Graceful

Graceful takes its puzzle from graph theory. The board is a tree: circles joined by lines, with no loops. You give each circle its own number, from 0 up to the number of lines, using no number twice. Every line then takes the difference between the two circles it joins, so a line between 7 and 2 is worth 5. You win when no two lines share a value, which means that with ten lines the values come out as exactly 1 to 10, each once. Lines that clash glow in the same colour, and you can undo as much as you like.

The idea is called graceful labelling. Every tree anyone has checked can be numbered this way, and mathematicians believe they all can, but the graceful tree conjecture has been open since the 1960s. Each daily is a small case of it.

There is one puzzle a day, numbered (Graceful #240, for example), the same for everybody. Monday opens easy and the week builds to Sunday's largest board, around twenty lines or more. Solving keeps a streak, and the result can be shared as a grid of blocks showing the shape of your solve without any numbers. The last seven days are always free to play; a single purchase, not a subscription, unlocks the full archive, and today's puzzle never needs it.

Graceful is made by Tim Davis and came out on the App Store in August 2026. It has no accounts, no ads and no timers pushing you, and works with no signal. It suits players who like number logic and a little real mathematics.