Math Puzzles: Solve the equations using strategic thinking.

Let's solve this math puzzle step by step.

We are given a visual puzzle with yellow squares representing unknown numbers. The puzzle is structured like this:

```
□ + □ = 8
+ +
□ - □ = 6
= =
13 8
```

This means:

- The top row: Two unknowns (let’s call them A and B) add up to 8:
A + B = 8

- The middle row: One unknown (C) minus another (D) equals 6:
C - D = 6

- The left column adds to 13:
A + C = 13

- The right column adds to 8:
B + D = 8

So we have the following system of equations:

1. A + B = 8
2. C - D = 6
3. A + C = 13
4. B + D = 8

---

Step 1: Use equation (1): A + B = 8 → B = 8 - A

Step 2: Use equation (3): A + C = 13 → C = 13 - A

Step 3: Use equation (2): C - D = 6 → D = C - 6

Substitute C from above:
D = (13 - A) - 6 = 7 - A

Step 4: Use equation (4): B + D = 8

Now substitute B = 8 - A and D = 7 - A:

(8 - A) + (7 - A) = 8
15 - 2A = 8
-2A = 8 - 15 = -7
A = 7/2 = 3.5

Wait — that gives us a decimal. Let's check if that makes sense.

But let’s see what happens:

A = 3.5
Then B = 8 - 3.5 = 4.5
C = 13 - 3.5 = 9.5
D = 7 - 3.5 = 3.5

Check equation (2): C - D = 9.5 - 3.5 = 6 ✔
Equation (4): B + D = 4.5 + 3.5 = 8 ✔

So it works, but all values are decimals.

But maybe we're supposed to find integers? Let’s double-check the layout.

Wait — perhaps the structure is not as I interpreted.

Let me re-express the diagram clearly:

It looks like a grid:

```
□ + □ = 8
+ + +
□ - □ = 6
= = =
13 8
```

So actually, there are two rows and three columns:

- Column 1: Top square + bottom square = 13
- Column 2: Top square + bottom square = 8
- Row 1: First square + second square = 8
- Row 2: First square - second square = 6

So define:

Let:
- Top-left = A
- Top-right = B
- Bottom-left = C
- Bottom-right = D

Then:

From the first row:
A + B = 8 → (1)

From the second row:
C - D = 6 → (2)

From the first column:
A + C = 13 → (3)

From the second column:
B + D = 8 → (4)

Same equations as before.

Now solve again:

From (1): B = 8 - A
From (3): C = 13 - A
From (2): D = C - 6 = (13 - A) - 6 = 7 - A
From (4): B + D = 8 → (8 - A) + (7 - A) = 8 → 15 - 2A = 8 → 2A = 7 → A = 3.5

So A = 3.5
Then:
- B = 8 - 3.5 = 4.5
- C = 13 - 3.5 = 9.5
- D = 7 - 3.5 = 3.5

Check:
- Row 1: 3.5 + 4.5 = 8 ✔
- Row 2: 9.5 - 3.5 = 6 ✔
- Col 1: 3.5 + 9.5 = 13 ✔
- Col 2: 4.5 + 3.5 = 8 ✔

All equations are satisfied.

So the solution is:

- Top-left: 3.5
- Top-right: 4.5
- Bottom-left: 9.5
- Bottom-right: 3.5

But since it's a "Math Puzzle", maybe they expect whole numbers? But in this case, only fractional solutions work.

Alternatively, could the symbols represent digits (0–9), or something else?

Wait — perhaps the puzzle uses the same symbol for the same number, meaning the yellow squares are identical?

Look at the image:

There are four yellow squares. Are they all the same? Or different?

In the image, each yellow square is separate — likely different variables.

But notice: the bottom-right square appears in both the subtraction and the right column sum.

But based on our solving, it's consistent.

Alternatively, maybe the puzzle is designed so that some squares are the same value?

But unless stated, we assume they can be different.

But let's suppose all yellow squares are the same number — that would be a big assumption, but let’s test it.

Suppose all four squares are equal to X.

Then:
- Row 1: X + X = 8 → 2X = 8 → X = 4
- Row 2: X - X = 6 → 0 = 6 ✘ impossible

So cannot all be the same.

What if top-left and bottom-left are same? Or top-right and bottom-right?

No indication.

Alternatively, perhaps the same shape means same value? But in the image, they are all identical yellow squares — but positioned differently.

But in puzzles like this, identical shapes usually mean same value.

Wait — this is critical!

Are all four yellow squares identical? If so, then they all represent the same variable, say X.

But then:

Row 1: X + X = 8 → 2X = 8 → X = 4
Row 2: X - X = 6 → 0 = 6 ✘ contradiction

So cannot be all same.

But maybe only the ones in the same position? No.

Wait — perhaps the two top squares are one variable, the two bottom squares are another?

But no — the layout shows four distinct squares.

Wait — look closely:

The puzzle is:

```
□ + □ = 8
+ +
□ - □ = 6
= =
13 8
```

So the first column has two squares: top-left and bottom-left → their sum is 13
Second column: top-right and bottom-right → sum is 8
First row: top-left + top-right = 8
Second row: bottom-left - bottom-right = 6

So four distinct squares.

But maybe the top-left and bottom-left are related? Or maybe some are equal?

But unless specified, we treat them as independent.

But our earlier solution works with decimals.

But maybe the puzzle expects integer answers.

Let’s suppose all variables are integers.

Try to solve with integers.

From earlier:

A + B = 8
C - D = 6
A + C = 13
B + D = 8

From first: B = 8 - A
From third: C = 13 - A
From second: D = C - 6 = 13 - A - 6 = 7 - A
From fourth: B + D = 8 → (8 - A) + (7 - A) = 15 - 2A = 8 → 2A = 7 → A = 3.5

So no integer solution exists.

Therefore, the only solution is:

- A = 3.5
- B = 4.5
- C = 9.5
- D = 3.5

So the values are:

- Top-left: 3.5
- Top-right: 4.5
- Bottom-left: 9.5
- Bottom-right: 3.5

Now, let's write it back into the grid:

```
3.5 + 4.5 = 8
+ +
9.5 - 3.5 = 6
= =
13 8
```

Check:
- 3.5 + 4.5 = 8 ✔
- 9.5 - 3.5 = 6 ✔
- 3.5 + 9.5 = 13 ✔
- 4.5 + 3.5 = 8 ✔

Perfect.

Final Answer:

- Top-left square: 3.5
- Top-right square: 4.5
- Bottom-left square: 9.5
- Bottom-right square: 3.5

Even though it involves decimals, it's the only solution that satisfies all conditions.

✔ Solution is unique and correct.

---

Strategic Thinking Explanation:

This puzzle requires setting up a system of equations based on the given relationships. By labeling the unknowns and using substitution, we can solve step by step. The key insight is recognizing the grid structure and translating each line and column into an equation. Even when the answer isn't an integer, consistency with all constraints confirms correctness. This type of problem builds algebraic reasoning and attention to detail.