240.—­THE UNION JACK.

[Illustration: 

+-------+    +-----
A  B    |    |    /
\   |    |   /
|\   \  |    |  /   /|
| \   \ |    | /   / |
|  \   \|    |/   /  |
|   \   |    /   /   |
|    \  |\  /|  /    |
+-----\-|-\/-|-/-----+
\| /\ |/
|/  \/
|\  /\
/| \/ |\
+-----/-|-/\-|-\-----+
|    /   /  \|  \    |
|   /   |    \   \   |
|  /   /|    |\   \  |
| /   / |    | \   \ |
|/   /  |    |  \   \|
/   |    |   \
/    |    |    \
-----+    +-----

]

There are just sixteen points (all on the outside) where three roads may be said to join.  These are called by mathematicians “odd nodes.”  There is a rule that tells us that in the case of a drawing like the present one, where there are sixteen odd nodes, it requires eight separate strokes or routes (that is, half as many as there are odd nodes) to complete it.  As we have to produce as much as possible with only one of these eight strokes, it is clearly necessary to contrive that the seven strokes from odd node to odd node shall be as short as possible.  Start at A and end at B, or go the reverse way.

241.—­THE DISSECTED CIRCLE.

[Illustration: 

/---------------\
/                 \
/  /------B         \
/  /       |     /^\  \
/  /  |\    |    /   \  \
/  /   | \   |   /     \  \
/  /    |  \  |  /  A    \  \
/  /     |   \ | /   |     \  \
| /      |    \|/    |      \ |
| | -----+-----*-----+----- | |
| | \    |    /|\    |    / | |
| |  \   |   / | \   |   /  | |
| |   \  |  /  |  \  |  /   | |
| |    \ | /   |   \ | /    | |
| |     \|/    |    \|/     | |
D-+------*-----+-----*----E | |
|     /|\    |    /|\     | |
|    / | \   |   / | \    | |
|   /  |  \  |  /  |  \   | |
|  /   |   \ | /   |   \  | |
| /    |    \|/    |    \ | |
| -----+-----*-----+----- | |
\      |    /|\    |      / |
\     |   / | \   |     /  /
\    |  /  |  \  |    /  /
\   | /   |   \ |   /  /
\  |/    |    \|  /  /
\       |       /  /
\------+------/  /
|        /
C-------/

]

It can be done in twelve continuous strokes, thus:  Start at A in the illustration, and eight strokes, forming the star, will bring you back to A; then one stroke round the circle to B, one stroke to C, one round the circle to D, and one final stroke to E—­twelve in all.  Of course, in practice the second circular stroke will be over the first one; it is separated in the diagram, and the points of the star not joined to the circle, to make the solution clear to the eye.

242.—­THE TUBE INSPECTOR’S PUZZLE.

The inspector need only travel nineteen miles if he starts at B and takes the following route:  BADGDEFIFCBEHKLIHGJK.  Thus the only portions of line travelled over twice are the two sections D to G and F to I. Of course, the route may be varied, but it cannot be shortened.

243.—­VISITING THE TOWNS.