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This is a classic magic square puzzle. The goal is to place the numbers 1 through 9 (each used exactly once) into a 3×3 grid so that the sum of the numbers in each row, column, and both diagonals equals 15.

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🔍 Step-by-Step Solution:

We need to fill a 3×3 grid with numbers 1–9, no repeats, such that every row, column, and diagonal adds up to 15.

Let’s denote the grid as:

```
[ A ] [ B ] [ C ]
[ D ] [ E ] [ F ]
[ G ] [ H ] [ I ]
```

We want:
- Rows: A+B+C = 15, D+E+F = 15, G+H+I = 15
- Columns: A+D+G = 15, B+E+H = 15, C+F+I = 15
- Diagonals: A+E+I = 15, C+E+G = 15

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🧠 Key Insight:

The sum of all numbers from 1 to 9 is:

$$
1 + 2 + 3 + \dots + 9 = \frac{9 \times 10}{2} = 45
$$

Since there are 3 rows, and each must sum to 15, total sum is $ 3 \times 15 = 45 $ — which matches! So it's consistent.

Now, since each row, column, and diagonal must sum to 15, we can use known properties of magic squares.

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✔ The Unique (up to rotation/reflection) 3×3 Magic Square

There is only one basic 3×3 magic square using digits 1–9 (ignoring rotations and reflections), and it looks like this:

```
8 1 6
3 5 7
4 9 2
```

Let’s verify:

#### Rows:
- 8 + 1 + 6 = 15 ✔
- 3 + 5 + 7 = 15 ✔
- 4 + 9 + 2 = 15 ✔

#### Columns:
- 8 + 3 + 4 = 15 ✔
- 1 + 5 + 9 = 15 ✔
- 6 + 7 + 2 = 15 ✔

#### Diagonals:
- 8 + 5 + 2 = 15 ✔
- 6 + 5 + 4 = 15 ✔

Perfect!

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📌 Final Answer:

Fill the grid like this:

```
[ 8 ] [ 1 ] [ 6 ]
[ 3 ] [ 5 ] [ 7 ]
[ 4 ] [ 9 ] [ 2 ]
```

This satisfies all conditions: every row, column, and diagonal sums to 15.

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💡 Tips for Solving It Yourself:

1. Center must be 5: Since it's used in 4 lines (middle row, middle column, both diagonals), it needs to be the median value. The average of 1–9 is 5, so 5 goes in the center.
2. Corner numbers are even (2, 4, 6, 8), and edge numbers are odd (1, 3, 7, 9), but not always — actually, in this case, corners are 8, 6, 4, 2 (even), edges are 1, 3, 7, 9 (odd). This pattern helps.
3. Use trial and error with logic: if you pick a number for a corner, it affects three lines.

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✔ You now have the solution! Just write the numbers in the boxes accordingly.