TetraCairos
Wei-Hwa Huang and Brendan Owens solved my various problems with tetracairos.  Related -- After being contacted by students of math teacher Odette De Meulemeester from Belgium, Livio Zucca has made a new page about covering areas with polyforms, you can see it here.  These are known as Farm problems, where the polyform is used to make a fence.  If you try to make two identical shapes with a large hole, to problem becomes even harder.  Also, trying to make a symmetrical shape with the largest possible hole is quite hard.

Some Solvers:  Brendan Owen and Wei-Hwa Huang solved my Tetracairo problems.  Wei-Hwa Huang has found at least five solutions for all the positions he's checked.

Gary Mulkey has solved the Cairo problem of Mark Thompson.  I've put his findings here.

material added august 1st

I regularly check the websites of Zucca, Vicher, and Clarke for polyform discoveries.  Livio and Brendan Owens recently made a list of the 17 tetracairos.  It's a rather nice set!

The 17 tetracairos. Larger Image.
Livio called them polycairos after the famous Cairo tessellation of equilateral pentagons.  I tried to solve the lovely figure in black for awhile.  Eventually, I pulled out the awesome tool known as Parity, and noticed it was impossible.  Do you see why?  The four free pentagons can be grouped together in five different ways, in theory.  It was a lovely enough set to play with that I'm having 54 sets cut.  If you'd like to order one, see my orders page.  I'll be selling my first sets while I'm in Los Angeles Aug 10-Aug14 for a puzzle convention.  Here, you can see Livio's list of the 55 pentacairos.

Brendan Owens shows how 16 tetracairos make a diamond.

The person at the forefront of Cairo puzzles is Mark Thompson.  He discovered the tetracairos several years ago.  A small puzzle of his -- Arrange the 5 tricairos into a shape with perimeter 21.  At his website, you can see several puzzles and games based on this tiling pattern.  He recently gave the same set as above to Kate Jones of Kadon Enterprises (the best place to get high quality puzzles).  His Cairo game looks like an excellent puzzle.  Can these 32 tiles be placed to completely match up?  I also found his Venn Polyominoes page intriguing.