Balanced Polyhex Triangles

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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

 

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