**From New Scientist #2363, 5th October 2002** [link]

We had a gathering of our very respectable and mathematically inclined family recently to celebrate the forthcoming centenary of its oldest surviving member.

Out of interest I pointed out to them that, given any odd number which does not end in 5, some multiple of it must consist simply of a string one ones. For example, if you look at 71 then the number 1111111… consisting of thirty-five ones is a multiple of it!

This set the family chattering and Alan wrote down the shortest string one ones which was a multiple of his age.

Alan’s father, Bob, then rubbed out a majority of the ones to leave the shortest string of ones which was a multiple of his age. Bob’s grandfather, Colin, then rubbed out a majority of the remaining ones to leave the shortest string of ones which was a multiple of his age.

How old is Bob?

[enigma1207]

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Enigma 1212for more fun with repdigits.To get to a unique solution I required at least 16 years between generations. (i.e. Bob must be at least 16 years older than Alan, and Colin must be at least 32 years older than Bob).

This Python program runs in 33ms.

Solution:Bob is 53.Bob’s repunit had 13 digits.

Alan is aged either 27 (and his repunit has 27 digits), or 29 (and his repunit has 28 digits).

Colin is aged 91 (and his repunit has 6 digits).

If we reduce the generation gap we can find an additional solutions. With a gap of 13 we get:

Alan is 47 (repunit has 46 digits) or 49 (repunit has 42 digits). Bob is 63 (repunit has 18 digits). Colin is 91 (repunit has 6 digits).