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# How Old Are the Children?

##### Age 11 to 14Challenge Level

There was a good postbag for this question.

All submissions got as far as listing the possible sets of three numbers with product 72. There are 12 such "triples", namely

[1, 1, 72], [1, 2, 36], [ 1, 3, 24], [1, 4, 18],

[1, 6, 12], [1, 8, 9], [2, 2, 18], [2, 3, 12],

[2, 4, 9], [2, 6, 6], [3, 3, 8] and [3, 4, 6].

But what about the door number? How does Mrs Smith's answer help Sally?

This part of the solution was a stumbling block for many entrants.

We assume that Sally knew the number. Now all the triples have a different sum EXCEPT [2,6,6] and [3,3,8] which both sum to 14. So the door number must be 14, otherwise the information would have told Sally the answer. It follows that the correct triple must be one of these two.

Finally, since Amanda is the youngest child, she is not a twin, and the correct triple is [2, 6, 6] rather than [3, 3, 8], so we know that Mrs Smith has a 2 year old called Amanda and twins aged 6.

There were many partially correct solutions and plenty of correct answers with explanations that were incomplete or plain false, but solutions from the following were both complete and correctly argued.

Jessica, Daniel and Mark of Jack Hunt School, Peterborough;

Alice, Ashley, Alice and Elisabeth of The Mount School, York;

Larissa, Hollie and Emma, also of The Mount School;

Rachel and Christiane, again of The Mount School;

Georgina of Davison High School, Worthing;

Rosie of Davison High School, Worthing;

James of Hethersett High School.