Consider a standard $8\times 8$ chessboard where a pawn is placed on each of the squares $d1,d2,d3,d4$ . Dissect the board into $4$ congruent pieces (reflections are allowed) such that each piece contains exactly one pawn.
Dissection of a chess board into 4 congruent pieces
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combinatorics
geometry
recreational-mathematics
puzzle
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2What are your thoughts? – 2012-12-22
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1Do the pieces have to be connected? – 2012-12-22
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0@MarkBennet: The problem sheet on which I found this problem is ambiguous but since the original formulation of the problem talks of a field which is supposed to be partitioned among 4 brothers I guess one should assume that the parts should be connected. – 2012-12-22
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0The question is interesting because there are forms of transposition cypher which operate by rotating an 8x8 grid with 16 "windows" in it through quarter-turns, writing letters of the message in sequence each time. There are 16 places in each quarter - and each is allocated to one of the four rotations, and that determines the whole pattern - and this replicates the not-necessarily-connected case. In fact it is potentially undesirable to have connected squares, because there will always be an orientation in which two of the letters in the original message stay together. – 2012-12-22
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0There are still sixteen distinct types of square (look at a quarter of the board divided horizontally and vertically). Assume each congruent piece needs to have one of each kind. Then you can attach one square to the central pawn. The second pawn down needs to be connected to one of the central four squares, and there is only one viable option. Then work on the third pawn, extending the existing configurations for the other two by symmetry. Then there is one option for the bottom pawn, and the pattern becomes clear. – 2012-12-22
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0@MarkBennet: I have problems to undestand your hint. Could you be a bit more explicit in your description? – 2012-12-22
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1Print out the diagram and draw the lines ... Top pawn has to be joined to the square to the left. Square to right of top pawn needs to be joined down then left ... other central squares follow by symmetry. – 2012-12-22
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4Try spirals that grow from each corner simultaneously. – 2012-12-23
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0@Greg: Care to post that as an answer? – 2013-01-02
1 Answers
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I think growing spirals from each corner simultaneously does the trick.
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3I edited your image to make it easier to see the pieces, you can use it in your answer if you like: http://i.stack.imgur.com/DUrtX.png – 2013-01-03
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0hooray, I was hoping someone would improve the picture - and you did even better than I'd hoped! Thank you – 2013-01-03
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2I think this problem goes back to Dudeney's Amusements in Mathematics. – 2013-01-04