin from square 14 but there is no coins) or to take coin from square 10 to square 3. Can you explain why?

• As a rule of thumb, if you google a term or concept you use in a question and you don't immediately find an openly accessible page that would explain it to people who don't know it, it would usually be a good idea to define it in the question. (In the present case, this applies to the game of "Nimble".) Mar 25, 2020 at 12:23
• @joriki Since "Ferguson" was mentioned, I guessed it was in Part I of Tom Ferguson's Game Theory text at math.ucla.edu/~tom/Game_Theory/Contents.html , and it is. But without the text being searchable I had to skim to find it's defined in exercise 3 on page 12 (in 2.6). I absolutely agree that a full explanation and source should be edited in the question so people don't have to go digging/in case links change. Mar 25, 2020 at 12:43
• I;m sorry :D I edit my question. So anyone know what is the answer? Mar 25, 2020 at 12:59
• Please follow @MarkS.' suggestion and provide an explanation of the game. Links often go dead at some point, and then the value to the site of any answer to your question would be lost. Mar 25, 2020 at 13:09
• Your two links both go to solutions so the problem is not given. Please put it and the rulels of Nimble into your question. You also change $4$ to $3$ in the list of positions in your post. Is that a typo? Mar 25, 2020 at 13:54

If the coin on square $$13$$ were on square $$14$$ instead, then the answers given by Ferguson would be the correct winning moves, so it seems likely that there's simply a misprint in the diagram.
With coins as they actually do appear in the diagram, the winning moves for the first player are to move $$13$$ to $$7$$, $$10$$ to $$0$$, or $$9$$ to $$3$$ (and, as the OP points out below, $$8$$ to $$2$$, which I missed).