Using your Head is Permitted

June 2014 riddle

A "domino" is a rectangular shape whose long dimension is twice the length of its short dimension. It can be created by gluing together two squares at one of their sides. With only two squares, no other shape can be created in this way.

With three squares, however, one already has two possibilities: the straight-line triomino and the corner triomino (a.k.a. "the corner").

Consider tiling a 64x64 square by corner triominoes (composed of three 1x1 squares each). Clearly, 64x64 does not divide by three, so it is impossible to tile the entire 64x64 square. One square must be left untiled. But which one?

Specifically: consider all possible tilings where corner triominoes cover all the 64x64 square except one 1x1 square. For which 1x1 squares are there such tilings where they are the only untiled square? For which squares are there no such tilings? Give a complete characterization and prove your answer.

List of solvers:

Yuping Luo (1 June 06:58)
Radu-Alexandru Todor (1 June 08:28)
Lian Wang (1 June 17:21)
Zhou Lili (1 June 22:14)
Joseph DeVincentis (2 June 00:45)
Guangda Huzhang (2 June 01:06)
Kan Shen (2 June 01:51)
Lv Jianhao (2 June 05:11)
Oded Margalit (3 June 08:05)
Yan Wang (3 June 11:14)
Dorian Nogneng (4 June 11:15)
Dharmadeep Muppalla (4 June 15:45)
Mukund R (4 June 18:43)
Harald Bögeholz (5 June 01:13)
Lorenz Reichel (5 June 05:39)
Daniel Bitin (5 June 22:33)
Naftali Peles (6 June 03:10)
Hakan Summakoǧlu (8 June 21:24)
Andreas Stiller (10 June 07:43)
Taketsuna Hisaji (14 June 06:29)
Jin Fan (15 June 17:57)
David Tenne (16 June 06:30)
Phil Muhm (18 June 01:42)
Zhao Ziwen (22 June 21:41)
Zhuo Wang (29 June 13:33)
Xiao Liu (30 June 08:01)
Itsik Horovitz (30 June 23:22)

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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