(In reply to
non-computer solution by xdog)
Here is my understanding of xdog's solution. Call it: xdog for dummies :-)
How do you use pairwise differences to find if a number is divisible by 11? I didn't know until i watched this very nice video that also explains why it works:
Now what to make of this line: "S3 has digits d to abs(d-5)" inclusive?
Ans: since there are 6 distinct digits with max-min=5, these can only be, in some order:
012345, 123456, 234567, 345678, or 456789. Fair enough.
Now the list any S3 number makes (if not 012345) will have the same pairwise differences (summed) if the digits were dropped down to the list 012345. This can be done for our examples above by subtracting 111111, or 222222 ... or 444444 from the number, because by doing so, we are effectively adding -1, -1, -1 and 1, 1, 1, to the differences, or 0. Likewise 2,2,2,-2,-2,-2 = 0 etc.
So finally, do any of the S3 permutations of 012345 have a sum of differences that make -11, 0 or 11? The largest sum of differences is 5+4+3 - 0-2-1 = 9 and the smallest -9.
So we are only left to ask about the sum of differences adding to 0. If we take pairwise differences of 3 even and 3 odd numbers, in any pairs, we will aways get an odd number, not 0. So none of S3's members are divisible by 11.
So maybe not within KISS, but I get it.
Edited on July 7, 2020, 6:15 pm
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