Strings of glass
balls, as shown in the
illustration, are to be fired at.

What is the total number of different ways in
which these sixteen balls may be broken, if we must always break the
lowest ball that remains on any string?

Thus, one way
would be to
break all the four balls on each string in succession, taking the
strings from left to right.

Another would be to break all the fourth
balls on the four strings first, then break the three remaining on the
first string, then take the balls on the three other strings
alternately from right to left, and so on.

There is such a vast number
of different ways (since every little variation of order makes a
different way) that one is apt to be at first impressed by the great
difficulty of the problem.

Yet it is really quite simple when once you
have hit on the proper method of attacking it.

How many different ways
are there?