# Tiling a hexagonal chessboard with “tribones”

A tribone is a tile made of three hexagons in a line.

A hexagonal chessboard is a hexagonal grid of 91 cells in the shape of a larger hexagon.

When 30 tribones are placed on a hexagonal chessboard without overlap, what are the possible locations of the uncovered space?

• are we gonna ignore symmetry? or the answer would be 6x? – Oray Aug 9 '17 at 20:45

Colour in all cells in the top horizontal row in Red, the second in Blue, then Green, Red, Blue and so on.

     R R R R R R
B B B B B B B
G G G G G G G G
R R R R R R R R R
B B B B B B B B B B
G G G G G G G G G G G
R R R R R R R R R R
B B B B B B B B B
G G G G G G G G
R R R R R R R
B B B B B B


Thanks to Mike Earnest for the diagram!

As you place tiles on the board, denote the number of each colour remaining uncovered at any stage as R, G, B.

Each piece must either cover 3 cells of a single colour, or one of each. Either way, the relative numbers of each colour modulo 3 remains unaltered, i.e. (R-G)%3, (G-B)%3 and (B-R)%3 are constant.

Initially there are 6+9+10+7=32 reds (=2 mod 3), and the same number of blue. There are 8+11+8=27 greens (=0 mod 3). So R=B mod 3 and G=R+1 mod 3.

Once there is only one cell left uncovered, that cell must therefore be Green. So it must be on row 3, 6 or 9.

By rotating through 60 degrees in either direction and applying the same logic, the uncovered cell must be one of the seven highlighted in the other answers - either the central cell or 3 rows in from any two adjoining edges.

• Thanks for the image - I can't see the edit history on this device so don't know who to thank :) – IanF1 Aug 11 '17 at 14:39
• That was me, nice job on the solution! – Mike Earnest Aug 12 '17 at 2:06

(rotate for blue dots)

I've reduced it to these hexagons:

Those labelled 2, because each 'tribone' must cover a 1, a 2 and a 3 and there are 30 1s, 30 3s but 31 2s.

• I reduced it to the same ones - I'm trying to find something to rule out the other 2s. – Deusovi Aug 9 '17 at 23:45

There are

7 uncovered place possibilities

as shown in the figure below:

I am working on how to show that these are only possible places by programming or by some logical-deduction and post it soon!