Answers 265- THE HOUSE NUMBERS
The numbers of the houses on each side will add up alike if the number of
the house be 1 and there are no other houses; if the number be 6, with
8 houses in all; if 35, with 49 houses; if 204, with 288 houses; if 1,189, with
1,681 houses; and so on. But it was known that there were more than 50 and
fewer than 500 houses, so we are limited to a single case, and the number of
the house must have been 204.
Find the integral solutions of --2-X2 + x = y2. Then we get the answers:
x = Number of houses. y = Number of particular house.
1............ .......... 1
8 ...................... 6
49.................... .. 35
288 ...................... 204
1,681 ...................... 1,189
and so on.
169. A NEW STREET PUZZLE
Brown's number must have been 84, and there were 119 houses. The num-
bers from 1 to 84 sum to 3,570 and those from 1 to 119 to 7,140, which is just
double, as stated.
Write out the successive solutions to the Pellian equation (explained on
page 164 in my book Amusements in Mathematics) 2x^2 - 1 = y2, thus:x ~
1
5 7
29 41
169 239
985 1,393and so on. Then the integral half of any value of x will give you the house
number and the integral half of y the total number of houses. Thus (ignoring
the values 0-0) we get 2-3, 14-20,84-119,492-696, etc.