Enigma 146: Around the clock
18 November 2013
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From New Scientist #1291, 4th February 1982 [link]
I recently amused my son with a little numerical pastime. I drew a circle and marked 12 equally spaced points around the circumference. By each of the points I wrote a digit: each of the digits from 0 to 9 occurred somewhere (and, of course, there were some repeats).
The exercise consisted of starting at the centre of the circle, drawing a straight line to one of the 12 points on the circumference, drawing another straight line from there to another of the 12 points, a line from there to another of the 12 points, and finally a line from there to another of the 12 points. We would then look at the four-figure number “spelt out” (we never started with a zero). We’d then start again at the centre.
With one such sheet we did three similar exercises. The four-figure numbers spelt out were, respectively:
a product of five consecutive integers;
a number divisible by 7, 11 and 13;
an odd perfect cube.
By coincidence each of the straight lines drawn in obtaining these three numbers was a whole number of inches long.
What was the perfect cube?