Is a p position if j t note that m n 1 and n n 2 1 3

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3Problem 8.9 from KPjjjjjjjj012345Here is an example game of Nimble.There are 6 slots, numbered 0 to 5,with 3 coins in slot 1, 1 in slot 3, and 4 in slot 4. One possible move is to take acoin from slot 4 and move it to 0. The question asks which starting position arein P? And my hint was to see the connection to Nim. Well, we pickonecoinand “reduce” it’s position by moving it to the left. And we can move it as farleft as 0. That is like picking a pile, and reducing the number of chips in thepile. Therefore, moving a coin from slot 4 to slot 0 is like taking a pile of size 4and reducing it to size 0.Thus, the game shown above is like a Nim game with eight piles of chips. 3piles have 1 chip each (because the coins on slot 1 can only move 1 space to theleft), one pile has 3 chips, and 4 piles have 4 chips each. The number of coins isthe number of piles, and the position of each is like the number of chips in eachpile. Therefore, we have a P position in Nimble if and only if the Nim sum ofthe number of chips in each pile is 0, or in Nimble-speak, the Nim-sum of thepositions of each of the coins is 0.Is the position shown above in N or in P? If in N, what would be a winningmove?2

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