Master · Worked puzzle
Cube roots that multiply to one
A reciprocal pair of cube roots supplies a tiny missing part of the target.
∛(□) × □ + □ × 8 = 33
2 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 2 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
Four times eight gets very close to thirty-three. Look for a root product equal to one.
Read the worked approach
- Choose four in the last blank to make 4 × 8 = 32. That leaves one for the cube-root product.
- The direct answer is ∛1 × 1 = 1. The full equation is therefore ∛1 × 1 + 4 × 8 = 33.
- The keypad also supplies 1/2 and ∛2. Their combination gives ∛(1/2) × ∛2 = ∛1 = 1. This is a different permitted expression, even though the product is the same.
- Checking every combination from the displayed keypad leaves these two solutions. Keep the half inside the printed cube root and the ∛2 choice in the following blank.
A common trap
Square-root and cube-root symbols are different operations. The small three on ∛ is part of the operator.
Carry this idea to the next puzzle
Reciprocals are useful with roots as well as fractions. Look for products of radicands that simplify to one.
Reveal all 2 solutions
One representative of each distinct solution is shown. Other accepted orders of the same addition or multiplication are not repeated.
- ∛(1) × 1 + 4 × 8 = 33
- ∛((1/2)) × (∛2) + 4 × 8 = 33
Check one answer, step by step
∛(1) × 1 + 4 × 8 = 33
- 1 × 1 = 1
- 4 × 8 = 32
- 1 + 32 = 33
The step check shows decimal approximations where needed; fraction and root keys above retain their exact written form.