Medium · Worked puzzle
Look just below thirty-four
Use a nine-digit remainder window to find five multiplication-and-addition answers.
□ × □ + □ = 34
5 distinct solutions · 3 blanks
Play this puzzleTry 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 5 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
The last blank can contribute between zero and nine, so the product cannot be far below thirty-four.
Read the worked approach
- Rearrange ab + c = 34 as ab = 34 − c. Since c is from zero to nine, the product must lie between 25 and 34 inclusive.
- The products in that interval made from two digits are 25, 27, 28, 30 and 32. Their unordered factor pairs are 5 × 5, 3 × 9, 4 × 7, 5 × 6 and 4 × 8.
- Subtract each product from thirty-four. The remainders are respectively 9, 7, 6, 4 and 2. Each remainder fits the last blank.
- Numbers such as 26 and 33 have factors outside the permitted range. Reversing any multiplication pair does not make an extra solution, leaving five distinct answers.
A common trap
Do not assume the last blank must be small. The square 5 × 5 needs the largest possible remainder, nine.
Carry this idea to the next puzzle
Search the multiplication table within a narrow interval, then calculate the last blank instead of guessing it.
Reveal all 5 solutions
One representative of each distinct solution is shown. Other accepted orders of the same addition or multiplication are not repeated.
- 9 × 3 + 7 = 34
- 7 × 4 + 6 = 34
- 8 × 4 + 2 = 34
- 5 × 5 + 9 = 34
- 6 × 5 + 4 = 34
Check one answer, step by step
9 × 3 + 7 = 34
- 9 × 3 = 27
- 27 + 7 = 34