This puzzle comes from a tweet by Ed Southall (@EdSouthall), who identified the following mathematical relationship:
3³ + 5³ + 2³ = 160 1³ + 6³ + 0³ = 217 2³ + 1³ + 7³ = 352 3³ + 5³ + 2³ = 160
The question (not posed, but I'm answering it anyway) is: Is this a unique relationship, or are there more like it?
And the answer is that there are a lot more like it. I have carried out a brief search and found a total of eight specific groups. Here they are, in the order that I discovered them (starting with 100, the first three-digit number).
Terminate at 1
1³ + 0³ + 0³ = 1
1³ = 1
It's not exactly earth-shattering, but here's the first case: terminates at 1. Example numbers which do this are: 100, 112, and 121.
Terminate at 153
1³ + 0³ + 2³ = 9
9³ = 729
7³ + 2³ + 9³ = 1080
1³ + 8³ = 513
5³ + 1³ + 3³ = 153
Other numbers which do this are: 108, 111, and 135. The example of 102 goes via 1080, while 105 follows a longer route:
1³ + 0³ + 5³ = 126
1³ + 2³ + 6³ = 225
1³ + 4³ + 1³ = 66
6³ + 6³ = 432
4³ + 3³ + 2³ = 99
9³ + 9³ = 1458
1³ +4³ + 5³ + 8³ = 702
7³ + 2³ = 351
3³ + 5³ + 1³ = 153
Loop with a low of 55
2³ + 8³ = 520 5³ + 2³ = 133
Terminate at 371
Terminate at 370
Summary
The eight conclusions of the 'cubic sums' puzzle for three-digit numbers (which I found) are:
Terminate at 153
Loop around 55 (55, 250, 153, 55)
Terminate at 371
Terminate at 370
Loop around 160 (160, 217, 352, 160)
There are probably other 'identities' but there's no pattern to them and they would only be found by trial and error (or an algorithm).
It's interesting (and probably coincidental) to note that 137 to 150 show a repeating pattern of their own - the results go through a sequence 371, 1543, 370). This is probably only a coincidence, but it's interesting how these results crop up!
