Easy · Worked puzzle
Why six arrangements make three solutions
A simple addition shows exactly how Mathoji counts unique answers.
□ + □ = 5
3 distinct solutions · 2 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 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
Start with zero and keep one side no larger than the other.
Read the worked approach
- Both blanks accept digits from zero through nine. Start with 0 + 5. Increase the first blank by one and reduce the second by one to keep the total unchanged.
- The next pairs are 1 + 4 and 2 + 3. All three are valid, and each uses a different pair of values.
- The next arrangement, 3 + 2, repeats the pair 2 + 3. The same is true of 4 + 1 and 5 + 0. Mathoji counts swapping the terms of this addition only once.
- A first value above five would need a negative second value. Those values are not on this keypad. The three pairs therefore cover the entire permitted range.
A common trap
Zero is a valid choice. Leaving out 0 + 5 would miss a solution even though the other pairs look more familiar.
Carry this idea to the next puzzle
An ordered list prevents both missed answers and repeated answers. The game accepts either order of each pair.
Reveal all 3 solutions
One representative of each distinct solution is shown. Other accepted orders of the same addition or multiplication are not repeated.
- 5 + 0 = 5
- 4 + 1 = 5
- 3 + 2 = 5
Check one answer, step by step
5 + 0 = 5
- 5 + 0 = 5