Choose a starting node, then click a connected node to walk along the edge between them. Each edge can be used only once. Use Undo to take back the last step or Reset Path to start the attempt over.

You succeed when every edge has been walked. If the countdown reaches zero first, the attempt fails.

There are three fixed layouts. Easy is a square with 4 nodes and 4 edges, Medium has 6 nodes and 10 edges, and Hard has 8 nodes and 12 edges. The time limits are 90, 60 and 30 seconds.

Where you start matters on Medium. That layout has two nodes with three edges each, the bottom-left and bottom-right corners, and any full route has to begin at one of them and finish at the other.

Your score starts at a base value: 500 on Easy, 900 on Medium and 1,400 on Hard. You lose 4 points for every second you take and gain 5 points for every second that was left on the clock, with a minimum of zero.

Your best score is saved in your browser under the username you pick. If you sign in, personal bests can also sync to your account.

On Easy and Hard, every node has an even number of edges. That has a useful consequence: once you start walking, the only place you can get stuck is back at your starting node. Every other node always offers a way out, because you used one edge to arrive and an even number leaves at least one more.

So the real danger is returning to the start too early. If you close a loop and the start has no unused edges left while some edges are still untouched elsewhere, undo back to the point where you chose to close the loop and take another branch.

On Medium, begin at a bottom corner. The walk will end at the other bottom corner, so you cannot close a loop there and you should not finish early anywhere else.

The puzzle comes from a famous problem solved by Leonhard Euler in 1736. The city of Königsberg had seven bridges, and the question was whether a walker could cross each exactly once. Euler showed it was impossible, and in doing so founded graph theory.

His rule is that a connected graph has a walk that uses every edge exactly once only if zero or two of its nodes have an odd number of edges. If none are odd, the walk can start anywhere and ends where it began, a so-called Eulerian circuit. Easy and Hard here have only even nodes, so you can start on any node. Medium has exactly two odd nodes, the bottom corners, so its walk must start at one and end at the other.

• Count the edges at each node before you click. Nodes with more edges are junctions you will pass through several times.
• Prefer outer edges first and leave central connections for later, so you still have a way to link the loops you create.
• Walk a loop and look at what is left. If the remaining edges are all attached to nodes you have already visited, you can usually pick them up on the way back.
• On Hard you only have 30 seconds, so decide on a route in your head, then click quickly.

Every second costs 4 points, plus you lose the 5-point bonus for time left. That is nine points per second, so one wasted second matters, but a reset costs you everything you have used so far. A few seconds of planning before the first click almost always beats starting fast and undoing.