The Answer in One Line
The squares from 1 to 50 are $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500$.
The first ten ($1^2$ to $10^2$) appear in nearly every algebra problem.
Quick Answer
Result: the squares from $1^2 = 1$ to $50^2 = 2500$.
Notation: $n^2 = n \times n$.
Method shown: direct multiplication and the squares-near-50 shortcut.
Pattern: the last digit of $n^2$ depends only on the last digit of $n$ (specifically the pattern $0, 1, 4, 9, 6, 5, 6, 9, 4, 1$).
Quick Reference Table — Squares 1 to 50
$n$ | $n^2$ | $n$ | $n^2$ | $n$ | $n^2$ | $n$ | $n^2$ | $n$ | $n^2$ |
|---|---|---|---|---|---|---|---|---|---|
1 | 1 | 11 | 121 | 21 | 441 | 31 | 961 | 41 | 1681 |
2 | 4 | 12 | 144 | 22 | 484 | 32 | 1024 | 42 | 1764 |
3 | 9 | 13 | 169 | 23 | 529 | 33 | 1089 | 43 | 1849 |
4 | 16 | 14 | 196 | 24 | 576 | 34 | 1156 | 44 | 1936 |
5 | 25 | 15 | 225 | 25 | 625 | 35 | 1225 | 45 | 2025 |
6 | 36 | 16 | 256 | 26 | 676 | 36 | 1296 | 46 | 2116 |
7 | 49 | 17 | 289 | 27 | 729 | 37 | 1369 | 47 | 2209 |
8 | 64 | 18 | 324 | 28 | 784 | 38 | 1444 | 48 | 2304 |
9 | 81 | 19 | 361 | 29 | 841 | 39 | 1521 | 49 | 2401 |
10 | 100 | 20 | 400 | 30 | 900 | 40 | 1600 | 50 | 2500 |
Where the Squares 1 to 50 Appear
The squares from 1 to 50 are the values most students see most often. They appear in the Pythagorean theorem (a leg of length 7 and a leg of length 24 produce a hypotenuse $\sqrt{49 + 576} = \sqrt{625} = 25$, the classic $7$–$24$–$25$ triple), in quadratic-equation solutions (every perfect-square discriminant in the table makes a quadratic factor over the integers), and in any SAT or ACT mental-maths section where speed depends on knowing the table by heart.
Concept Definition — Perfect Squares
A perfect square is an integer that equals some integer multiplied by itself. The square of $n$ is $n^2 = n \cdot n$.
For positive integers, $n^2$ is also the area of a square with side length $n$. The geometric interpretation is built into the name.
How to Compute Squares Quickly
Method 1 — Direct multiplication
$n \times n$.
For two-digit $n$: $34^2 = 34 \times 34 = 1156$ (use long multiplication or column form).
Method 2 — The squares-near-50 trick
For $n = 50 \pm k$:
$$n^2 = 2500 + 100k \cdot (\text{sign}) + k^2.$$
More precisely, $(50 + k)^2 = 2500 + 100k + k^2$ and $(50 - k)^2 = 2500 - 100k + k^2$.
Example. $47^2 = (50 - 3)^2 = 2500 - 300 + 9 = 2209$. Faster than long multiplication.
Method 3 — Squares of multiples of 5
$(5n)^2 = 25 n^2$. So $45^2 = 25 \cdot 81 = 2025$.
Method 4 — Last-digit pattern
The last digit of $n^2$ follows the cycle $0, 1, 4, 9, 6, 5, 6, 9, 4, 1$ as the last digit of $n$ goes $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$. Useful as a sanity check.
Common Mistakes With Square 1 to 50
1. Confusing $n^2$ with $2n$.
Where it slips in: Student writes $7^2 = 14$.
Don't do this: $n^2 = 2n$.
The correct way: $n^2 = n \times n$. $7^2 = 49$.
2. Off-by-one in the table.
Where it slips in: Student writes $13^2 = 144$ (mixing with $12^2$).
Don't do this: Trust memory without checking neighbours.
The correct way: $13^2 = 169$. $12^2 = 144$. Adjacent squares differ by $2n + 1$ — useful sanity check.
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