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9×9 · Misc

Counting Circles Sudoku

Each circled cell's digit equals the number of circles across the grid that contain that same digit.

3 3 3 1

An example Counting Circles puzzle: each circled digit says how many circles hold it — three circles show 3, and one circle shows 1.

Counting Circles Sudoku draws circles on some cells with a wonderfully self-referential rule: the digit inside a circle equals how many circles on the whole grid contain that same digit. A circle showing 3 is telling you there are exactly three circles holding a 3 — itself included. Cells without a circle don't count at all.

The rule: the digit counts its own circles

Read any circled digit as a tally. If a circle holds 3, then across the grid exactly three circled cells contain a 3. If it holds 1, it is the only circle with a 1. Below, four circles read 1, 3, 3, 3 — consistent, because one circle holds the 1 and three circles hold a 3.

◯1◯3◯3◯3

Four circles reading 1, 3, 3, 3: one circle holds the 1, and three circles hold a 3 — consistent.

A worked example, step by step

Suppose one circle is already known to be a 1.

◯1◯?◯?◯?

Step 0. One circle is a 1; three others are unknown.

Step 1 — the 1 is unique. A circle of 1 says exactly one circle holds a 1. Since this circle is that one, no other circle can be a 1.

◯1◯no 1◯no 1◯no 1

Step 1. The 1 is the only circle allowed to be a 1 — the others can't.

Step 2 — count the rest. The three remaining circles must be self-consistent: if one of them is a 3, then exactly three circles hold a 3, so all three must be 3s. The tally forces them together, and the whole set reads 1, 3, 3, 3.

Why the counting is so tight

Every circled digit is both a value and a constraint on the whole set of circles, so they lock into a small number of consistent patterns. A high digit like 8 is rare — it would need eight circles holding an 8 — while low digits appear in tight bunches (a lone 1, a pair of 2s, a trio of 3s). Counting the circles of each value is usually the fastest way in.

Solving strategy

  • Read every circle as a tally. A digit d means exactly d circles hold d.
  • Start with the 1. A circled 1 is unique — it bans that value from all other circles.
  • Group the counts. Digits appear in bunches of their own size: one 1, two 2s, three 3s, and so on.
  • Feed the grid. Each circled digit placed behaves like a given for its row, column and box.

Common mistakes

  • Counting uncircled cells. Only circled cells count toward the tally — a 3 elsewhere is irrelevant.
  • Forgetting the circle counts itself. A circle of 3 is one of the three 3s it refers to.
  • Allowing too many. A digit d can appear in at most d circles — and must appear in exactly d.
  • Ignoring ordinary Sudoku. Circled cells still obey their rows, columns and boxes.

Can I play it here?

Not yet — Counting Circles Sudoku isn't one of the puzzle types SudokuStreak generates for now. But you can learn its rules above, and there's plenty here to sink your teeth into meanwhile: