Expert · Worked puzzle

Six answers from two factor cases

Solve a sum-times-difference puzzle by undoing its final division.

Mathoji.com · 16 October 2026 · Math checked against the game’s rules

(□ + □) × (□ − □) ÷ 7 = 18

6 distinct solutions · 4 blanks

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Before you play

Each blank takes one digit from 0 to 9. Choices may be repeated.
Follow the printed brackets and the order of operations. Each division must be exact. Division by zero is never allowed.
Find all 6 distinct solutions. Reordering terms of the same addition or multiplication counts once. Subtraction, division and bracket structure still matter.
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Show a starting hint

Multiply eighteen by seven, then find factor pairs that fit a sum and a difference.

Read the worked approach

The numerator must be 18 × 7 = 126. Let S be the first sum and D the second difference. S is at most eighteen and a positive D is at most nine.
The only fitting assignments are S = 14, D = 9 and S = 18, D = 7. Other factor pairs of 126 exceed one of those bounds.
For S = 14, the unordered sums are 5 + 9, 6 + 8 and 7 + 7. Pair each with the only difference of nine, 9 − 0. That gives three solutions.
For S = 18, use 9 + 9. A difference of seven can be 7 − 0, 8 − 1 or 9 − 2. These give three more solutions. Every numerator is 126, so the final division is exact.

A difference of seven has more than one digit pair. Stopping at 7 − 0 misses two answers.

Factor cases tell you which totals are possible; listing every pair for each total completes the search.

Reveal all 6 solutions

One representative of each distinct solution is shown. Other accepted orders of the same addition or multiplication are not repeated.

(9 + 5) × (9 − 0) ÷ 7 = 18
(8 + 6) × (9 − 0) ÷ 7 = 18
(7 + 7) × (9 − 0) ÷ 7 = 18
(9 + 9) × (7 − 0) ÷ 7 = 18
(9 + 9) × (8 − 1) ÷ 7 = 18
(9 + 9) × (9 − 2) ÷ 7 = 18

(9 + 5) × (9 − 0) ÷ 7 = 18

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