How does the one line puzzle work?
Each figure is a set of points joined by lines. Tap a point to start, then tap the next point along a line, and the line is drawn. Every line has to be drawn exactly once, and you can never jump: each new line starts where the last one ended. Undo takes back the last line and Restart clears the figure. There are 6 figures, from a square with one diagonal to a window with fourteen lines.
How is your result calculated?
Your result is the time from pressing Start to drawing the last line of the last figure. Restarts and undos aren't penalised, but both are counted and shown with your time.
Why can some shapes be drawn in one line and others not?
Count the lines meeting at each point. A figure can be drawn in one line exactly when it is all in one piece and the number of points where an odd number of lines meet is zero or two. If there are two, you must start at one of them and you will finish at the other. If there are none, you can start anywhere and you will end where you began.
The reason fits in a few sentences. Every time your line passes through a point, it uses two of the lines there: one in, one out. Only the start and the finish get a single line without a partner. So only those two points can have an odd count. This rule comes from Leonhard Euler, who used it in 1736 to show that nobody could walk across each of the seven bridges of Königsberg exactly once.
Every figure on this page passes Euler's test, and the site's tests check that each one really can be drawn. After 2 restarts on the same figure, the points with an odd number of lines are marked for you.
How do you solve one line puzzles?
- Find the odd points first. If two points have an odd number of lines, start at one of them. Starting anywhere else can never work.
- Leave bridges for last. If drawing a line would cut the remaining lines into two separate pieces, take another line first if you can.
- Try the house. The figure called the house, a square with a roof and both diagonals, is a classic: in Germany children draw it while saying "Das ist das Haus vom Ni-ko-laus", one syllable per line.
This is a recreational puzzle. For another puzzle about a route, try the Trail Making Test; for another classic logic puzzle, the River Crossing Puzzle.