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 How old did you say they were? (Posted on 2005-03-03)

The hostess, at her 20th wedding anniversary party, tells you that her youngest child likes her to pose this problem to guests, and she proceeds to explain: "I normally ask guests to determine the ages of my three children, given the sum and products of their ages. Since Smith gave an incorrect answer to the problem tonight and Jones gave an incorrect answer at the party two years ago, I'll let you off the hook."

Your response is "No need to tell me more, their ages are..."

 See The Solution Submitted by Erik O. Rating: 3.0000 (10 votes)

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 solution | Comment 1 of 31

Assuming that all the current ages are unique integers between 5 and 20, the following are all those whose combination of current sum and product are not unique:

` `
`                   --former- -ages-  sum  prod sum  prod 5  9 14 28    630 22    252 5 12 18 35   1080 29    480 5 14 16 35   1120 29    504 5 15 16 36   1200 30    546 6  7 15 28    630 22    260 6  9 20 35   1080 29    504 6 10 14 30    840 24    384 6 10 20 36   1200 30    576 6 12 14 32   1008 26    480 6 14 15 35   1260 29    624 6 15 16 37   1440 31    728 7  8 15 30    840 24    390 7  8 20 35   1120 29    540 7  9 16 32   1008 26    490 7 10 18 35   1260 29    640 8  9 20 37   1440 31    756 8 12 15 35   1440 29    780 8 15 18 41   2160 35   1248 9 10 16 35   1440 29    784 9 12 20 41   2160 35   1260 9 15 16 40   2160 34   127410 12 18 40   2160 34   1280`

The following are those that are non-unique combination sum and product two years ago:

` 5 10 12 27    600 21    240 5 11 18 34    990 28    432 5 12 14 31    840 25    360 5 14 16 35   1120 29    504 6  7 14 27    588 21    240 6  8 17 31    816 25    360 6  8 20 34    960 28    432 6  9 20 35   1080 29    504 6 10 17 33   1020 27    480 6 11 12 29    792 23    360 6 12 16 34   1152 28    560 6 14 17 37   1428 31    720 7  8 14 29    784 23    360 7  8 18 33   1008 27    480 7  9 18 34   1134 28    560 7 10 20 37   1400 31    720 7 11 16 34   1232 28    630 8  9 17 34   1224 28    630 8 12 16 36   1536 30    840 8 14 16 38   1792 32   1008 8 16 17 41   2176 35   1260 9 10 17 36   1530 30    840 9 11 18 38   1782 32   1008 9 12 20 41   2160 35   126010 14 17 41   2380 35   144011 12 18 41   2376 35   144011 17 18 46   3366 40   216012 14 20 46   3360 40   2160`

Only 6  9 20 appears on both lists, and is the set of ages.

 Posted by Charlie on 2005-03-03 20:47:33

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