Number grid puzzle rules

What is a number grid puzzle?

A number grid puzzle is a square grid where each row and each column forms an arithmetic equation. The grid alternates between number cells (squares) and operation cells (circles), ending each row and column with an equals sign and a result. Some cells are given; the rest are blank. Your task is to fill in the blanks so every equation is correct.

The concept is sometimes called a math grid puzzle, an arithmetic puzzle, or loosely a cross number puzzle — though "cross number" and "crossnumber" can also refer to crossword-shaped puzzles with mathematical clues, which is a different format.

The rules

1. Each row is an equation. Read left to right: number, operation, number, operation, ... = result. Whatever the equation works out to must equal the number after the equals sign.
2. Each column is also an equation. Read top to bottom using the same rule.
3. Ordinary order of operations. The usual precedence applies (PEMDAS/BODMAS): × and ÷ before + and −. 2 + 3 × 4 means 2 + 12 = 14, not (2 + 3) × 4 = 20.
4. Division is always exact. No fractions appear. If a division would produce a non-integer, that combination of values is wrong.
5. The numbers inside the grid are a fixed set, each used once. Exactly like sudoku, where every box holds each of 1–9. A 3×3 grid uses each of 1–9, a 4×4 uses each of 1–16, a 5×5 each of 1–25, and a 6×6 each of −12 to −1 and 1 to 24. The results after the equals signs are not part of the set.
6. One unique solution. Every puzzle has exactly one correct way to fill all the blanks — and the fixed set is part of what guarantees it, because on some grids a blank is pinned by nothing else.

Which numbers go in the grid: the pool

The numbers you write inside the grid are not free. Each level has one pool, the same pool on every puzzle at that level, and a solved grid uses each number in it exactly once. Think of it the way you think about sudoku: you do not merely know the digits cannot repeat, you know they are 1 to 9.

Only the inside counts. The pool covers the block of numbers you fill in. The results down the right-hand side and along the bottom are totals — whatever the arithmetic makes them. They sit outside the pool, they can be large or negative, and two of them can be the same number.

Why "no repeats" is not a strong enough way to say it. On some easy grids a blank sits in a row whose total is also blank and a column whose total is also blank. Nothing in the arithmetic pins it: write 42 there and every equation still comes out right, and 42 does not repeat any number on the board. Only "use each of 1 to 9 exactly once" rules it out and leaves one answer standing. So the rule is about which numbers, not just about repeats.

Cell types

• Number cells (square shape) — hold the pool numbers. On easy and medium grids that is 1–9; on Einstein grids it is −12 to −1 and 1 to 24. Each one is used exactly once.
• Operation cells (circular shape) — hold one of +, −, ×, or ÷. On easy and medium grids all operations are given. From hard onward, some are blank.
• Equals cells — always pre-filled. They separate the equation from its result. A green checkmark appears when the equation is correct; a red ✗ when it is not.
• Result cells (square, at the end of each row/column) — hold the value the equation should equal. Sometimes given, sometimes blank. These are totals, so they are outside the pool and two of them may be the same.

Worked example (3×3)

Suppose you see this easy grid (blanks shown as ?):

  9  −  4  +  2  =  7
  +     +     +
  8  +  1  −  5  =  4
  +     +     +
  6  +  3  −  ?  =  ?
  =     =     =
  23    8     ?

Check what is already there. Row 1 is 9 − 4 + 2 = 7. Row 2 is 8 + 1 − 5 = 4. Column 1 is 9 + 8 + 6 = 23 and column 2 is 4 + 1 + 3 = 8. All correct, so the given numbers hold together.

The last blank is not pinned by arithmetic. Its row reads 6 + 3 − ? = ? and its column reads 2 + 5 + ? = ?. Both results are blank, so any value works: put 42 in and the row comes to −33, the column comes to 49, and nothing is contradicted. 42 does not repeat a number on the board either.

The pool decides it. An easy grid uses each of 1 to 9 exactly once. Already on the board: 9, 4, 2, 8, 1, 5, 6, 3. The only number left is 7, so that is the answer — and the two blank results follow from it: 6 + 3 − 7 = 2 and 2 + 5 + 7 = 14.

Two habits come out of this. Cross-check every value against both its row and its column, and keep a running list of which pool numbers you have already placed. On the bigger grids that list does as much work as the arithmetic.

Operations explained

In a three-operand equation like 8 + 4 ÷ 2 = 10, the division goes first: 4 ÷ 2 = 2, then 8 + 2 = 10. Reading order decides which numbers sit where; precedence decides which operation is worked out first.

Grid sizes by difficulty

Solving strategies

• Single-blank equations first. If a row or column has one unknown, compute it directly. This often reveals another single-blank equation somewhere else.
• Work backwards from the result. If a row reads ? × 3 = 12, the unknown is 12 ÷ 3 = 4.
• Tick off the pool as you go. Write out the level's numbers (1–9 on a 3×3, 1–16 on a 4×4, and so on) and cross each one off as it appears. A blank that the equations cannot settle is often settled by what is left on that list, and on the last few cells it can settle them outright.
• Use division as a constraint. Division must produce an integer. If a position only allows a ÷ that would result in a fraction, that value or operation is wrong.
• For blank operations: try each one. Usually only one or two produce an integer and match the crossing equation.
• On large grids: collapse each × and ÷ into a single value first. A 5-operand equation then becomes a short chain of additions and subtractions, and knowing the first three values narrows the last two considerably.

Number Grid vs KenKen vs Kakuro

All three involve filling numbers into grids using arithmetic, but the formats differ:

Number Grid is the most accessible of the three if you just want to practice basic arithmetic. All three limit which numbers you may use; the difference is where. Number Grid spends one pool across the whole grid, KenKen repeats 1–N in every row and column, and Kakuro restarts 1–9 in every run.

Frequently asked questions

Related puzzle rules

• KenKen rules — Cage arithmetic with Latin square constraints
• Kakuro rules — Cross-sum number puzzles
• Killer Sudoku rules — Sudoku with cage sums
• Futoshiki rules — Latin square with inequality constraints