Medium · Worked puzzle

Find eight answers by factoring ninety

Group the search by the outside multiplier instead of guessing three blanks independently.

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

(□ + □) × □ + 7 = 97

8 distinct solutions · 3 blanks

Try the board first, or open the hints below at your own pace. No answers are shown until you choose to reveal them.

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 8 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

Subtract seven first. The sum in brackets can be at most eighteen.

Read the worked approach

Subtract seven from 97 to obtain (a + b)c = 90. Since a + b is at most eighteen and c is a digit, only c = 5, 6 or 9 can work.
If c = 9, the sum is ten. Its five unordered digit pairs are 1 + 9, 2 + 8, 3 + 7, 4 + 6 and 5 + 5. Zero would need a partner of ten, which is not allowed.
If c = 6, the sum is fifteen: 6 + 9 and 7 + 8. If c = 5, the sum is eighteen: only 9 + 9 fits.
The groups contain five, two and one answers, giving eight solutions. Keep the outside multiplier in its own position; it is not interchangeable with a single term inside the sum.

A factor of ninety is not automatically a legal multiplier. For example, c = 3 would demand a sum of thirty.

Divide a large search into disjoint cases. Counting the solutions in each case also tells you when you are finished.

Reveal all 8 solutions

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

(9 + 1) × 9 + 7 = 97
(8 + 2) × 9 + 7 = 97
(7 + 3) × 9 + 7 = 97
(6 + 4) × 9 + 7 = 97
(5 + 5) × 9 + 7 = 97
(9 + 6) × 6 + 7 = 97
(8 + 7) × 6 + 7 = 97
(9 + 9) × 5 + 7 = 97

(9 + 1) × 9 + 7 = 97

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