The figure below show the 14 pieces that can be made composing four triangles. The six pieces on the top row are symmetric, ie, coincide with teir mirror images. Those on the bottom line are asymmetric.

Trimini pieces

Overall the 14 pieces contain 56 triangles, therefore they could nicely fit on the 4x7 rectangular board:

4x7 rectangular board

Unfortunately, this is not so ! Can you prove that it is not possible to arrange the 14 triminis on the 4x7 board?

Give this problem a try before continuing reading. You may cut the trimini pieces out of a piece of cardboard and play with them.

Hint: a non-solution

The 4x7 board has 14 black squares and 14 white squares. If we can show that
it is impossible to color the triangles of the pieces in such a way that half of
them are black and half are white, then we have a proof that the 4x7 covering is not possible.

The pieces A, B, C, D, E, F, G, I, J, L, M, and N always cover two black triangles and two white triangles. There remain the odd pieces H and K which cover three triangles of one color and
one of the other.

If the odd pieces (H and K) were in odd number we would be done. Unfortunately they are in even number.
So this approach does not lead us to a proof of impossibility of the cover.

A solution

The previous idea can be applied to another coloring of the board, based on triangles instead of squares, as shown in the figure below.

triangular coloring

In this coloring the pieces A, B, C, E, G, H, I, L always have one unit square black and one unit white.
The pieces, F, J, K, M, and N, have 3/4 units of one color and 5/4 of the other.
Finally the piece D can be either one unit black and one white, or 3/2 one color
and 1/2 the other.

Counting the color units we have that one of the two equations must have an integer solution for the covering to be possible:

  8 * 1 + x * 0.75 + (1-x) * 1.25 + 1 = 8 * 1 + (1-x) * 0.75 + x * 1.25 + 1
  8 * 1 + x * 0.75 + (1-x) * 1.25 + 1.5 = 8 * 1 + (1-x) * 0.75 + x * 1.25 + 0.5

The first equation has solution x = 1/2.
The second equation has solution x = 3/2.

This proves that the 4x7 rectagle covering is not possible.

Marco Corvi - 2014