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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.

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?

Triangular Tantaliser

Draw all the possible distinct triangles on a 4 x 4 dotty grid. Convince me that you have all possible triangles.


Age 11 to 14
Challenge Level

We received two different explanations to this problem.

Danny Ng, 16, from Milliken Mills High School in Canada sent us this solution:

"Turning a left-handed helix over does not make it become a right handed helix.

When you observe a bolt at eye level, you see the parallel threads.
There are two kinds of threads:
1) go "down-right", from upper-left down to lower-right
2) go "down-left", from upper-right down to lower-left.
If you draw a line that is "down-right" on a piece of paper(the line goes from upper-left down to the lower-right) and turn the paper upside down (just like turning a left-handed helix over) you will notice the line is still "down-right" and it never goes "down-left"."

Harry (Xiaotian) Li (Age: 15) from Riccarton High School, Christchurch, New Zealand, sent us this solution:

"Consider, for example, the spiralling direction of a right-handed one:
To screw down (from top) turn clockwise.
To screw up (from top) turn anti-clockwise.
After that we turn it over (top to bottom).
To screw it down, still turn clockwise.
To screw it up, still turn anti-clockwise.

And also, two kind of helix look different (opposite each other). When you turn just one of them over they are still opposite, which confirms that it has not changed.

Therefore, when we turn a left-handed helix over (top to bottom). It does not become a right-handed helix."

Paul Marcinkowski, aged 14, from the Harwich School in GB convinced himself by making a helix, and contributed:

"No because I have made a left handed helix and whatever way you turn it up it is still a left handed helix."