**From New Scientist #1414, 26th July 1984** [link]

I have in mind a five-figure number. It satisfies just one of the statements in each of the triples below.

The sum of its digits is not a multiple of 6.

It is divisible by a number whose units digit is 3.

Its middle digit is odd.

The sum of its digits is odd.

It has a factor which is not palindromic.

It is not divisible by 1001.

It has two or more different prime factors.

It is not a perfect square.

It is not divisible by 5.

What is the number?

Due to industrial action **New Scientist** was not published for 5 weeks between 19th June 1984 and 19th July 1984.

This brings the total number of *Enigma* puzzles available on the site to 804, just over 45% of all *Enigma* puzzles published.

[enigma267]

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I think the puzzle is flawed, in that there are multiple solutions.

This program examines all 5 digit numbers. It runs in 1.7s.

Solution:The number is either 10201 or 14641.The published solution is 10201.

I think the minimal way to fix this puzzle is to change the 6th statement to be:

While that does yield a unique solution to the problem it is clearly not the one the setter had in mind, as 10201 is eliminated (because it is divisible by 101), leaving 14641 as the only possible solution.

Hi Jim,

Why not the number 36481 can not be an answer, too? Thanks.

For 36481 two of the statements in the second triple are true, we need numbers that have exactly one true statement in each triple.

2.1) The sum of the digits is 22, so this statement is false.

2.2) The divisors are 1, 191 and 36481, so (unless we exclude the number itself) then it does have a divisor which is not palindromic, so this statement is true.

2.3) It is not divisible by 1001, so this statement is true.

If we took statement 2.2 to exclude the actual number itself (so only consider “proper divisors”), then 36481 would be another potential solution (along with the two we’ve already found).