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

by : $157$

 * 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, 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!


 * Dudeney-Modern-Puzzles-157.png


 * 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.