Rocket chalks 3 + 4 × 2 on the garage wall. "Fourteen," he says at once. "Three plus four is seven, times two."
"I got eleven," says Raven. "Four times two is eight, plus three. What do you notice? Same numbers, two answers."
"Then the puzzle is broken," Rocket says. "Half the street will say fourteen."
Nova hovers beside the chalk line. "A puzzle needs one answer," she says. "So mathematicians agreed on one order, long ago."
"Like rules for a board game," Raven says. "Everyone plays the same way or nobody can win."
Nova dims her light to a steady glow. "Parentheses first. Then powers. Then multiply and divide. Then add and subtract."
Rocket wipes out the fourteen. "Eleven," he says. "Fine. But I want parentheses on the next one."
A numerical expression is numbers joined by operation signs. It names exactly one number, but only if everyone works it out the same way.
The agreed order has four steps. Work from left to right inside each step.
| Step | Expression | What we did |
|---|---|---|
| Start | 2 + 3 × 4² | Look for parentheses: none. |
| Powers | 2 + 3 × 16 | 4² = 16 |
| Multiply | 2 + 48 | 3 × 16 = 48 |
| Add | 50 | 2 + 48 = 50 |
Parentheses change everything. 3 + 4 × 2 = 11, but (3 + 4) × 2 = 14. The parentheses say "add first".
A power sits inside its own step. 3 × 2² = 12, because 2² is worked out before the 3 multiplies it.
| Sentence | True or false? |
|---|---|
| 3 + 4 × 2 = 14 | ? |
| (3 + 4) × 2 = 14 | ? |
| 10 - 2 × 3 = 4 | ? |
| 8 ÷ 2 + 2 = 2 | ? |
One order, one answer. Tomorrow is Puzzle Lab: folding, stacking and squaring with real paper and real beans.