Subtract your way to the root

The idea

Successively subtract 1, 3, 5, … from a natural number: reaching exactly 0 shows it is a perfect square whose square root is the number of steps; crossing past 0 shows it is not a perfect square.

Start at 25

Every perfect square is a sum of odd numbers starting from 1. For example, 25 = 1 + 3 + 5 + 7 + 9.

Start at 25 and subtract 1, then 3, then 5, … What happens?

• We land exactly on 0
• We never get near 0
• We pass 0 and go negative

Subtract the odd numbers

Press the button to subtract the next odd number. Test 25 first, then 38.

Exactly 0, or past 0

Subtract 1, 3, 5, … from a number. Reaching exactly 0 shows it is a perfect square, and its square root is the number of steps. Crossing past 0 shows it is not a perfect square.

• Number | What happens | Verdict
• 25 | 25 − 1 − 3 − 5 − 7 − 9 = 0 (5 steps) | Perfect square, √25 = 5
• 38 | … 13 − 11 = 2, then 2 − 13 = −11 | Not a perfect square

Notes

Keep subtracting 1, 3, 5, …: reaching exactly 0 means a perfect square (the number of steps is its square root); crossing past 0 means not a perfect square.

Check yourself

Subtracting 1, 3, 5, … from 30 gives 29, 26, 21, 14, 5, −6. What does this tell us?

50 − 1 − 3 − 5 − 7 − 9 − 11 − 13 = ?

Answer: 1

50 − 49 = 1. The next odd number is 15, and 1 − 15 = −14 is past 0, so 50 is not a perfect square.

A number reaches exactly 0 on the 12th subtraction of 1, 3, 5, … What is the number?

Answer: 144

Twelve odd numbers add to 12² = 144, so the number is 144 and √144 = 12.

• 30 is a perfect square with root 6. We never landed on 0. We went from 5 to −6, past 0.
• 30 is not a perfect square — correct. Yes! We crossed past 0 without landing on it.
• 30 is a perfect square with root 5. The count of steps only gives a root if we land exactly on 0.