Suppose the numbers set for you are the following:
29971
14563
76382
37797
80130
You ask leave to add a like quantity of numbers; in doing this, you
take care that each of the figures you set down make 9 with each of the
figures that have been given for you.
70028
85436
23617
62202
19869
――――――
499995
――――――
The first figure being 2, you must set 7; the second being 9, (which
completes the number wanted) you must set a cypher (0); the third being
the same, operate as before; the fourth being 7, set down 2; the fifth
being 1, set down 8.
The second row beginning by 1, your first figure will be 8; the second
number being 4, set down 5; the third being 5, put down 4; the fourth
being 6, you must set down 3; the fifth being 3, set down 6.
As the third row begins by 7, begin yours by 2; under the 6 lay 3, then
1 under the 8, and 7 under the 2.
For the fourth row, set 6 under the 3, 2 under the first 7, and another
2 under the other 7; a 0 under the 9, and 2 under the 7, which complete
this row.
You are to do the same for the fifth row, putting 1 under the 8, 9
under the 0, 8 under the 1, 6 under the 3, and 9 under the 0.
Then desire some of the company to cast up these ten sums, and it will
be found that the product of the whole addition will form the sum of
499995.
In order to come to this combination, you need only fix the number of
figures that will compose each row, and determine the number of rows;
then to reckon each row for 9, as has been shewn above.
You may likewise present this addition, by saying, that it is the
total amount of ten rows, composed of five figures each; out of which
five rows will be set by the person who chuses to do it; then multiply
secretly as many times 9 as you are to set rows of five figures;
therefore multiply 5 times 9 by 5, which will give you the sum of
499995.
The person having set his numbers, you are to add your five rows,
taking care that every number you set will make 9 with that to which
it corresponds; which being done, you are to ask any one to cast the
whole sum up, and the product will be the same as the sum you set down
before-hand.
If it were requisite to employ other numbers instead of that of 9, you
should, in order to succeed, warn the persons who chuse to set the
figures, to be attentive that their numbers do not exceed that agreed
upon.