**From New Scientist #1214, 14th August 1980** [link]

It might be said — indeed it has been said — that Uncle Bungle has a genius for getting things wrong. This has not hitherto ruffled the imperturbability of his temper, but the other day, perhaps as a result of excessive pulling of a leg that has perceptibly lengthened through the years, something snapped. “All right”, he said crossly, “if you say I get things wrong I’ll get things wrong”. And he proceeded to produce a cross-number puzzle in which every single clue was incorrect. And this time there was no mistake in the mistakes. (There are no 0’s in the solution). The puzzle was as follows:

**Across:**

* 1-2-3* Not a multiple of 3 and of 5 and of 7.

*4-5-6* The sum of the digits is greater than 5.

*7-8-9* The last digit is less than the sum of the first two.

**Down:**

*1-4-7* The sum of the digits is less than 19.

**2-5-8** The sum of the digits is greater than the sum of digits of 3 down.

**3-6** An odd number.

Find Uncle Bungle’s solution.

[enigma71]

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Solution:Uncle Bungle’s solution is:Here’s a solution using the

CrossFigure()solver fromenigma.py(originally written forEnigma 1760). It runs in 60ms.Just for the record, let me point that there may be a slight glitch at the hints, namely the 1-across. 945 is the unique solution where 1-A is both a multiple of 3, 5 and 7. But the negation of “A and B and C” should be “A or B or C” (uncle Bungle would still be wrong if he answered a multiple of 5 who isn’t a multiple of 7, for instance), but in that case there would be plenty of solutions…

I viewed the clue of 1-2-3 as “not(a multiple of 3 and of 5 and of 7)” so the negation is “a multiple of 3 and of 5 and of 7”. Which give us a unique solution that is the same as the published solution. So I think this is probably what the setter intended.

If you view it as “not(a multiple of 3) and not(a multiple of 5) and not(a multiple of 7)” then the negation is indeed “(a multiple of 3) or (a multiple of 5) or (a multiple of 7)”, and there are 21 solutions. The top row being one of: 912, 915, 917, 918, 924, 925, 927, 933, 935, 936, 938, 939, 945, 948, 955, 957, 959, 966, 969, 978, 999.

Quite right, Jim. Your explanation settles this question quite well. With that, I can’t see anything else than “not (a multiple of 3, 5 and 7)” (with the condition being only one and the three numbers being just a list of numbers not a list of conditions). Thank you.