How to play Sum Circles

Place 1 to 9 once each so every circle equals the total of the four squares around it. Also known as Circle sums, 3×3 sum square.

▶ Play Sum Circles ⊞ Open the solver

The rules

Each number from 1 to 9 is used exactly once (1 to 16 in the 4×4 version).
A circle sits where four squares meet and shows their total.
In the colour version, the squares of each colour add up to that colour’s total.
Given numbers can’t be changed.

Playing Sum Circles on Puzzle Solvers

Choose 3×3 circles, 3×3 with colours, or the big 4×4 (1–16), and a difficulty, then press New Puzzle.
Tap a square, then a number on the pad (or type it). Or tap a number first, then the square for it.
Circles turn green when their four squares add up, and red if they’ve gone over.
Hint lists the ways a circle can be made and which numbers that rules out.

Open the Sum Circles game →

Using the Sum Circles solver

Choose the version, type each circle total (and colour total) and any given numbers — or press Sample.
Press Solve and step through: the ways each circle can be made, the only digit or only place left, and what-ifs.

Open the solver →

A worked example

A circle shows 11. Four different numbers that make 11 must be 1+2+3+5 — there’s no other way.

So the four squares around that circle hold 1, 2, 3 and 5, and no other square can. If the corner of that circle also belongs to a circle of 26, it must be the 5.

Strategies

Extreme totals. The smallest possible circle is 10 (1+2+3+4) and the largest 30 (6+7+8+9). Totals near these have very few ways to be made.
The middle square. In the 3×3 grid the centre touches all four circles, the edge squares touch two, and the corners just one.
Add all four circles. The four circles together count the centre four times, each edge square twice and each corner once — a quick check on what the centre can be.
Shared squares. Two neighbouring circles share two squares, so the difference between their totals is the difference between their other two squares.
Colours cut the options. A colour total over two squares has only a few pairs — combine that with a circle to pin a square down.

Common mistakes

Using a number twice — every number appears once.
Forgetting that the centre square counts in every circle.
Adding the colour squares but forgetting one hidden behind a circle.

Where it comes from

Circle-total puzzles on a 3×3 grid were devised by Jai Gomer of Kobayaashi Studios and became a daily favourite in British newspapers from 2010. Versions with coloured groups and bigger grids followed.

Frequently asked questions

What maths does it practise?

Adding four numbers quickly, and partitioning: finding all the ways a total can be made from different numbers.

Is there always one answer?

Yes — every puzzle here is checked to have exactly one solution.

What is the 4×4 version?

Sixteen squares holding 1 to 16 with nine circles — the same rules on a bigger grid.

How do I solve one from a newspaper?

Open the Sum Circles Solver, type the circle totals and press Solve.

Keep going

All guides: How to play every puzzle and game.