THE "T" CARD PUZZLE.
(
Problems Concerning Games.)
An entertaining little puzzle with cards is to take the nine cards of a
suit, from ace to nine inclusive, and arrange them in the form of the
letter "T," as shown in the illustration, so that the pips in the
horizontal line shall count the same as those in the column. In the
example given they add up twenty-three both ways. Now, it is quite easy
to get a single correct arrangement. The puzzle is to discover in just
how many different ways it may be done. Though the number is high, the
solution is not really difficult if we attack the puzzle in the right
manner. The reverse way obtained by reflecting the illustration in a
mirror we will not count as different, but all other changes in the
relative positions of the cards will here count. How many different ways
are there?
Answer:
If we remove the ace, the remaining cards may he divided into two groups
(each adding up alike) in four ways; if we remove 3, there are three
ways; if 5, there are four ways; if 7, there are three ways; and if we
remove 9, there are four ways of making two equal groups. There are thus
eighteen different ways of grouping, and if we take any one of these and
keep the odd card (that I have called "removed") at the head of the
column, then one set of numbers can be varied in order in twenty-four
ways in the column and the other four twenty-four ways in the
horizontal, or together they may be varied in 24 x 24 = 576 ways. And as
there are eighteen such cases, we multiply this number by 18 and get
10,368, the correct number of ways of placing the cards. As this number
includes the reflections, we must divide by 2, but we have also to
remember that every horizontal row can change places with a vertical
row, necessitating our multiplying by 2; so one operation cancels the
other.
Informational
