n that problem, which I have seen puzzle some excellent
accountants.

Think of a number; multiply by 3; if the result is odd, add 1 and divide
by 2; multiply by 3; if result be odd, add 1, and again divide by 2. By
how many 9's is the result divisible?

On receipt of that information you at once give the number thought of.
One of the most puzzling features of the trick is that no 9's are
obtainable in the result should either 1, 2, or 3 be thought of, as the
following will show:--

Number thought of 1 2 3
multiply by 3 3 3
--- ---
3 9
Add 1 1
--- --- ---
Divide by 2 4 6 10
2 3 5
Multiply by 3 3 3
--- ---
9 15
Add 1 1
--- --- ---
Divide by 2 6 10 16
3 5 8

As will be seen, none of these results is divisible by 9, yet the number
thought of is correctly given in each instance.

SOLUTION.--When the number thought of is multiplied by 3, you ask the
question, "Is the result odd or even?" If the answer is "odd," make a
mental note of _one_; then proceed. "Add one and divide by two. Is the
result odd or even?" If the answer is again "odd," make a mental note of
_two_; and proceed. "Add one and divide by two. How many nines are
obtainable in the result? I do not want to know what the surplus is."

The above figures illustrate that when 1 is the number thought of there
is only an addition of 1. When 2 is the figure, no addition is required
to the first result; but the second result being 9, 1 is added and _two_
noted, which, of course, is the figure thought of. When 3 is thought of
two additions are necessary, one to the 9 and one to the 15, making a
total of _three_ to be remembered, which represents the original number.
When 4 or any succeeding number is thought of the final result is always
divisible by 9, and in your mental calculation each 9 must represent 4,
to which you add the figures you have previously noted.

EXAMPLES.

Numb