Trace every line exactly once without lifting your pencil.
Six of these can be traced without lifting your pencil, and six can't. How can you tell which?
Click empty space to add a dot. Click two dots to join them; joining them again adds a second line. Click a line to remove it. Drag a dot to move it.
Forget the shapes. Look only at the dots, and at how many lines meet at each one. What do the six figures that work have in common?
Count the lines that meet at each dot. The secret is whether the count is odd or even.
Each time your pencil passes through a dot, it uses two lines there: one in, one out. So at every dot except the start and the finish, the lines pair up, and the count is even.
No odd dots: start anywhere, and you finish where you began. Two odd dots: start at one and finish at the other. More than two: it can't be done.
Of the five Platonic solids, only the octahedron can be traced. It has four lines at every corner; the others have three or five.
In 1736 Leonhard Euler showed that no walk could cross the seven bridges of Königsberg exactly once, and graph theory began. Only five bridges stand at those sites today, so the walk is now possible. German children draw the House of St. Nicholas one syllable per line: Das ist das Haus vom Ni-ko-laus. Mail and snowplow routes are planned the same way.