The Second Book of Mathematical Puzzles and Diversions

CHAPTER FOUR
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Digital Roots
J OT DOWN your telephone number. Scramble the order
of the digits in any way you please to form a new num-
ber, then subtract the smaller number from the larger. Add
all the digits in the answer. Now place your finger on the
star in the circle of mysterious symbols [Fig. 191 and count
them clockwise around the circle, calling the star 1, the tri-
angle 2 and so on until you reach the number that was the
final step in the procedure given above. Your count is sure
to end on the spiral.
The operation of this little trick is not hard to understand,
and it provides a painless introduction to the concept of
numerical congruence formulated by the great German
mathematician Carl Friedrich Gauss. If two numbers have
the same remainder when divided by a given number called
k, they are said to be congruent modulo k. The number k is