MATHEMATICAL RECREATIONS 197

in which we have the } terms if » > 2, but not if n = 2.
Then

n—1
d+d =@—-1)+@F—=2r4+@—=24+1) X r*+0-7" 4 7-p»
k=2
=D+ —Dr4(r—1) X r1 4 0pn ot
k=3

=114 0rm 4 (r — 1) i =D L (r=2)r+(r—1)
k=3

We may put our results in words as follows:

Th. If r1s the radix and d is a number of m digits (m > 1),
of which the first may be 0 and such that the sum of the first and
last digits is v — 1 and all the other digits, if there are any, are
equal tor — 1, if also d’ is the number got from d by interchanging
its first and last digits; then
if m = 2,d+ d' is a number of two digits both of which are
equal to r — 1;
if m > 2,d+ d is a number of m + 1 digits, of which the
Jirst two are 1, 0, the last two r — 2 and v — 1, and the others,
if there are any, are equal to r — 1.

 

Examples. Radix ten. 18 099 495 6993
81 990 594 3996
99 1089 1089 10989

Radix two. 10 1110 11110

01 0111 01111

11 10101 101101

483. In case the radix is greater than 2, we can put our

result in another form.
We have as in § 482

d+d = (r — 1)(1 + 2‘27"’—1-}— l-r”)
k=2