# 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.