MATHEMATICAL RECREATIONS 193

Then m = n + 1, where n = 1, and

a=ao+ ar+ -+ aar”t + aa.r”

n+1
Y ap_grtt
k=1

Il

n
= ao+ 2 Gpa?"t 4 a1,
k=2

where we have the middle terms if » > 1, but not if » = 1.
Let @’ be the number obtaind from e by interchanging the

first and last digits.
Then

n
a = a, + 2 arar*t + apr®
k=2

~ We wish to subtract @’ from a. In order that the digits of
the minuend may all be surely greater than the corresponding
digits of the subtrahend, we will find it convenient to add the

number 2
o
k=1

or n+1

Z pk—1

k=2

to both the minuend and the subtrahend, which addition will
not change the difference.
For the minuend we will write this number in the form

n
p ot
=2

in which we have the second part if z > 1, but notif n = 1;
for the subtrahend we will write this number in the form

n
Z yk—1 + 7'",
k=2

in which we have the first paft if n > 1, butnotif » = 1.