Hard · Worked puzzle

A difference has a smaller ceiling

Use the different ranges of subtraction and addition to place the factors of 117.

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

(□ − □) × (□ + □) = 117

3 distinct solutions · 4 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 3 distinct solutions. Reordering terms of the same addition or multiplication counts once. Subtraction, division and bracket structure still matter.
Play without an account. Progress stays in this browser; the Lobby’s Continue button resumes an unfinished game. Web Hint and Reveal are free.

Show a starting hint

A positive difference of two keypad digits is at most nine, while their sum can reach eighteen.

Read the worked approach

Let D = a − b and S = c + d. Because the target is positive and the sum is nonnegative, D must be positive too. We need DS = 117.
The factor pairs are 1 × 117, 3 × 39 and 9 × 13. The first two demand a sum above eighteen, so only D = 9 and S = 13 are possible.
A difference of nine can only be 9 − 0. A sum of thirteen has three unordered possibilities: 4 + 9, 5 + 8 and 6 + 7.
Combine the fixed difference with each sum. Reversing the sum repeats a solution; reversing the subtraction gives a negative result and is invalid.

Subtraction is not interchangeable. The positions of 9 and 0 in the first bracket are essential.

Use an operation’s range as well as the target’s factors. A sum and a difference need not accept the same factor.

Reveal all 3 solutions

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

(9 − 0) × (9 + 4) = 117

Found a problem with an explanation? Tell us which step needs a look.