Problem of the Month (July 2005)

This month we consider no-touch tilings. These are reptile tilings in which no two congruent shapes are adjacent. Karl Scherer and Patrick Hamlyn completely classified no-touch tilings of n × n squares into smaller integer-sided squares: they exist for n=16, n=18, and n≥22. We say that these are the no-touch numbers of a square. n=16 (Karl Scherer) n=18 (Patrick Hamlyn) n=22 (Patrick Hamlyn) n=23 (Karl Scherer) n=24 (Patrick Hamlyn) n=25 (Karl Scherer) n=26 (Karl Scherer) n=27 (Karl Scherer) n=28 (Karl Scherer)

This month we ask which other reptiles have no-touch tilings, and for those that do, what are their no-touch numbers?

Can you show that equilateral triangles and cubes do not have no-touch tilings? Can you find a no-touch tiling of a bent triomino? Can you show that that dominoes have no-touch tilings for n≥5? What are the no-touch numbers of other rectangles? Other polyominoes? Polyiamond or polytan trapezoids? Other shapes?

We continue the recent tradition of offering a \$10 prize to solver offering the best solutions.

Joseph DeVincentis proved that tans, equilateral triangles, and cubes do not have no-touch tilings.

Tino Jonker noted that all integer-sided right triangles have non-touch tilings, because the altitude to the hypotenuse divides the triangle into two similar triangles. He was also able to prove that almost all tritans have no-touch tilings.

ShapeKnown No-Touch NumbersUnknown 16, 18, 22+ 5+ 14+ 34, 4428-33, 35-43, 45+ 16, 18, 21+ 16, 18-20, 22+ 15, 16, 18+ 7, 10-4647+ 23+ none 7, 10, 11, 13+ 19, 24, 25, 27-3031+ 17, 18, 2021+ 7, 10, 13-2425+ 1920+

Here are 1×2 rectangle no-touch tilings that do not come from square or smaller tilings: n=5 n=6 n=7 n=8 n=9 n=11 n=13 n=17 n=19

Here are 1×3 rectangle no-touch tilings that do not come from square tilings: n=14 (Patrick Hamlyn) n=15 n=17 (Patrick Hamlyn) n=19 (Patrick Hamlyn) n=20 n=21

Here are 1×4 and 1×5 rectangle no-touch tilings that do not come from square tilings: n=21 (Patrick Hamlyn) n=19 (Patrick Hamlyn) n=20 (Patrick Hamlyn)

Here are 2×3 rectangle no-touch tilings that do not come from square tilings: n=15 (Patrick Hamlyn) n=19 (Patrick Hamlyn) n=20 (Patrick Hamlyn) n=21 (Patrick Hamlyn)

Here are small bent triomino no-touch tilings: n=34 (Patrick Hamlyn) n=44

Here are small tritan no-touch tilings that do not come from smaller tilings: n=7 (Patrick Hamlyn) n=10 (Patrick Hamlyn) n=11 (Patrick Hamlyn) n=12 (Patrick Hamlyn) n=13 (Patrick Hamlyn) n=15 (Patrick Hamlyn) n=16 (Joseph DeVincentis) n=17 (Joseph DeVincentis) n=18 (Patrick Hamlyn) n=19 (Joseph DeVincentis) n=23 (Patrick Hamlyn) n=25 (Patrick Hamlyn) n=27 (Patrick Hamlyn) n=29 (Joseph DeVincentis) n=31 (Patrick Hamlyn) n=37 (Patrick Hamlyn) n=41 (Patrick Hamlyn) n=43 (Patrick Hamlyn)

Here are small triamond no-touch tilings that do not come from smaller tilings: n=7 (Patrick Hamlyn) n=10 n=11 (Patrick Hamlyn) n=13 n=15 n=16 n=17 n=18 n=19 n=22

Here are small straight pentiamond no-touch tilings: n=19 (Patrick Hamlyn) n=24 (Patrick Hamlyn) n=25 (Patrick Hamlyn) n=27 (Patrick Hamlyn) n=28 (Patrick Hamlyn) n=29 (Patrick Hamlyn) n=30 (Patrick Hamlyn)

Here are small octiamond no-touch tilings: n=17 (Patrick Hamlyn) n=18 (Patrick Hamlyn) n=20 (Patrick Hamlyn)

Here are small tridrafter no-touch tilings: n=7 (Patrick Hamlyn) n=10 (Patrick Hamlyn) n=13 (Patrick Hamlyn) n=15 (Patrick Hamlyn) n=16 (Patrick Hamlyn) n=17 (Patrick Hamlyn) n=18 (Patrick Hamlyn) n=19 (Patrick Hamlyn) n=22 (Patrick Hamlyn) n=23 (Patrick Hamlyn) n=24 (Patrick Hamlyn)

Here are small pentadrafter no-touch tilings: n=19 (Patrick Hamlyn)

And the winner of the \$10 prize is Patrick Hamlyn.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 7/30/05.