Easy2 solutions
Two ways to reach eighteen
□ × 9 + □ = 18
Use the size of the last blank to narrow a multiplication puzzle to two possibilities.
Explore puzzleA closer look
Every answer has a way in.
Fifteen selected puzzles. A specific strategy for each. Start with a hint, explore the reasoning, or head straight to the board.
Easy4 solutions
Undo the division first
(□ + □) ÷ 3 = 4
Turn a bracketed division into a short list of number pairs.
Explore puzzleEasy3 solutions
Why six arrangements make three solutions
□ + □ = 5
A simple addition shows exactly how Mathoji counts unique answers.
Explore puzzleMedium2 solutions
An even remainder narrows the search
□ × □ − □ × 2 = 62
Combine parity and bounds to solve a three-blank equation efficiently.
Explore puzzleMedium8 solutions
Find eight answers by factoring ninety
(□ + □) × □ + 7 = 97
Group the search by the outside multiplier instead of guessing three blanks independently.
Explore puzzleMedium5 solutions
Look just below thirty-four
□ × □ + □ = 34
Use a nine-digit remainder window to find five multiplication-and-addition answers.
Explore puzzleHard5 solutions
Three factors, one small subtraction
□ × □ × □ − □ = 74
Organise a four-blank puzzle by product and avoid counting rearrangements.
Explore puzzleHard3 solutions
A large target with only three answers
(□ + □) × (□ + □) = 221
Factor the target before filling the two sums in brackets.
Explore puzzleHard3 solutions
A difference has a smaller ceiling
(□ − □) × (□ + □) = 117
Use the different ranges of subtraction and addition to place the factors of 117.
Explore puzzleExpert3 solutions
Why the divisor must be one
□ × (□ + □) ÷ □ = 78
A maximum numerator rules out most of the final blank’s values.
Explore puzzleExpert2 solutions
Strip away the outside operations
(□ × □ + □) ÷ □ + 4 = 77
Reduce a nested Expert equation to two nearby multiplication facts.
Explore puzzleExpert6 solutions
Six answers from two factor cases
(□ + □) × (□ − □) ÷ 7 = 18
Solve a sum-times-difference puzzle by undoing its final division.
Explore puzzleMaster4 solutions
A fraction can be the missing multiplier
(□ + □) × □ × 8 = 60
Use the Master keypad’s halves to reach a target between whole-number products.
Explore puzzleMaster3 solutions
Three ways for a root product to make two
√(□) × □ + □ × 5 = 12
A square root can combine with another root to give a whole-number result.
Explore puzzleMaster2 solutions
Cube roots that multiply to one
∛(□) × □ + □ × 8 = 33
A reciprocal pair of cube roots supplies a tiny missing part of the target.
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