## David Shield's Proofs

David Shield, a retired mathematician from Australia, became engaged in the Sphinx puzzles through the student work published here. He set to work building a formal proof of the number of solutions of various sizes. Part of his work is reproduced here. David has attempted to write his proof in a form that Junior Secondary students could follow.
I have checked, and feel reasonably confident that there are only 16 ways of making a Size 4 sphinx. It is instructive to note that 8 of them can be obtained by combining each of the four ways of making a Size 3 sphinx with each of the 2 ways of making a 3-by-4 parallelogram.
The other 8 ways all start as in this diagram...

My estimate of the number of Size 5 sphinxes has stabilised at 153, but I still wouldn't consider it reliable. For anyone interested in checking, I have classified the solutions according to the positions of the first few small sphinxes at the upper left. For the first and third, only four of the twelve conceivable orientations of a small sphinx are actually possible; I have labelled them:

 (a) (b) (c) (d)

Once the first has been placed, the position of the second is compulsory, so it is of no help in classifying solutions. The 'third' is the next to take up edge space to the left and below. With the Size 4 sphinx diagrams above, the last 8 would all be classified as 1a 3a, numbering as follows:

The first eight include two of 1d 3a,

two of 1d 3b,
four of 1d 3c,
and none of 1d 3d.

Got the system?
My count of solutions for size 5 is:

 1a3a ...0 1b3a ...27 1c3a ...0 1d3a ...0 3b ...0 3b ...10 3b ...12 3b ...0 3c ...2 3c ...20 3c ...48 3c ...0 3d ...0 3d ...0 3d ...0 3d ...34 Total ...153
It is comparatively easy to describe 56 of them:

There are:

• 2 ways of completing the parallelogram in the head;
• 4 ways of completing the size 3 sphinx at lower left;
• 7 ways of completing the remaining shape (6 of them include a 5-by-6 parallelogram).

So 56 ways of completing a size 5 sphinx using these internal borders.
Another fruitful subdivision has the lower left sphinx turned around:

With these internal borders, there are 1 x 4 x 6 = 24 ways of constructing a size 5 sphinx.

Looking at the sequence so far:
S2 = 1
S3 = 4
S4 = 16
S5 = 153 (?)
I feel inclined to speculate that S6 is well over 1000. Two thousand wouldn't surprise me. But I don't feel inclined to check it by hand!

Regards, David.