Non-Touch Tilings

Some shapes can be tiled with smaller similar copies with the property that two congruent tiles do not touch. The smallest ways of doing this are shown for some shapes:

Square

16.

Found by Karl Scherer, 2001
Proved unique by Patrick Hamlyn, 2002

18.

Proved unique by Patrick Hamlyn, 2002

23.

Found by Karl Scherer, 2001
Proved unique by Patrick Hamlyn, 2002

24.

Proved unique by Patrick Hamlyn, 2002

26.

There is one other solution, by flipping over the left rectangle
Found by Karl Scherer, 2001
Proved unique by Patrick Hamlyn, 2002

27.

First one found by Karl Scherer, 2001, Second one found by Patrick Hamlyn, 2002

28.

First one found by Karl Scherer, 2001, Second one found by Patrick Hamlyn, 2002

29.

Found by Patrick Hamlyn, 2002

Triamond

There are 5 solutions with these tiles
Found by Patrick Hamlyn, 2002

Straight Pentiamond

19.

Found by Patrick Hamlyn, 2002

24.

Found by Patrick Hamlyn, 2002

25.

There are 18 solutions, 10 like the top and 8 like the bottom
Found by Patrick Hamlyn, 2002

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