twice the length of a single square of the chessboard, and whose width is
the length of one square of the chessboard, so that each tile covers exactly
two adjacent squares of the chessboard.
It is easy to cover the chessboard exactly with 32 tiles, so that all squares
are covered and no tile extends beyond the boundary of the chessboard.
Since each row can be covered by laying four tiles end to end, do that for
each of the eight rows. Now, suppose that you remove the two squares at
the ends of a long diagonal from the chessboard; these might be the
square at the left end of the back row and the square at the right end of
the front row. This leaves a board that has only 62 squares remaining.
Can you cover this board exactly with 31 tiles, so that every square is cov-
ered?
As you might suspect from the lead-in to this section, or from some ex-
perimentation, this cannot be done; there is a simple, and elegant, reason
for this. Imagine that the chessboard is colored in the usual way, with al-
ternating black and red squares. Each tile covers precisely 1 black square
and 1 red square, so the 31 tiles will cover 31 black squares and 31 red
squares. If you look at a chessboard, the square at the left end of the back
row and the square at the right end of the front row have the same color
(we’ll assume they are both black), so removing them leaves a board with
32 red squares and 30 black squares—which the 31 tiles cannot cover. It’s
a simple matter of counting; the clever part is seeing what to count.
One of the reasons for the power of both science and mathematics is
that once a productive line of reasoning is established, there is a rush to
extend the range of problems to which the line of reasoning applies. The
above problem might be classed as a “hidden pattern”—it is obvious that
each tile covers two squares, but without the coloring pattern normally
associated with chessboards, it is not an easy problem to solve. Discover-
ing the hidden pattern is often the key to mathematical and scientific
discoveries.
When There Is No Music out There
We are all familiar with the concept of writer’s block: the inability of a
writer to come up with a good idea. The same thing can happen to math-
ematicians and scientists, but there is another type of block that exists for
the mathematician or scientist for which there is no analogy from the
arts. A mathematician or scientist may work on a problem that has no
answer. A composer might be able to come to grips with the idea that, at
the moment, he is incapable of composing music, but he would never ac-
cept the idea that there simply is no music out there to compose. Mathe-
xv Introduction