154 Combinatorial & Topological Problemsof the three houses, A, B, and C, without any pipe crossing another. Take
your pencil and draw lines showing how this should be done. You will soon
find yourself landed in difficulties.- CROSSING THE LINES
There is a little puzzle about which, for many years, I have perpetually re-
ceived enquiries as to its possibility of solution. You are asked to draw the
diagram in Figure 1 (exclusive of the little crosses) with three continuous
strokes of the pencil, with out removing the pencil from the paper during a
stroke, or going over a line twice.
As generally understood, it is quite impossible. Wherever I have placed a
cross there is an "odd node," and the law for all such cases is that half
as many lines will be necessary as there are odd nodes-that is, points from
which you can depart in an odd number of ways. Here we have, as indicated,~rr I~ I~
eight nodes, from each of which you can proceed in three directions (an odd
number), and, therefore,Jour lines will be required. But, as I have shown in
my book of Amusements, it may be solved by a trick, overriding the condi-
tions as understood. You first fold the paper, and with a thick lead pencil
draw CD and EF, in Figure 2, with a single stroke. Then draw the line from
A to B as the second stroke, and GH as the third!
During the last few years this puzzle has taken a new form. You are given