*Spoiler for New Scientist Enigma 1777 “Around the houses” *(Follow the link to see the puzzle.)

Kind of an odd puzzle and I wasn’t entirely sure how to interpret some of the wording.

But I think it’s pretty easy. First, if there are *any* houses 90 degrees apart then the number of houses must be a multiple of 4. Number of houses *N*=4*n.*

Second, if there are any digit reversal pairs 90 degrees apart then the house numbers differ by *n*: (10*a*+*b*)–(10*b*+*a*)=*n*. Then *n*=9*a*–9*b*. So *n* is a multiple of 9, and *N*=4*n*=36*k.*

Then there are only two possibilities, *N*=36 or 72. Examining these two — and I assume we don’t use leading zeros so 10 is not a reversal of 01, for instance, though in fact allowing that doesn’t permit any solution to the puzzle — we find:

For *N*=36, there are 2 90-degree pairs ((12, 21) and (23, 32)) and 1 180-degree pairs ((13, 31)).

For *N*=72, there are 5 90-degree pairs ((13, 31), (17, 71), (24, 42), (35, 53), and (46, 64)) and 2 180-degree pairs ((15, 51) and (26, 62)).

Now, does “the higher digit of the lowest-numbered house involved” refer to the *larger* digit or the *digit in the higher place*? If the latter then in either case it’s 1 and that’s the difference between the two counts for the *N*=36 case. If the former, then it’s 2 in the *N*=36 case, which isn’t the count difference, or 3 in the *N*=72 case, which is.

I *think* they mean the larger digit, in which case the answer is 72.

I suppose one should check the cases of *N*=18, 54, and 90, for which the counts would be 0 for 90 degrees apart (there are no houses 90 degrees apart) and nonzero for 180 degrees, but even if those cases turn up more solutions, I doubt if they’re what was intended… and besides, I’m not that interested!