One Line

The trick is where you start

One Line feels like trial and error until you learn the rule that solves most levels before you draw anything. Count the lines meeting at each dot. A dot where an odd number of lines meet is an odd dot. A shape can be traced in one unbroken stroke only if it has zero odd dots or exactly two. If it has two, you must start at one of them, and you will always finish at the other. So don't start in the middle of a symmetric shape and hope. Find the odd dots first, start on one, and the path usually falls into place.

If every dot is even, the rule is kinder still. Start anywhere you like and your stroke will always loop back to where it began. That's why some levels close on themselves while others force you corner to corner. All-even levels are the forgiving ones, since a failed attempt only means your route was wrong, never your starting dot. The Hint button points you at a sound starting dot, which is really just handing you one of the odd ones.

Try it on the little house

German children learn this rule from a doodle called the House of Santa Claus: a square, two roof lines, and both diagonals of the square, drawn in one stroke of eight lines. Count the meetings and the secret shows itself. The two bottom corners each touch three lines, so they're the odd dots. Every other point touches an even number. Start at a bottom corner and the house draws itself; start anywhere else and it always fails, no matter how clever the route. People have even counted the possibilities: from one bottom corner there are exactly 44 different orders to draw it. Every level in this game is the same kind of object wearing a different costume, so the same counting cracks it.

Don't burn your bridges

Knowing the starting dot doesn't finish the busy levels, so here are the habits that do. A dot touched by only one line is odd by definition, so it has to be the start or the finish of your whole stroke. If a level has a lone spur sticking out, begin there or plan to end there, never in between. And borrow the big rule from Fleury's method, a recipe for one-stroke drawing published back in 1883: never cross a bridge too soon. A bridge is a line that is the only link between two parts of the shape. Once you cross it you can't come back, so finish everything on your side before you do.

• You can pass through a dot as many times as you like. Only the lines are limited, each drawn exactly once.
• When two routes look equal, take the one that doesn't strand a far corner you still need.
• Stuck halfway? Reset and start from the other odd dot. Direction alone is often the whole fix.

Busy junctions deserve a little extra thought too. A dot with four or six lines will be visited more than once, and each visit uses up one way in and one way out. On your last visit, make sure leaving doesn't cut you off from lines you still owe. Slowing down at the big crossings is worth more than speed anywhere else, and it's what turns the last of the twelve levels from a maze into a checklist.

Euler solved your level in 1736

This game is the mathematics of the Seven Bridges of Konigsberg. That old Prussian city (today Kaliningrad, Russia) had seven bridges joining its riverbanks and two islands, and townsfolk wondered whether a single walk could cross every bridge exactly once. In 1736 Leonhard Euler proved it couldn't, using the very counting you now use: all four pieces of land had an odd number of bridges touching them, and anything more than two odd spots makes a route impossible. His short proof founded graph theory, the branch of math that now sits underneath route planners and network design. Every level of One Line asks Konigsberg's question with a happier answer, and the odd-dot rule you're using to crack it is Euler's own, dressed up as a game.