Answer: The prices of my pieces of furniture were 313, 317 and 373 euros
The winner is John Lutz of Denistone, New South Wales, Australia. There were 209 entries.
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Worked answer:
The primes to be used are 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97.
Since we need only consider three-digit primes, which can be x/yy and xx/y, the only first digits may be 2, 3, 5 or 7.
The third digit must be prime, but cannot be 2 or 5, so can only be 3 or 7, so two-digit primes can only end in 3 or 7.
The second of the three digits could only be 1, 3, 7 or 9, so the first digit of the two-digit primes could not include 2, 4, 5, 6 or 8.
Hence, the two-digit primes can only be 13, 17, 37, 73 or 97. In addition,
- if the first of the three digits were 2, the second could not be 1 or 7
- if the first of the three digits were 3, the second could not be 3 or 9
- if the first of the three digits were 5, the second could not be 1 or 7
- if the first of the three digits were 7, the second could not be 7
The available three-digit numbers are thus:
237* 297* 313 317 373 537* 597* 713 717* 737‡ 797
It will be quickly evident that those marked * are multiples of 3, and that marked ‡ is a multiple of 11.
Hence, we need only check the following numbers for for primacy: 313, 317, 373, 713 and 797
In fact, 713 = 23 × 31 but the rest are prime.
The possible sums of three are shown in the diagram, top-right.
1003 = 17 × 59, but the other three are primes.
So the prices, in ascending order, were 313, 317 and 373 euros.
