Here is last week’s puzzle, as originally posted:
Perfect numbers are numbers that equal the sum of their divisors. For example, the number 6 can be divided evenly by 1, 2, and 3, and 6 = 1 + 2 + 3.
Almost perfect number pairs are pairs of numbers that equal the sum of their partner’s divisors. For example, 1184 and 1210 are an almost perfect number pair.
1184 = 1+2+5+10+11+22+55+110+121+242+605, the sum of the divisors of 1210.
1210 = 1+2+4+8+16+32+37+74+148+296+592, the sum of the divisors of 1184.This week’s GeekDad Puzzle of the Week is straightforward: How many perfect numbers or almost perfect number pairs are there below 50,000, and what are they?
This week’s solution:
6 (perfect number)
28 (perfect number)
220, 284
496 (perfect number)
1184, 1210
2620, 2924
5020, 5564
6232, 6368
8128 (perfect number)
10744, 10856
12285, 14595
17296, 18416
In retrospect, this week’s puzzle was “googlable,” if only you knew the right terms. The set of numbers requested, above, are amicable numbers if they come in pairs.
In researching this puzzle, it turns out that numbers are amicable if there are two that sum to each other’s proper divisors; they are social numbers if they sum not as a pair, but as a chain, like: 12496, 14288, 15472, 14536, 14264. That is, the numbers that divide 12496 (1, 2, 4, 8, 11, 16, 22, 44, 71, 88, 142, 176, 284, 568, 781, 1136, 1562, 3124, and 6248) add up to 14288, the numbers that divide 14288 add up to 15472, … and the numbers that divide 14264 add up to 12496.
Congratulations to Jonathon Peever for submitting one of the many correct answer submitted! He will soon be the proud owner of a $50 ThinkGeek gift certificate.
For those of you who sent in a solution (or are just ThinkGeek shoppers) please feel free to use discount code GEEKDAD33NG for $10 off a ThinkGeek order of $50 or more. Thanks to all that posted a solution, and good luck with Garth’s upcoming brainteaser!
