**From New Scientist #2866, 26th May 2012** [link]

I woke up one morning last summer near the time of the solstice, and noticed that the clock in the darkened bedroom indicated a time either between 3.40 and 3.45am or between 8.15 and 8.20am – I couldn’t tell which.

When I mentioned this to my eccentric uncle, he produced a defunct clock, which he had altered by making the minute and hour hands identical. He then arranged the hands to a particular setting which corresponded to my bedroom dilemma, and told me that it would be impossible to distinguish, by appearance alone, which of two particular times was being shown by the clock.

Assume that at noon both hands on my uncle’s clock point exactly to the 12 marker. Tell me, to the nearest second, what is the difference (less than 4 hours 40 minutes) between the two times which could have been indicated by my uncle’s clock?

[enigma1699]

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Some simultaneous equations give the solution. Here’s a Python program that computes the answer. It runs in 37ms.

Solution:The difference between the times is 4 hours, 36 minutes and 55 seconds.Note:The exact time of the Summer Solstice in 2011 was 17:16 UTC (on 21st June 2011), which would be 18:16 BST.Or you can use

SymPyto solve the simultaneous equations: