July 2018 little math problem

[Valentin Albillo]

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Hi, Albert Chan:

I'm on the very verge of going on a trip to start my summer vacations so I haven't had much time, if any, to devote to this nice problem by pier4r.

From the top of my head I just had the time to concoct a quik'n'dirty 6-lines of HP-71B code (main program) which calls another simple 6-line subprogram to perform an essentially brute-force search with a few refinements, which finds all solutions very quickly but I have no time to post it here right now, I must go.

Nevertheless, a cursory read of the already existing posts made me realize that yet another symmetry is lacking from all solutions so far (if I read the posts correctly, read them really fast).

Albert Chan Wrote:Add back missing digits to confirm above 12 cases. All confirmed ! Each solution actually represent 8 solutions, by head swap, tail swap and reverse the digits.

Close but no cigar. Actually there are only 6 primary solutions, not 12, each one of them representing 16 derived solutions, not 8.

You mention head swap, tail swap and reverse the digits but you missed 10's-complementing the digits:

• if (d1)(d2)...(d9) is a solution with sum S then (10-d1)(10-d2)...(10-d9) is a solution as well, with sum 30-S.

This halves the number of primary solutions from 12 to just 6, all the remaining 90 are directly derived from this 6.

Albert Chan Wrote:SUM SOLUTIONS
=== =========
13 391847256
14 284917635
14 194827536
14 482935617
14 581674329
14 671584239
16 178625934
16 187634925
16 394571826
16 475382619
16 574391628
17 458362917

Your first 6, say, can be considered the 6 primary solutions. But the remaining 6 are actually derived from them and so aren't primary. For instance, your first solution is:

      13 391847256

and your last (12th) solution can be derived from it by simply 10-complementing it (and in your case, reversing the digits), like this:

      [13, 391847256 ] -> [30-13, (10-3)(10-9)(10-1)...(10-2)(10-5)(10-6)] -> [17, 719263854]

and after reversing the digits we get: 17 458362917, which is your 12th solution indeed. In the same fashion you can derive your 11th solution from the 2nd primary, the 10th solution from the 3rd primary, and so on until deriving your 7th solution from the 6th primary.

Sorry but absolutely no time for more, will be back next September, have a nice summer.

Regards.
V.
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