Mathematical_Recreations-Kraitchik-2e

318 Mathematical Recreations
cards each. This merely permutes the rows and columns of
the first distribution, as in Figure 177, so that when you are
told in which row the card now lies, you know that it lies in
the column whose number was given for the row in the previous
distribution.
You can also ask your victim in what order he wishes his
card to appear. Suppose he answers that it is to be the sixteenth
card. Express 16 - 1 = 15 in the ternary scale, that
is, as 120. When he has indicated the pile after the first deal,
pick up the cards so that there are 0 piles above the selected
FIGURE 178.
pile. The second time pick up the cards so that there are 2
piles over the selected pile. The third time place 1 pile over
the selected pile. On the fourth deal the selected card will be
in position (0 + 2·3 + 1·9) + 1 = 16. Goormaghtigh found
that a similar method is possible with 45, 52, ... cards.
Gergonne generalized the game so that one uses mm cards
dealt in m piles of mm-l each. His result is: If a, b, c,· .. are
the numbers of the piles in which the selected card lies, then
the position N of the card in the mth deal is given by
N = 1 - a + bm - cm2 + dm3 • •• , if m is even;
N = a - bm + cm2 - dm3 • • " if m is odd.
12. To GUESS Two CARDS. Deal out n(n + 1) cards in
pairs and ask someone to select one of the pairs. Collect the