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Answer: The numbers were 413, 568 and 927

The winner is Johan Uys of Bellville, South Africa. There were 232 entries.

Worked answer

Nonno’s number is at most 9–, Otto’s is at most 8–, and Hepta’s is at most 7–.

A quick run through using the repeat button on a calculator shows that Hepta’s number is one of these:

(H) 217 259 287 329 371 413 427 469 497 581 623 679 721 749 763 791.

So Hepta’s is at least 2–, Otto’s is at least 3–, and Nonno’s is at least 4– (and it cannot be 423).

Again the calculator quickly yields the following possibilities:

(O) 328 376 472 536 568 584 632 712 824 856 872;

(N) 531 549 639 657 873 927 963 981.

Where is the digit 1 going to be?

If it’s in (O) we’ll have ?/712/? giving ?/712/963 with no possible (H).

If it’s in (N) we’ll have ?/?/531 (giving ?/472/531 with no possible (H)) or 427/536/981 (but the product of those three begins and ends with a 2).

So we can conclude that the digit 1 is in (H).

The revised list of choices for (H) is therefore: (H) 217 371 413 581 721 791.

Where is the digit 4 going to be?

If it’s in (H) we’ll have 413/?/? giving 413/?/657 or 413/?/927. The only possible triples are 413/568/927 and 413/856/927.

If it’s in (O) we’ll have ?/472/? or ?/584/? or ?/824/?. The first has no possible (H), the second leads to 217/584/? or 371/584/?, and the third leads to ?/824/963. The only possible triples are 217/584/639 and 217/584/963.

If it’s in (N) we’ll have ?/?/549 giving ?/376/549 or ?/328/549 with no (H).

So now we look at the product of the three numbers in each case highlighted in bold above. Because a calculator only handles 8 digits we look at:

41.3 x 56.8 x 92.7 ≈ 217459.37

41.3 x 85.6 x 92.7 ≈ 327720.46 (so the boys’ product will be 3277..)

21.7 x 58.4 x 63.9 = 80979.192 (so the boys’ product has only 8 digits)

21.7 x 58.4 x 96.3 ≈ 122039.06 (so the boys’ product will be 122…)

Multiplying out 413 x 568 x 927 by long multiplication (or by using the above answer plus the fact that 13 x 68 x 27 ends in 68) shows that the product is 217,459,368 which does indeed use all the digits 1-9 exactly once.