Puzzlemaster

COSTEL HARNASZ teaches maths in Manchester. His card carries his name and co-ordinates on one side, as you'd expect; the flip side is the same, but in mirror writing. The letters on the front and the back have been placed by the printers so that they are exactly superposed, as I discovered by holding the card up to the light.

What is psychologically interesting about this half-unexpected object is not that people are thrown by it, but that they seem compelled to rationaliseand to suggest how it has come about. Impulsive explanations of Harnasz's ice-breaker range from "Oh, it's in Russian" to "It's transparent!"

I tried it at home. One visitor thought the ink had leaked through the card. My secretary suggested the printer had stacked the cards while wet so that the backs bore the reverse of the print on the cards with which they had been in contact.

My own first impression had been equally daft: the cards were slices off a solid bar with the print running through it like seaside rock.

That unguarded moment of mental fumbling as the subject gets to grip with something unfamiliar gives an insight into thinking in general. The mind finds ambiguity disturbing. It tries to assimilate the unusual to an already familiar framework even at the expense of consistency: if the card had really been transparent you'd have been able to see through it; if the mirror writing were due to ink seepage or stacking while wet, it would have been smudgier.

The production of wild surmises and analogies is an essential side of creative problem-solving. The other is the disciplined appraisal of all these loony "what-ifs". The one makes mathematics and science creative; the other makes them applicable. Without divergent creativity there would be no new ideas: without the convergent rigour of proof and reality checks we'd have too many ideas and we wouldn't know whether they were useful or not.

In science we need both: to have wild ideas and to be sceptical of them. The impractical explanations offered to Harnasz's card are a sign of healthy creativity. Being satisfied with them is not. It shows a lack of the scepticism necessary to scientific thinking.

I think it was Kierkegaard who kept a post-horn on his desk as an embodiment of the impossibility of true repetition. I keep Harnasz's card on my desk because it reminds me of so many things.

Solutions to last week's problems

a) There are 720 ways of six passengers taking up their seats in a six- seater train compartment. (Imagine seating each in turn. For each of the six ways of seating A there are five ways of seating B; then four for C, and so on, resulting in 6 x 5 x 4 x 3 x 2 x 1 = 720 different ways.)

b) 720. Imagine one of the passengers is the invisible man and you will "see" this reduces to the same problem as a).

c) Erich Weiss, or Houdini, died on Hallowe'en 1926. What was his first name?

Points to ponder

1. If the number of seating arrangements for six passengers in a six- seater compartment is the same as for five, why is it not the same for four?

2. A friend had a Harnasz-type card printed with just his first name on it. He was disappointed to find that both sides read the same. What was it?