Enigma 1778: Around the houses
4 December 2013
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From New Scientist #2946, 7th December 2013 [link]
Fewer than 100 houses have been built on the edge of a circular green, equally-spaced around the whole perimeter and numbered 1, 2, … in order. I have counted the number of pairs of houses with their first and second digits exchanged that are either 90 degrees or 180 degrees apart around the circle. The numerical difference between the counts for 90 degrees and for 180 degrees apart is the same as the higher digit of the lowest-numbered house involved.
How many houses are there in all?