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Programming Enigma Puzzles

18 December 2017

Posted by on **From New Scientist #2239, 20th May 2000** [link]

A semi-prime is the product of two prime numbers. Harry, Tom and I each chose a set of three two-digit semi-primes which formed an arithmetic progression, such that the six factors of the three semi-primes in any one set were all different. The first (lowest) number in each of our sets was identical.

We then each chose another such set. This time the middle number in each of our sets was identical.

We then each chose another such set. This time the last (highest) number in each of our sets was identical, and of the other two semi-primes in my set one also appeared in Harry’s set and the other also appeared in Tom’s set.

The nine sets that we chose between us were all different, but one semi-prime appeared in all three sets that I chose.

List the three sets that I chose in the order in which I chose them, with the numbers in each set in ascending order.

[enigma1083]

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It’s a bit fiddly to check all the necessary conditions, but at the end of it you can determine all of the sequences (although we can’t distinguish between Tom and Harry, so we don’t know who chose which of their possible sequences).

This Python program runs in 45ms.

Run:[ @repl.it ]Solution:Dick’s sequences are: 1st (77, 86, 95); 2nd (21, 58, 95); 3rd (91, 93, 95).Dick’s first sequence of numbers was 77 (7 × 11), 86 (2 × 43), 95 (5 × 19), with a common difference of 9.

Harry and Tom’s first sequences were: 77 (7 × 11), 82 (2 × 41), 87 (3 × 29), with a common difference of 5; and 77 (7 × 11), 85 (5 × 17), 93 (3 × 31) with a common difference of 8.

All of these sequences start with 77.

Dick’s second sequence was: 21 (3 × 7), 58 (2 × 29), 95 (5 × 19), with a common difference of 37.

Harry and Tom’s second sequences were: 39 (3 × 13), 58 (2 × 29), 77 (7 × 11), with a common difference of 19; and 51 (3 × 17), 58 (2 × 29), 65 (5 × 13), with a common difference of 7.

All of these sequences have 58 in the middle.

Dick’s third sequence was: 91 (7 × 13), 93 (3 × 31), 95 (5 × 19), with a common difference of 2.

Harry and Tom’s third sequences were: 87 (3 × 29), 91 (7 × 13), 95 (5 × 19), with a common difference of 4; 93 (3 × 31), 94 (2 × 47), 95 (5 × 19), with a common difference of 1.

All of these sequences end with 95, and the first and second elements of Dick’s sequence (91 and 93) appear in Harry and Tom’s sequences.

Each of Dick’s sequences ends with the number 95.