Graceful Tree

Put the numbers on the dots so that no two lines have the same difference — and find out where 1 and the biggest number have to go.

The Graceful Tree board

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Grades
2–6
Subjects
Adding & Subtracting, Logic
Kind
Puzzle
Cost
Free, no account, no ads

What it is

The board is a drawing of dots joined by lines, and a tray holds one odd number for every dot: 1, 3, 5 and on up. Drag the numbers onto the dots, one on each. Every line then has a difference, the bigger number at its ends take away the smaller, and the drawing is finished when no two lines have the same one. Drop a number on a dot that already has one and the two swap places. The drawing is springy: a number landing gives its dot a push, and the dot wobbles and settles back. Done counts the differences up from 2, lighting each line as it gets there, and stops at the first difference it finds twice.

The rules

Put the numbers on the dots so that every line has a different difference.

Three dots joined by two lines, and the numbers 1, 3 and 5 in a row under them. The 3 goes onto the left dot, the 1 onto the middle dot and the 5 onto the right dot.315
Put one number on every dot.
Three dots numbered 3, 1 and 5. The line from 3 to 1 lights up and 2 is written on it; then the line from 1 to 5 lights up and 4 is written on it.31524
A line's difference is the bigger number take away the smaller.
Three dots numbered 1, 3 and 5: both lines have the difference 2, and both turn red. Then the 1 and the 3 swap places, and the lines have the differences 2 and 4.513222224
No two lines may have the same difference.

What it gives you to practise

Subtraction in the two times table, and a reason to do a lot of it. An odd number take away an odd number is always even, so the differences are 2, 4, 6 and so on, and every number put down is checked against each of its neighbours. Then the reasoning: on a drawing in one piece the biggest difference can be made only one way, the biggest number beside 1, so those two must share a line, and the next difference down narrows things again. A starfish is solved by one rule at any size, its middle dot 1 or the biggest number, and so is a snake, zig-zagging 1, the biggest, 3, the next biggest down. Help writes each line’s difference on it and keeps a strip of the differences still to be made.

In a classroom

A finished drawing is what mathematicians call a graceful labelling (Alexander Rosa, 1967), with each number doubled and one added to make it odd. Every tree, a drawing in one piece with no loops, is conjectured to have one; nobody has proved it, and computers have checked every tree up to 35 dots. The 37 boards run from four-dot trees through starfish, snakes and bigger trees to Fossils: sixteen drawings in two or more pieces, like a skeleton dug up in bits, with loops, most of them with a few numbers already written in that cannot be moved. Where the biggest number can go is a good first question for a class. The board prints as the bare drawing, with any numbers it gives, for a pencil.

Share or embed

The whole puzzle at one address, with every board in it. Whoever opens it stays inside it, and their progress stays in their own browser.

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