## Teaser 3003: All that glitters

**From The Sunday Times, 12th April 2020** [link]

My aunt has a collection of sovereigns, and she set me a challenge:

“You can have the coins if you can work out the dates, which (in increasing order) are equally spaced and all in the 20th century. The number of coins is an odd prime. The highest common factor of each pair of dates is an odd prime. The sum of the number of factors of each of the dates (including 1 and the date itself) is an odd prime.”

I worked out the dates, though the gift was much less valuable than I’d hoped.

What were the dates?

[teaser3003]

## Jim Randell 5:46 pm

on9 April 2020 Permalink |I assumed the dates we are looking for are the years in the 20th century for each coin.

This Python program runs in 93ms.

Run:[ @repl.it ]Solution:The dates of the coins were: 1903, 1936, 1969.Manually (as suggested by Robert):

Most number have divisors that come in pairs, so have an even number of divisors. The exception is the square numbers, which have an odd number of divisors (see:

Puzzle #08).So, in order for the sum of the divisors of the dates to be odd, the list of dates must include an odd number of square numbers. And in the range 1901 – 2000 there is only one square number, 1936. So that must be one of the dates.

1936 factorises as: (2^4)(11^2), so the other dates must have a GCD with 1936 of 11.

For numbers less than 1936, we get: 1925, 1903. For numbers greater than 1936 we get: 1947, 1969, 1991.

Looking for arithmetic sequences containing 1936, with a number of elements that is an odd prime we get:

Only the second of these has a divisor sum that is prime, and

gcd(1903, 1969) = 11so this satisfies all the required conditions and gives the answer.LikeLike

## Robert Brown 9:08 pm

on9 April 2020 Permalink |The only numbers with odd numbers of factors are perfect squares. There is only one of these in the 20th century, and that date has only has one factor >1 that’s an odd prime. Quite easy to find the answer by inspection.

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## Jim Randell 7:53 am

on10 April 2020 Permalink |@Robert: That’s a neat insight which quickly leads to a manual solution.

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