Enigma 1721

Spoiler for New Scientist Enigma 1721: “Odds and evens” (Follow the link to see the puzzle.)

Rewrite this as

   AbC
   dEf
   ---
gHiJKL
  MnOP
  QrST
------
uvWxYZ

where uppercase (lowercase) letters denote even (odd) digits. The only sensible way I can see to interpret this is to assume gHiJKL = d * 100 * AbC, MnOP = E * 10 * AbC, QrST = f * 1 * AbC. I’d write such a multiplication as

   AbC
   dEf
   ---
  QrST
  MnOP
gHiJKL
------ 
uvWxYZ

myself. With that assumption, K = L = P = 0; Z = T. That gives us:

   AbC
   dEf
   ---
gHiJ00
  MnO0
  QrST
------
uvWxYT

Now AbC * f = QrST ≥ 2100, and f ≤ 9, so AbC ≥ 234. Then AbC * E = MnO ≤ 898, so E = 2 and A = 2 or 4.

Now look at MnO = 2 * AbC. To make the middle digit of the product odd, there must be a carry of 1 from 2 * C, so C = 6 or 8; to make the first digit even there must be no carry from 1 + 2 * n. So n = 2 * b + 1 ≤ 9. This means b = 1 or 3, n = 3 or 7. Then since AbC ≥ 234:

AbC   =  236  238  416  418  436  438
MnO   =  472  476  832  836  872  876
3*AbC =  708  714 1248 1254 1308 1314 
5*AbC = 1180 1190 2080 2090 2180 2190
7*AbC = 1652 1666 2912 2926 3052 3066
9*AbC = 2124 2142 3744 3762 3924 3942

The numbers in blue are EOEE (so could be QrST) and the ones in green are OEOE (so could be gHiJ). We see there are only three possibilities:

   236     418     436
   729     327     725
   ---     ---     ---
165200  125400  305200
  4720    8360    8720
  2124    2926    2180
------  ------  ------
172044  136686  316100

and only the third of these is OOEOEE. So the two 3-figure numbers are 436 and 725.

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