Master · Worked puzzle
A fraction can be the missing multiplier
Use the Master keypad’s halves to reach a target between whole-number products.
(□ + □) × □ × 8 = 60
4 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 accepts one of these complete expressions: 012341/21/32/33/2√2√3∛2. Choices may be repeated.
- Follow the printed brackets and the order of operations. Fractional intermediate values are allowed. Division by zero is never allowed.
- Find all 4 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
Divide sixty by eight. The bracketed sum times the third choice must equal seven and a half.
Read the worked approach
- Undo the fixed multiplier of eight: (a + b)c = 60 ÷ 8 = 15/2. The blanks use the displayed Master choices, not every digit or every possible fraction.
- If c = 3/2, the sum must be five. Two available unordered pairs make five: 1 + 4 and 2 + 3. Multiplying either sum by 3/2 gives 15/2.
- If c = 3, the sum must be 5/2. Two available pairs work: 1 + 3/2 and 2 + 1/2. Multiplying by three again gives 15/2.
- The game’s exhaustive keypad check gives four distinct solutions, listed below. Each multiplies 15/2 by eight to make sixty; swapping the terms of its sum repeats that solution.
A common trap
The keypad choices are complete expressions. Selecting 3/2 fills one blank; it does not place three and two into separate blanks.
Carry this idea to the next puzzle
Use exact fractions while reasoning. Seven and a half is 15/2, which makes the useful multipliers easier to spot.
Reveal all 4 solutions
One representative of each distinct solution is shown. Other accepted orders of the same addition or multiplication are not repeated.
- (4 + 1) × (3/2) × 8 = 60
- ((3/2) + 1) × 3 × 8 = 60
- (3 + 2) × (3/2) × 8 = 60
- ((1/2) + 2) × 3 × 8 = 60
Check one answer, step by step
(4 + 1) × (3/2) × 8 = 60
- 4 + 1 = 5
- 5 × 1.5 = 7.5
- 7.5 × 8 = 60
The step check shows decimal approximations where needed; fraction and root keys above retain their exact written form.