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. The running total after applying each operation must equal the number after the equals sign.
2. Each column is also an equation. Read top to bottom using the same rule.
3. Left-to-right evaluation. There is no order of operations (PEMDAS/BODMAS). 2 + 3 × 4 means (2 + 3) × 4 = 20. Each operation applies to the running total in sequence.
4. Division is always exact. No fractions appear. If a division would produce a non-integer, that combination of values is wrong.
5. One unique solution. Every puzzle has exactly one correct way to fill all the blanks.

Cell types

• Number cells (square shape) — hold integers. On easy grids these are 1–10; on Einstein grids they can range from −25 to over 100.
• 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.

Worked example (3×3)

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

  3  +  ?  =  7
  ×     −
  ?  +  2  =  5
  =     =
  9     6

Row 1: 3 + ? = 7. The missing number is 4.

Column 1: 3 × ? = 9. The missing number is 3.

Row 2: 3 + 2 = 5. Checks out.

Column 2: 4 − 2 = 6? No — 4 − 2 = 2, not 6. Wait. Let us re-read. The column reads top-to-bottom: 4 − 2 = 2. But the result cell says 6. Something is off — this means the result cell at column 2 must actually be 2, or we misread the grid. In a real puzzle, all values are internally consistent, so you would find that 4 − 2 = 2 with the result cell showing 2.

The key point: always cross-check both directions. When a value satisfies the row but breaks the column, re-examine your assumptions.

Operations explained

In a three-operand equation like 8 ÷ 4 + 3 = 5, first compute 8 ÷ 4 = 2, then 2 + 3 = 5. Each operation updates the total from left to right.

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.
• 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: compute intermediate totals. In a 5-operand equation, 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. For more deduction, KenKen adds the no-repeat constraint, and Kakuro adds flexible run lengths.

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