Using your Head is Permitted

March 2014 riddle

Consider the problem of tiling an n-by-n chessboard by polyomino pieces that are k-by-1 in size (copies of the straight-line polyomino of size k). Every one of the k pieces of each polyomino tile must align exactly with one of the chessboard squares.

Clearly, it is not always possible to tile the entire board. For example, n2 may not even be a multiple of k. However, there is always some arrangement such that the least number of chessboard squares are left un-tiled. Let m=m(n,k) be this number.

This month's question: prove that for any choice of n and k, the value of m is always a perfect square.

List of solvers:

Radu-Alexandru Todor (2 March 00:15)
Lorenz Reichel (2 March 01:09)
Lian Wang (2 March 16:11)
Jan Fricke (4 March 00:02)
Dan Dima (4 March 03:45)
Joseph DeVincentis (6 March 02:11)
Dharmadeep Muppalla (16 March 03:49)
Guangda Huzhang (19 March 00:54)
Thomas Mack (21 March 02:19)
Daniel Bitin (22 March 00:35)
Itsik Horovitz (31 March 00:25)

Elegant and original solutions can be submitted to the puzzlemaster at Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.

The solution will be published at the end of the month.


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