TAOCP 7.2.2.1 Exercise 313

Section 7.2.2.1: Dancing Links

Exercise 313. ▶ [29] [29] (Hexiamonds of hexiamonds.) A "whirl," shown here, is an interesting dodeciamond that tiles the plane in a remarkably beautiful way.

If each triangle '$\triangle$' of a hexiamond is replaced by a whirl, in how many ways can the resulting 72-triangle shape be packed with the full set of hexiamonds? (Exercise 289 discusses the analogous problem for pentiamonds.)

Consider also using "flipped whirls," the left-right reflections of each whirl.

Verified: no
Solve time: 4m28s

I cannot give a corrected numerical solution to this exercise without performing the actual enumeration. The previous response was wrong in treating the missing figure as a reason to stop, because the figure is part of the exercise data. However, a valid solution must still contain the output of the exact-cover enumeration, and I do not have a verified count for this particular $72$-triangle whirl instance.

A correct solution would need to proceed as follows:

1. Encode the triangular lattice cells of the whirl dodeciamond from the figure.
2. Form the inflated region. If the original hexiamond has triangle set

$$ H={t_1,\ldots,t_6}, $$

replace each $t_i$ by a translated copy of the $12$-triangle whirl $W$. The resulting region is

$$ R=\bigcup_{i=1}^{6}(W+t_i), $$

with

$$ |R|=6\cdot12=72 $$

unit triangles.

1. Generate all legal placements of the twelve hexiamonds in $R$. A placement is determined by a hexiamond type together with a lattice translation and one of its allowed orientations. The exact-cover matrix has:

• $72$ primary colum