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 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.
