2nd Grade Math Puzzles - Free Printable

This is a classic logic puzzle — essentially a 4x4 version of the “N-Queens” problem, where you must place 4 pieces (here, “Captain Salamanders”) on a 4x4 grid such that no two are in the same row, column, or diagonal.

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## 🔍 Understanding the Rules

You must place 4 salamanders on the 4x4 grid so that:

✔ Each salamander is in a different row
✔ Each salamander is in a different column
✔ No two salamanders share a diagonal

This is exactly like placing 4 queens on a 4x4 chessboard so none attack each other.

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## 🧠 Strategy

We’ll label the grid with rows 1–4 (top to bottom) and columns A–D (left to right).

We need to find all possible arrangements where:

- One salamander per row
- One salamander per column
- No two share a diagonal (i.e., for any two positions (r1, c1) and (r2, c2), |r1 - r2| ≠ |c1 - c2|)

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## ✔ Solution 1: First Valid Arrangement

Let’s try placing them as follows:

- Row 1 → Column B (1,B)
- Row 2 → Column D (2,D)
- Row 3 → Column A (3,A)
- Row 4 → Column C (4,C)

Check:

- Rows: All different ✔️
- Columns: B, D, A, C — all different ✔️
- Diagonals:
- (1,B) & (2,D): |1-2|=1, |B-D|=2 → 1≠2 ✔️
- (1,B) & (3,A): |1-3|=2, |B-A|=1 → 2≠1 ✔️
- (1,B) & (4,C): |1-4|=3, |B-C|=1 → 3≠1 ✔️
- (2,D) & (3,A): |2-3|=1, |D-A|=3 → 1≠3 ✔️
- (2,D) & (4,C): |2-4|=2, |D-C|=1 → 2≠1 ✔️
- (3,A) & (4,C): |3-4|=1, |A-C|=2 → 1≠2 ✔️

✔ All good!

So one solution is:

```
Row 1: _ X _ _
Row 2: _ _ _ X
Row 3: X _ _ _
Row 4: _ _ X _
```

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## ✔ Solution 2: Second Valid Arrangement

Now let’s try another arrangement:

- Row 1 → Column C (1,C)
- Row 2 → Column A (2,A)
- Row 3 → Column D (3,D)
- Row 4 → Column B (4,B)

Check:

- Rows: 1,2,3,4 ✔️
- Columns: C, A, D, B ✔️
- Diagonals:
- (1,C) & (2,A): |1-2|=1, |C-A|=2 → 1≠2 ✔️
- (1,C) & (3,D): |1-3|=2, |C-D|=1 → 2≠1 ✔️
- (1,C) & (4,B): |1-4|=3, |C-B|=1 → 3≠1 ✔️
- (2,A) & (3,D): |2-3|=1, |A-D|=3 → 1≠3 ✔️
- (2,A) & (4,B): |2-4|=2, |A-B|=1 → 2≠1 ✔️
- (3,D) & (4,B): |3-4|=1, |D-B|=2 → 1≠2 ✔️

✔ All good!

Second solution:

```
Row 1: _ _ X _
Row 2: X _ _ _
Row 3: _ _ _ X
Row 4: _ X _ _
```

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## 🎯 Final Answer

There are exactly 2 solutions to this 4x4 Salamander Line-Up Puzzle. Here they are visually:

🟩 Solution 1:

```
_ X _ _
_ _ _ X
X _ _ _
_ _ X _
```

🟩 Solution 2:

```
_ _ X _
X _ _ _
_ _ _ X
_ X _ _
```

These are the only two ways to place 4 Captain Salamanders on a 4x4 board without sharing any row, column, or diagonal.

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## 💡 Pro Tip

If you’re solving this manually, start by placing one salamander in the first row, then try valid spots in the next row avoiding conflicts, and backtrack if you get stuck. This is called backtracking algorithm — used in computer science for N-Queens problems!

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✔ You’ve now solved both solutions! Great job!