Exploring the Iceberg: Sizes 5, 6, 7

At this point a mathematician might ask:
What happens if we look for other size Sphinxes?

After all, the original growth pattern of Sizes 1, 2, 4 begged the question of Size 3. Since the next one in the original sequence is Size 8, it is reasonable to seek Sizes 5, 6, & 7.

Louise Anderton's Solutions

Having conquered the Size 3 Sphinx, Louise Anderton (from Gnosall, England, and 14 at the time) accepted the challenge of constructing the Size 5 (if it existed). Her solution (which she wants me to tell you was arrived at with only a little help from her mum, ZoŽ) can be found when you return to the Iceberg of the Sphinx page. Here you will also find that, rather than tackle her homework during the Easter break, Louise continued her constructions to solve both the Size 6 and Size 7 Sphinxes.

We encourage you to search for them yourself
before looking at Louise's solutions below.

Louise Anderton

Green Line
Sphinx 5 - Louise Anderton
Louise Anderton's Size 5 Solution

Sphinx 6 - Louise Anderton
Louise Anderton's Size 6 Solution

Sphinx 7 - Louise Anderton
Louise Anderton's Size 7 Solution

After Louise made inroads into this problem, the students at Thorne Grammar, near Doncaster, England also became hooked.

Hypotheses

From England

Andy Martin was Head of Department at Thorne Grammar. Andy was always looking for ways to involve more students in problem solving. Noting the interest in Sphinx that was developing across the school he engaged the Design & Technology Department to mass produce Sphinx shapes. The result was over 1000 pieces made from waste plastic cut to fit 20mm isometric paper.

This extract from a fax to the Task Centre gives some idea of the students' enthusiasm.

I enclose work from Sarah Hutchinson who found 4 Size 3 Sphinx solutions. She has conjectured 9 Size 5, 16 Size 7 to relate the sequences of square numbers and primes.
Andy also sent 3 solutions for Size 5 and 3 for Size 9, from Steven, Catherine and Richard in Year 8, and copies of a Size 5 solution from Sarah Steadman & Catherine Finnegan and a Size 7 solution from Steven Tootel.

More than a year later we received another set of solutions from Andy. In the interim, Year 7 & 8 classes had been working on Sarah's hypothesis of 9 Size 5 Sphinxes. They found 10 and Andy was able to illustrate the mathematical process of disproof by counter example.

With this remarkable work came:

  • 10 different solutions for the Size 7 Sphinx AND
  • an attempt at formal proof by a Year 9 boy, Paul Elliott, that argued at least 92 solutions to the Size 7 Sphinx and suggested a direction to find even more!
An example of one Size 7 is shown in this photograph...

Size 7 Sphinx

...and Paul's proof can be found in the Sphinx Album.

From Sweden

A second hypothesis came from Johan ÷berg, a university student, who conjectured that the number of solutions for Size N sphinx is:
(N -1)^2 ... ie: (N-1) squared.