As it turns out, the toy has proved far more intriguing to me (and other adult visitors) than it has to my children! It’s flexibility allows it to mutate between an equilateral triangular shape and a 3D framework (where its six component pieces form half of the 12 edges of a cube).

Purely for the intellectual and creative challenge, I set out to produce an animation of the metamorphosis from equilateral triangle to cube-frame. My eventual success surprised even me. I intend to make some further refinements but the current result is shown below. [Do not adjust your speakers. I have yet to attempt to add any kind of soundtrack to the animation as yet!]

These ideas were then used to create more realistic images in POVray (a FREE ray-tracing program that allows complex 3D forms to be created by amalgamating mathematical objects in various ways ). Whilst this particular animation is mathematically complex, it is incredibly simple to produce some impressive objects in POVray and animate them in all manner of ways. The interested reader is welcome to inquire further but is advised that POVray is an enjoyable but time-consuming hobby!

Multiple freeze-frame images from POVray were then compiled using Windows Movie Maker (which I presume is part of the Microsoft Windows package as I certainly didn’t purchase it specifically).

The mathematically-minded reader is invited to solve the following problem:

If the length of each of the six components is considered to be 1 unit (and for simplicity we are measuring each component from the centre of one hinge to the centre of the hinge at the other end):

1. What is the distance from the centre of the red-blue hinge to the centre of the green-yellow hinge when the model is in the equilateral triangle formation?
2. What is the distance from the centre of the red-blue hinge to the centre of the green-yellow hinge when the model is in the cube-frame formation?
3. What happens to this distance as the model is flexed?

I have yet to answer the third question, but the answers to the first two questions have led me to an intriguing (but seemingly improbable) conjecture that I’ve yet to prove or disprove. I would be very pleased and impressed if anybody can confirm my results for questions 1 and 2, guess at my conjecture and help me out in any way with question 3!

You ask for two random numbers in the nineties. (You needn’t let on in advance what you intend to do with the numbers). Perhaps, 93 and 98 are given. Almost immediately, you could write:

93 × 98 = 9114

You have no difficulty writing the answer from left-to-right rather than right-to-left. In other words, you are not mentally reproducing some pencil and paper method of multiplication that requires the carrying of digits.

With practice, this trick could be performed orally. The obvious advantage of performing this trick orally is that (if performed well) it will clearly show you to be quicker. A second advantage is that, if you perform the trick more than once, your audience will not still be able to see several examples from which to discern the trick for themselves and steal your thunder. Of course, if you are performing this in front of a mathematics class, you may well wish for a student to learn the trick in which case writing the sum on the board will be helpful.

The intelligent reader might wish to work out how this trick can be performed for him/herself – perhaps by considering several examples or else by applying an algebraic approach to gaining understanding. If so, read no further (at least for now).

How to perform the trick

When you are given the numbers (or when you write them down), try not to think of the numbers themselves. Instead, immediately think how much less than 100 each number is.

So, if the numbers 91 and 96 are to be multiplied, you should be thinking of 9 (100 – 9 = 91) and 4 (100 – 4 = 96). You should be able to think this when writing down “91 × 96 =”. With these numbers you must perform two small feats:

1. Add them (9 + 4 = 13) and subtract from 100 (100 – 13 = 87). If you are writing the trick, write the first two digits of the answer: “87″.

2. Multiply them (9 × 4 = 36). If you are writing the trick write the last two digits: “36″.

91 × 96 = 8736

Why the trick works

It is possible to be convinced that the trick works by seeing many examples, but understanding tricks such as this one provides a useful opportunity to utilise algebra.

If the two numbers given are less than 100 by a and b respectively then we are trying to perform:

(100 – a) × (100 – b)

Expanding these brackets using “F.O.I.L.” or an equivalent process gives:

(100 – a) × (100 – b) = 100 × 100 – 100 × b – 100 × a + a × b

= 10000 – 100b – 100a + ab

=10000 – 100(a+b) + ab

=100(100 – (a+b)) + ab

With a little thought, it should be obvious that 100 – (a + b) forms the first two digits of the answer and that ab gives the last two.

Provided we are working with numbers in the nineties, a and b will be less than 10. This ensures that ab is less than 100.

We could apply this technique with other numbers fairly close to 100, but some carrying might be necessary and slow the performance.

The interested reader could confirm, for instance, that:

• products with one number very close to 100 and a second number not very close such as 97 × 67 could be performed and no carrying would be necessary.

• products such as 91 × 89 could also be performed without the need for carrying.

• products such as 92 × 84, 87 × 89, etc. could also be performed, but care would need to be taken as carrying would be required.