# Sunday Times Teaser 2997 – Cemetery Lottery Symmetry

*by Stephen Hogg*

#### Published Sunday March 01 2020 (link)

Our local cemetery conservation lottery tickets have four numbers for a line. Using eight different numbers, my two lines have several symmetries. For each line: just one number is odd; there is one number from each of the ranges 1-9, 10-19, 20-29 and 30-39, in that order; the sum of the four numbers equals that sum for the other line; excluding 1 and the numbers themselves, the 1st and 2nd numbers share just one factor — as do the 2nd and 3rd (a different factor) and the 3rd and 4th (another different factor) and, finally, the 4th and 1st.

Printed one line directly above the other, my top line includes the largest of the eight numbers.

What is the bottom line?

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I originally thought that this might be possible by showing that the Greatest Common Divisor (GCD) of adjacent numbers on a line was prime but it turns out that the teaser constraints make this a bit more complicated since prime powers have to be checked as well. So I moved to a solution based on the divisors of the numbers. There is also a potential confusion in the use of the term ‘factor’ since there is a concern about whether this means any factor or only prime factors. Nevertheless the solution is fairly straightforward and this program finds it in less than 3 milliseconds.