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SECTION SUMMARYSOUMENDU SUNDAR MUKHERJEENote:In this note I plan to record summary of each section. I will try to update it weekly. If you havesuggestions, comments or questions, please email me at[email protected].Section 1 (September 6th):We went over a few impartial combinatorial games, includingNimble,Dominoes.Nimble: Here you havenboxes in a row, numbered 0 ton-1 from the left.Each boxi1contains a certain numberxiof coins. In a move a player can select one of the coins from one of theboxes numbered 1, . . . , n-1 and move it to some other box to the left. The player who moves thelast coin wins.Solution sketch: After some hit and miss, we arrived at the following idea — for each coin in thei-th box consider having a Nim pile of sizei. Then moving a coin from boxito boxjcorrespondsto taking (i-j) chips away from the Nim pile corresponding to the coin being moved. Thus, byBouton’s theorem, (x1, . . . , xn-1) is inPif and only ifn-1Xi=1i⊕ · · · ⊕i|{z}ximany= 0.Dominoes: Here we have an 1×nboard, which is to be tiled with 1×2 tiles, also called dominoes.In each move, a player places a domino in some empty place, if any. The player who moves last wins.Solution sketch: We made a preliminary table for smaller values ofn. Based on this we made then12345678typePNNNPNNN

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