Math Puzzles for 1st Grade - Free Printable

You're given a 3x3 grid of circles, where each circle is connected to its neighbors (up, down, left, right — not diagonally). The goal is to color each circle red, yellow, or blue such that no two connected (joining) circles have the same color.

This is a classic graph coloring problem — specifically, coloring a 3x3 grid graph with 3 colors so that adjacent nodes (circles) have different colors.

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## ✔ Step-by-Step Solution:

🔍 Step 1: Understand the structure

The grid looks like this (positions labeled for reference):

```
A — B — C
| | |
D — E — F
| | |
G — H — I
```

Each letter represents a circle. Adjacent means sharing an edge (not diagonal).

So:
- A is connected to B and D
- B is connected to A, C, E
- C is connected to B and F
- D is connected to A, E, G
- E is connected to B, D, F, H
- F is connected to C, E, I
- G is connected to D and H
- H is connected to E, G, I
- I is connected to F and H

---

🎨 Step 2: Use a valid 3-coloring pattern

Since it’s a bipartite-like grid (like a chessboard), we can use a checkerboard-style coloring, but since we have 3 colors, we can extend it.

Actually, a 3x3 grid is not bipartite because it contains odd-length cycles (e.g., A-B-E-D-A is a 4-cycle → even; but look at A-B-C-F-I-H-G-D-A — that’s 8, still even). Wait — actually, all cycles in a grid are even-length, so the 3x3 grid IS bipartite!

> 💡 Important: A grid graph is always bipartite — you can color it with just 2 colors in a checkerboard pattern.

But the problem says to use 3 colors: red, yellow, blue — and allows using all three. So we don’t *have* to use only 2 — we can use 3 if we want, as long as no adjacent circles share the same color.

So one simple way is to use only 2 colors (say, red and blue) in a checkerboard pattern — that will satisfy the condition. But since the problem asks for 4 different ways, we’ll need to vary the color assignments.

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## ✔ Let’s find 4 different valid colorings

We’ll create 4 different assignments using the 3 colors, ensuring adjacent circles are differently colored.

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🟢 Coloring 1: Checkerboard with Red & Blue

```
R — B — R
| | |
B — R — B
| | |
R — B — R
```

✔ All adjacent circles are different colors.

---

🟡 Coloring 2: Checkerboard with Yellow & Blue

```
Y — B — Y
| | |
B — Y — B
| | |
Y — B — Y
```

✔ Valid.

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🔵 Coloring 3: Checkerboard with Red & Yellow

```
R — Y — R
| | |
Y — R — Y
| | |
R — Y — R
```

✔ Valid.

---

🌈 Coloring 4: Use all 3 colors — more creative!

Let’s try to introduce the third color somewhere without breaking rules.

Start with:

```
R — B — Y
| | |
B — Y — R
| | |
Y — R — B
```

Check adjacents:

- Row 1: R-B (ok), B-Y (ok)
- Row 2: B-Y (ok), Y-R (ok)
- Row 3: Y-R (ok), R-B (ok)
- Columns:
- Col1: R-B (ok), B-Y (ok)
- Col2: B-Y (ok), Y-R (ok)
- Col3: Y-R (ok), R-B (ok)
- Center E (Y) is connected to B, D, F, H → all different: B, B, R, R? Wait — D is B, F is R, B is B, H is R → so E=Y is fine (different from all).

Wait — B (position B) is connected to A=R, C=Y, E=Y → B is blue → different from R and Y → OK.

Similarly, check position D=B: connected to A=R, E=Y, G=Y → all different → OK.

Position H=R: connected to E=Y, G=Y, I=B → different → OK.

✔ This is a valid 3-coloring using all 3 colors!

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## ✔ Final Answer: 4 Different Ways

Here are 4 distinct valid colorings:

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🖍️ Way 1:

```
Red Blue Red
Blue Red Blue
Red Blue Red
```

---

🖍️ Way 2:

```
Yellow Blue Yellow
Blue Yellow Blue
Yellow Blue Yellow
```

---

🖍️ Way 3:

```
Red Yellow Red
Yellow Red Yellow
Red Yellow Red
```

---

🖍️ Way 4 (Uses all 3 colors):

```
Red Blue Yellow
Blue Yellow Red
Yellow Red Blue
```

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## 📝 Notes:

- You can also create variations by rotating or reflecting these patterns — but since the problem doesn’t specify symmetry, we count them as different if the color assignments differ.
- These 4 are clearly distinct and satisfy the “no adjacent same color” rule.
- Since the grid is small and symmetric, there are many more than 4 possible solutions — but 4 is what’s asked.

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✔ You’ve now solved the problem! You can shade any of these 4 patterns into the 4 grids on your worksheet.

Let me know if you’d like to see how to draw them visually or generate more variations!