How do I square numbers in my head?
Squaring looks like brute-force multiplication, but two patterns turn most squares into a few quick steps. One handles anything ending in 5; the other handles everything else by rounding to a friendly base.
These are recall-and-adjust tricks, not new arithmetic. Practice them in short bursts and they become reflexes — SIXTY's adaptive difficulty will keep nudging you toward the squares you find slow.
How do I square a number ending in 5?
Take the tens digit n, multiply it by the next number up (n+1), and stick 25 on the end. The result is always exact.
For 35²: the tens digit is 3, so 3 × 4 = 12, append 25 → 1225. For 65²: 6 × 7 = 42, append 25 → 4225. It works for larger numbers too: 115² uses 11 × 12 = 132 → 13225.
- 35²: n = 3, so 3 × 4 = 12
- Append 25 → 1225
- 65²: n = 6, so 6 × 7 = 42
- Append 25 → 4225
What's the round-to-a-base method for other numbers?
Round the number to a friendly base, multiply the base by the number an equal distance the other side, then add the square of the gap. In symbols, n² = (n − d)(n + d) + d².
For 48², round up to 50 and down to 46 (each 2 away): 50 × 46 = 2300, plus 2² = 4, giving 2304. For 53², use 50 × 56 = 2800, plus 3² = 9, giving 2809.
- 48²: base 50, gap 2 → 50 × 46 = 2300
- Add gap²: 2² = 4 → 2304
- 53²: base 50, gap 3 → 50 × 56 = 2800
- Add gap²: 3² = 9 → 2809
Why does adding the square of the gap work?
It comes straight from (n − d)(n + d) = n² − d². That product is short by exactly d², so you add d² back to land on n².
You don't need to recite the algebra mid-calculation. Just remember the rhythm: cross-multiply to the base, then add the little square.
How do I square numbers near 100?
For numbers just below 100, find how far below (the deficit d), subtract that deficit from the number to get the leading part, and append d² as the last two digits.
For 97²: deficit is 3, so 97 − 3 = 94 leads, and 3² = 09 fills the end → 9409. For 96²: 96 − 4 = 92, and 4² = 16 → 9216. Keep the appended square two digits wide.
- 97²: deficit 3, 97 − 3 = 94
- Append 3² as two digits: 09 → 9409
- 96²: deficit 4, 96 − 4 = 92
- Append 4² as two digits: 16 → 9216
How do I square numbers just above 100?
Mirror the trick: take the excess e above 100, add it to the number for the leading part, and append e² as two digits.
For 103²: excess 3, so 103 + 3 = 106, and 3² = 09 → 10609. For 108²: 108 + 8 = 116, and 8² = 64 → 11664.
How do I check a square is right?
Estimate the ballpark first. 48² is near 2500 (since 50² = 2500), so 2304 is plausible while 230 or 23000 are not.
A digit-sum check helps for slips: the digit root of a square is always 1, 4, 7, or 9. 2304 sums to 9, which is valid; a result summing to 5 would signal an error.
Reading is review. Recall is what sticks.
Start a 60-second sprint →