The Second Book of Mathematical Puzzles and Diversions

I)i.rlitnl Roots 47
The key numbers in the strategy are those which have
digital roots that are the same as the digital root of the goal.
If you can score a number in this series, or permanently pre-
vent your opponent from doing so, you have a certain win.
For example, the game is often played with the goal of 31,
which has a digital root of 4. The only way the first player
can force a win is by rolling a 4. Thereafter he either plays
to get back in the series 4-13-22-31, or plays so that his op-
ponent cannot enter it. Preventing an opponent from enter-
ing the series is somewhat tricky, so I shall content myself
with saying only that one must either play to five below a
key number, leaving the 5 on the top or the bottom of the
die ; or to four or three below, or one above, leaving the 4 on
the top or the bottom.
There is always one roll, and sometimes two or three,
which will guarantee a win for the first player, except when
the digital root of the goal happens to be 9. In such cases
the second player can always force a win. When the goal is
chosen at random, the odds of winning greatly favor the
second player. If the first player chooses the goal, what
should be the digital root of the number he picks in order to
have the best chance of winning?
A large number of self-working card tricks depend on the
properties of digital roots. In my opinion the best is a trick
currently sold in magic shops as a four-page typescript titled
"Remembering the Future." It was invented by Stewart
James of Courtright, Ontario, a magician who has probably
devised more high-quality mathematical card tricks than
anyone who ever lived. The trick is explained here with
James's permission.
From a thoroughly shuffled deck you remove nine cards
with values from ace to 9, arranging them in sequence with
the ace on top. Show the audience what you have done, then
explain that you will cut this packet so that no one will know