A couple of submitted solutions on the theme of isolating small pieces in tilings of polyominoes of mixed sizes:

George Sicherman solved problem #44. (Tile a certain torus with the 1–5-ominoes such that the 1–4-ominoes do not touch each other. Finding a solution where none of the smaller pieces even meet at corners was optional, but well appreciated!) This solution contains four “crossroads”, or points where four polyominoes meet. These are sometimes considered an aesthetic flaw in a polyomino tiling, and whether or not you agree with that, finding solutions without them tends to be good for an extra challenge.

Problem **#44a**: find a solution for problem #44 without crossroads.

Jaap Scherphuis analyzed this challenge that was included with my Agincourt puzzle: find a tiling of an 8×8 square where none of the dominoes and trominoes touch each other or the outer edge. He found that there were 32 solutions. In just one of these, none of the smaller pieces meet at a corner:

I don’t feel that I acknowledge other puzzle creators enough in this space, so I want to give a shout out to Kadon for their Mini-Iamond Ring puzzle, which contains all of the 2–5-iamonds, and includes as a challenge isolating all of the different sizes of pieces:

“Aesthetic flaw”??

Some people are never satisfied!

All right, here is a solution with the tatami property:

http://userpages.monmouth.com/~colonel/tor1AT.png