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Balanced Polyhex Triangles

The challenge is to tile a solid triangle using equal numbers of two or more polyhexes.

 

Balanced Trihex Triangles

A + B	A + C	B + C

A + B + C

 

Balanced Tetrahex Triangles

 

These puzzles cannot have a solution unless the set contains either the 1 (I) or 6 (Q) piece, without which it is impossible to complete a single side of any triangle. Thanks to George Sicherman who sent several improved and new solutions.

 

Two Pieces
1 + 2	1 + 3 (GS)	1 + 4	1 + 5	1 + 6	1 + 7 (GS)
		?

2 + 6	3 + 6	4 + 6 (GS)	5 + 6	6 + 7

 

Three Pieces (all other combinations are impossible)
1 + 2 + 3	1 + 2 + 4	1 + 2 + 5	1 + 2 + 6	1 + 2 + 7
1 + 3 + 4	1 + 3 + 5	1 + 3 + 6	1 + 3 + 7	1 + 4 + 5
1 + 4 + 6	1 + 4 + 7	1 + 5 + 6	1 + 5 + 7	1 + 6 + 7
2 + 3 + 6	2 + 4 + 6	2 + 5 + 6	2 + 6 + 7	3 + 4 + 6
3 + 5 + 6	3 + 6 + 7	4 + 5 + 6	4 + 6 + 7	5 + 6 + 7

Four Pieces (all other combinations are impossible)
1 + 2 +  3 + 4	1 + 2 +  3 + 5	1 + 2 +  3 + 6
1 + 2 +  3 + 7	1 + 2 +  4 + 5	1 + 2 +  4 + 6
1 + 2 +  4 + 7	1 + 2 +  5 + 6	1 + 2 +  5 + 7
1 + 2 +  6 + 7	1 + 3 +  4 + 5	1 + 3 +  4 + 6
1 + 3 +  4 + 7	1 + 3 +  5 + 6	1 + 3 +  5 + 7
?
1 + 3 +  6 + 7	1 + 4 +  5 + 6	1 + 4 +  5 + 7
1 + 4 +  6 + 7	1 + 5 +  6 + 7	2 + 3 +  4 + 6
2 + 3 +  5 + 6	2 + 3 +  6 + 7	2 + 4 +  5 + 6
2 + 4 +  6 + 7	2 + 5 +  6 + 7	3 + 4 +  5 + 6
3 + 4 +  6 + 7	3 + 5 +  6 + 7	4 + 5 +  6 + 7

Five Pieces (2+3+4+5+7 is impossible)
1 + 2 + 3 + 4 + 5	1 + 2 + 3 + 4 + 6	1 + 2 + 3 + 4 + 7	1 + 2 + 3 + 5 + 6	1 + 2 + 3 + 5 + 7
1 + 2 + 3 + 6 + 7	1 + 2 + 4 + 5 + 6	1 + 2 + 4 + 5 + 7	1 + 2 + 4 + 6 + 7	1 + 2 + 5 + 6 + 7
1 + 3 + 4 + 5 + 6	1 + 3 + 4 + 5 + 7	1 + 3 + 4 + 6 + 7	1 + 3 + 5 + 6 + 7	1 + 4 + 5 + 6 + 7
2 + 3 + 4 + 5 + 6	2 + 3 + 4 + 6 + 7	2 + 3 + 5 + 6 + 7	2 + 4 + 5 + 6 + 7	3 + 4 + 5 + 6 + 7

Six Pieces
1 + 2 + 3 + 4 + 5 + 6	1 + 2 + 3 + 4 + 5 + 7	1 + 2 + 3 + 4 + 6 + 7	1 + 2 + 3 + 5 + 6 + 7
1 + 2 + 4 + 5 + 6 + 7	1 + 3 + 4 + 5 + 6 + 7	2 + 3 + 4 + 5 + 6 + 7

Seven Pieces

1 + 2 + 3 + 4 + 5 + 6 + 7

Tri- and Tetrahex Triangles

Tetrahex + One Trihex
1 + A	2 + A	3 + A	4 + A	5 + A (GS)	6 + A	7 + A
			X
1 + B	2 + B	3 + B	4 + B	5 + B	6 + B	7 + B
	X	X	X	X		X
1 + C	2 + C	3 + C (GS)	4 + C	5 + C	6 + C	7 + C
			X

 

 

Tetrahex + Multiple Trihexes
1 + A + B	1 + A + C	1 + B + C	1 + A + B + C
2 + A + B	2 + A + C (GS)	2 + B + C	2 + A + B + C
3 + A + B	3 + A + C (GS)	3 + B + C	3 + A + B + C
4 + A + B	4 + A + C	4 + B + C	4 + A + B + C
5 + A + B	5 + A + C (GS)	5 + B + C	5 + A + B + C
6 + A + B	6 + A + C	6 + B + C	6 + A + B + C
7 + A + B	7 + A + C	7 + B + C	7 + A + B + C