Solution to In Their Prime

Did you have a go at my In their Prime puzzle in Puzzlebomb yet? How about the other puzzles?

If your answer to these questions is “No”, then please turn to  page 13091204281 of the internet to have a go at the July Puzzlebomb.

If the answer is “Yes”, you can check the July solutions, including the numeric solution to In their Prime. But how did you find the solution? You may have successfully used trial and error, but that’s not usually very enlightening; you may have programmed a computer to do the dirty work for you (the puzzle was hand designed, but I did check the answer was unique with a bit of code). As a responsible puzzle-setter, I came up with the following possible proof of the solution. I’d be interested to hear from anyone who had a different method.

You’ve memorised the rules, right? Or you’ve got Puzzlebomb printed out or handily in another tab?
1) If you follow a gene through the family tree, each must have a different pairing in the three generations, so each prime needs three pairings with products under 120, so we certainly need the product with 2,3 and 5 to be less than the maximum, that is, 5p \leq 120, or p \leq 24.
2) Again, studying the diagram, you can see that no two prime genes can have same three partners. That means that at most one of 19 and 23, which only work with 2,3 and 5, can be used in the puzzle. As 19 has already been placed in the puzzle, you can eliminate 23, leaving the eight primes from 2 to 19.
3) If two of lower four {2,3,5,7} are paired together, then two of higher four {11,13,17,19} must also be paired, and their product would exceed 120. So a lower prime must always go with a higher prime. A \{2,3,5,7\} \times \{11,13,17,19\} times table is pretty necessary from now on.
4) Since 7 can never be paired with 19, so, looking up a generation you can eliminate the mid-right female (the sister of the marked mid-left male) having that gene, otherwise one of their parents would have 7 \times 19 > 120 as a check-out age. Similarly, looking a generation down you can eliminate the mid-left female (wife of the mid-left male) otherwise a child would have this disallowed  level of superannuation (above 120). So the mid-right male must have the ‘7 gene’. His wife has higher death age, and this would be impossible unless 7 is paired with 11. His wife’s age involving either the 13 or 17 gene, and glancing again at the table, the only higher option is 85.

5) By similar reasoning, the mid-left pair must be 38 and 39.
6) It’s now straightforward to fill out the other generations: the males below inherit the lowest and second highest of their parents’ prime genes, while the same is true of the top generation: the grandfathers each share the lowest and second highest of the parents’ genes.
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