Mathematical_Recreations-Kraitchik-2e

270 Mathematical Recreations
The reader will probably have noticed that the rule for
forming this table is just like the rule for forming the binomial
coefficients by Pascal's triangle. So it is not surprising
that the numbers in the third row from the bottom are the
coefficients in the expansion of (a + b)2, those in the fourth
row are the coefficients of (a + b)3, and so on. The connection
lies deeper than this, however, and it is this deeper
meaning that we want to discuss.
Suppose we use the symbol u to mean that the man is 1
column to the right of his original position, and use u-1 to
mean that he is 1 column to the left. In the same way u 2 will
mean 2 to the right, u-2 will mean 2 to the left, and so on.
Also, UO(= 1) will indicate no change of column. Now let
us use addition to represent the logical notion" or." x-1 + x
will then mean: "1 column to the left or 1 column to the
right," which is precisely the statement of the alternatives
presented to the man at every move. (We do not need to
bother about what row he moves to, since we know that he
always moves up one row.) Next we can indicate the fact
that he makes more than one move by using multiplication
to denote the logical relation" and." Thus, if he makes two
moves he makes one move and he makes another move, and
we can show this by writing (u- 1 +- u)(u-1 + u), that is,
(u-1 + U)2. Expressed without abbreviation this says: "He
moves one column to the left or right and he moves again
one column to the left or right."
Suppose we multiply this expression out (first without
collecting terms) and try to interpret the result in the same
logical terms. (u-1 + U)2 = u-2 + u-1u + uu-1 + u 2 then may
be interpreted: "Either he moves 2 columns to the left or he
moves 1 column to the left and 1 to the right or he moves 1
to the right and 1 to the left or he moves 2 to the right."
But just as u-1u = uu-1 = Uo = 1, so "1 to the left and 1 to
the right" has the same result as "1 to the right and 1 to the