Henry Ernest Dudeney/Modern Puzzles/157 - Crossing the Lines

Modern Puzzles by Henry Ernest Dudeney: $157$

Crossing the Lines

There is a little puzzle about which, for many years, I have perpetually received enquiries as to its possibility of solution.

You are asked to draw the diagram of Figure $1$ (exclusive of the little crosses) with three continuous strokes of the pencil,

without 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, $8$ odd nodes, from each of which you can proceed in three directions (an odd number),

and therefore, four lines will be required.

But, as I have shown in my book of Amusements in Mathematics, it may be solved by a trick, overriding the conditions 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 the same diagram and asked to start where you like and try to pass through every short line comprising the figure,

once and once only, without crossing your own path.

Figure $3$ will make quite clear what is meant.

It is an attempted solution, but it fails because the line from $K$ to $L$ has not been crossed.

We might have crossed it instead of $KM$, but that would be no better.

Is it possible?

Many who write to me about the puzzle say that though they have satisfied themselves as a "pious opinion", that it cannot be done,

yet they see no way whatever of proving the impossibility, which is quite another matter.

I will show my way of settling the question.

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Unicursal and Route Problems: $157$. -- Crossing the Lines
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Combinatorial & Topological Problems: Route & Network Puzzles: $414$. Crossing the Lines