Enigma 1186: Always or never a semi-prime
4 January 2016
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From New Scientist #2342, 11th May 2002
At snooker a player scores 1 point for potting one of the 15 red balls, but 2, 3, 4, 5, 6 or 7 points for potting one of the 6 “colours”.
Whyte potted his first red, then his first colour, then his second red, then his second colour, and so on until he had potted all 15 reds, each followed by a colour. Since the colours are at this stage always put back on the table after being potted, the same colour can be potted repeatedly.
After Whyte had potted each of the 15 colours his cumulative score called by the referee was always a semi-prime. A semi-prime is the product of two prime numbers; the square of a prime number counts as a semi-prime.
After potting the 15 reds and 15 colours a player tries to pot (in this order) the balls scoring 2, 3, 4, 5, 6 and 7 points. Whyte did so to complete a total “clearance”, but his cumulative score after each of those six pots was never a semi-prime.
What was his final score?
Thanks to Hugh Casement for providing a complete transcript for this puzzle.