*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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