Master · Worked puzzle

Three ways for a root product to make two

A square root can combine with another root to give a whole-number result.

Mathoji.com · 16 October 2026 · Math checked against the game’s rules

√(□) × □ + □ × 5 = 12

3 distinct solutions · 3 blanks

Try 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 3 distinct solutions. Reordering terms of the same addition or multiplication counts once. Subtraction, division and bracket structure still matter.
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Show a starting hint

Try two in the last blank. What must the first part contribute to twelve?

Read the worked approach

With two in the last blank, the final term is 2 × 5 = 10. The root product must supply the remaining two.
One way is √1 × 2 = 2. Another is √4 × 1 = 2. Both roots simplify to whole numbers before the multiplication.
A third way is √2 × √2 = 2. Put 2 inside the printed root and choose √2 for the next blank. Two identical positive square roots multiply to their radicand.
The exhaustive check of this puzzle’s allowed keypad choices yields these three distinct solutions. They have the same numerical result but different root-and-operand structures, so each earns a separate row.

The first blank already sits inside a square-root symbol. Choosing √2 there would create √(√2), not √2.

Read the printed operators before choosing an expression key. A whole-number answer can come from irrational intermediate values.

Reveal all 3 solutions

One representative of each distinct solution is shown. Other accepted orders of the same addition or multiplication are not repeated.

√(1) × 2 + 2 × 5 = 12

The step check shows decimal approximations where needed; fraction and root keys above retain their exact written form.

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