44 THE FASCINATION OF NUMBERS

similarly restricted. Indeed, within each section of eight
numbers, every digit from 1 to 8 appears. But the digits
nevertheless conform to a definite pattern within each section
and within each row. Thus, multiples of 7 always have their
roots in the order 7, 5, 3, 1, 8, 6, 4, 2, g repeated indefinitely.

(¢) Roots appearing in the first row of each section are

repeated in reverse order in the eighth row. The same re-
| lationship occurs between the second row and the seventh
‘ row, between the third and sixth rows and between the
fourth and fifth rows.

Further investigation shows that where numbers can be
related to each other as, for example, the numbers in ‘shape’
series can be related, then their roots also bear a fixed
relationship to each other.

The roots of triangular numbers form a regular pattern.
The first nine triangular numbers have roots in the following
order: 1, 3, 6, 1, 6, 3, 1, 9, 9, and all the higher triangular
numbers have roots which repeat this group in exactly the
same order.

!fi'flww"fi"“flwk\'-‘Mmmmwmmm‘w e

Number Root | Number Root | Number Root | Number Root

 

| |
38 I 1 ‘ I 190 1 | 406 I
= 3 3R 3 210 l 435 3
6 Bl g l gt L0 S0 G Bk
3 10 I 91 Loofs mes I 496 I
3 15 6 105 6 | 276 6 528 6
£ 21 3 120 3 300 3 | 561 3
£ 28 I 136 I 325 I 595 I
; 36 9 153 9 r 351 9 1 630 9
= 45 9 171 9 | 378 9 666 9

The roots of hexagonal numbers are even more selective,
there being only two possible roots:

Number Root } Number Root " Number Root
1 1 37 1 [ 127 1
7 7 61 Yiviod o yaln 7
19 I 91 I } 217 I

To conclude, there follow some further examples of the
persistence of g as a digital root.