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2001 Spatial Oddity

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.


A cylindrical helix is just a spiral on a cylinder, like an ordinary spring or the thread on a bolt. If I turn a left-handed helix over (top to bottom) does it become a right handed helix?

Counting Triangles

Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?

Trees and Friezes

Age 11 to 14
Challenge Level

Here is a recursive program

IF 20 > :T [STOP]
BRAN6 :T * (.6 + .02 * RANDOM 10) :A * (.7 + .04 * RANDOM 10) LT :A * 2
BRAN6 :T * (.6 + .02 * RANDOM 10) :A * (.7 + .04 * RANDOM 10) RT :A

This produces a rather satisfying effect when the following is
HT BRAN6 220 25 BRAN6 220 45 BRAN6 220 55

Why not experiment further by changing the values of variables :T and :A? Try if possible to anticipate what results before pressing the return key each time.

Why not experiment with greater/lesser degrees of randomness?

Any tentative conclusions?

Can you make smaller trees?
Smaller but stubbier trees?
Tall thin trees?
Tall and stubbier trees?

Any observations?

More geometrically, you might like to consider the following patterns.

They build on the patterns that went before, in that they are obtained from them by introducing a reflection or glide or both. There are 12 distinct patterns and each can be described according to an international system of coding.

The first seven are below. As before the invitation is for you to replicate these patterns (or with a much more flamboyant motif) using some elegant programming and to study these patterns yet further.