MATHEMATICAL RECREATIONS 191

The last statement of the set is obtaind from the given form
by putting / = r — 3 and is

N=14+1r41224 - +1r~2 4 (r — (r — 2))r—3
r—3
+@—1) % (r—k—2)r
k=r—3
141741724 oo 1t 2r 3 4 (r — 1)r3
1 + 17 _|_ 172 + R + 1pr—4 + 2pr—3 + pr—2 — pr—3
14+1r4+ 1724 -0 1=t 4 193 4 172

That is, N is a number consisting of » — 1 ones in the
notation whose radix is 7.
If » = 3, the given expression is meaningless.

OTHER CURIOUS PROPERTIES OF NUMBERS (1)

477. Write any number of three digits, of which the first

is greater than the last, sy . T .oy o o ot 170
Interchange the first and last digits................ 071
Subtract. —_

099
Considering this difference also to be a number of three

digits, interchange its first and last digits.............. 990
Add this number to the preceding. —_
The result will- alwaps he: = T s ot s 1089
Another example: :

582
285
297
792
1089

() Expanded from an article on “A Curious Property of Numbers”
in the Scientific American Supplement, June 12, 1915, p. 371, New York,
Munn and Co., by permission.