The puzzle was to
divide the circular field into
four equal parts by three walls, each wall being of exactly the same
There are two essential difficulties in this problem.
These are: (1) the thickness of the walls, and (2) the condition that
these walls are three in number.
As to the first point, since we are told that the walls are brick
walls, we clearly cannot ignore their thickness, while we have to find
a solution that will equally work, whether the walls be of a thickness
of one, two, three, or more bricks.
The second point
requires a little more
How are we to distinguish between a wall and walls?
straight wall without any bend in it, no matter how long, cannot ever
become "walls," if it is neither broken nor intersected in any
Also our circular field is clearly enclosed by one wall. But if it had
happened to be a square or a triangular enclosure, would there be
respectively four and three walls or only one enclosing wall in each
It is true that we speak of "the four walls" of a square building
or garden, but this is only a conventional way of saying "the four
If you were speaking of the actual brickwork, you would say, "I
am going to enclose this square garden with a wall."
Angles clearly do
not affect the question, for we may have a zigzag wall just as well as
a straight one, and the Great Wall of China is a good example of a wall
with plenty of angles.
Now, if you look at Diagrams 1, 2, and 3, you
may be puzzled to declare whether there are in each case two or four
new walls; but you cannot call them three, as required in our
The intersection either affects the question or it does not affect it.
If you tie two
pieces of string firmly together,
or splice them in a nautical manner, they become "one piece of
If you simply let them lie across one another or overlap, they remain
"two pieces of string."
It is all a question of joining and welding.
may similarly be held that if two walls be built into one
another—I might almost say, if they be made
homogeneous—they become one wall, in which case Diagrams 1,
2, and 3 might each be said to show one wall or two, if it be indicated
that the four ends only touch, and are not really built into, the outer
The objection to
Diagram 4 is that although it
shows the three required walls (assuming the ends are not built into
the outer circular wall), yet it is only absolutely correct when we
assume the walls to have no thickness.
A brick has thickness, and
therefore the fact throws the whole method out and renders it only
Diagram 5 shows,
perhaps, the only correct and
perfectly satisfactory solution.
It will be noticed that, in addition
to the circular wall, there are three new walls, which touch (and so
enclose) but are not built into one another.
This solution may be
adapted to any desired thickness of wall, and its correctness as to
area and length of wall space is so obvious that it is unnecessary to
I will, however, just say that the semicircular piece of
ground that each tenant gives to his neighbour is exactly equal to the
semicircular piece that his neighbour gives to him, while any section
of wall space found in one garden is precisely repeated in all the
Of course there is an infinite number of ways in which this
solution may be correctly varied.