Enigma 360: For the time boing
2 September 2016
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From New Scientist #1509, 22nd May 1986 [link]
“This is my favourite clock,” said Mr Fescu, the Curator of the House of Clocks. “It has a curious mechanism which prevents it from stopping or being started except exactly on the hour. It only chimes on the hour and normally emits a number of bongs equal to the hour it is striking. But if it stops and is restarted at a later hour less than 12 hours on it emits all the chimes it missed between the stopping and starting times, not counting those of the hour at which it is restarted.”
“You mean if it stops at 9:00 and is restarted at 8:00 it emits 70 bongs?”
“Precisely,” said my guide, clutching and enormous key. “And what is more interesting is that certain numbers of bongs are special, in that they occur between one unique pair of stopping and starting times only. If you heard 70 bongs it could only signify that the clock stopped at 9:00 and was restarted at 8:00. Given the time at which the clock has stopped this time, there is only one hour at which I could rewind it to yield a magic number of bongs, and Lo!” he said, hopping from foot to foot and gesticulating at the church clock just visible in the distance, “that hour is arrived.”
So saying he moved the hands to the right time, wound the mechanism up and kicked it, whereupon it emitted an uneven number of bongs and a prime number at that.
What was the time; when had the clock stopped; how many bongs did the clock emit?