178 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

 

 

 

 

 

 

 

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Let @ and b be two digits of the left hand border and top
border respectivly whose sum is greater than 10.

Then, since 10 < a 4+ b < 20, ¢ +d and e + f both are.
digits. § 455.

Also, since ¢ + f = 10 — (¢ + d), e + f is the complement
of ¢ + d.

That is, e 4+ f and ¢ + d are complementary digits.

But in this case e + f is the units digit of a + b. § 456.

That is, the units digit of ¢ 4+ b is complementary to ¢ + d.

Now a + b and ¢ + d are symmetrically situated with re-
spect to the diagonal line of 10’s.

For the a'th row from the top of the table is the ¢’'th row
from the bottom, the b'th column from the left is the d’th
column from the right, etc.

That is, the order of any row or column counting from one
side of the square is the complement of its order counting from
the opposit side.

Now @ + b is in the a’th row from the top and the b’th
column from the left; ¢ + d, which equals d 4+ ¢, is in the
d’th row from the top and the ¢'th column from the left, that
is, the b'th row from the bottom and the a’th column from