Solve the shape value puzzle using the given grid and equations.

Let's solve this puzzle step by step.

We are given a 4×4 grid with symbols:
- Circle (○)
- Star (☆)
- Triangle (△)

Each row and column has a total sum, and we need to find the value of each symbol based on these sums.

Step 1: Assign variables

Let:
- ○ = x
- ☆ = y
- △ = z

We’ll use the row and column totals to form equations.

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Step 2: Use the row sums

Row 1: ○ + △ + △ + ☆ = 16
→ x + z + z + y = 16 → x + 2z + y = 16 → (1)

Row 2: ☆ + ○ + ○ + ○ = 14
→ y + x + x + x = 14 → y + 3x = 14 → (2)

Row 3: △ + ○ + ☆ + △ = 16
→ z + x + y + z = 16 → x + y + 2z = 16 → (3)

Row 4: ☆ + ☆ + ☆ + ☆ = 20
→ 4y = 20 → y = 5 → (4)

Great! We now know:
y = 5

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Step 3: Plug y = 5 into other equations

From equation (2):
y + 3x = 14 → 5 + 3x = 14 → 3x = 9 → x = 3

So:
x = 3

Now plug x = 3 and y = 5 into equation (1):
x + 2z + y = 16 → 3 + 2z + 5 = 16 → 8 + 2z = 16 → 2z = 8 → z = 4

So:
z = 4

Let’s verify with equation (3):
x + y + 2z = 3 + 5 + 2×4 = 8 + 8 = 16 ✔

Perfect.

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Step 4: Check column sums

Now verify the column totals:

Column 1: ○ + ☆ + △ + ☆ = x + y + z + y = 3 + 5 + 4 + 5 = 17 ✔

Column 2: △ + ○ + ○ + ☆ = z + x + x + y = 4 + 3 + 3 + 5 = 15 ✔

Column 3: △ + ○ + ☆ + ☆ = z + x + y + y = 4 + 3 + 5 + 5 = 17 ✔

Column 4: ☆ + ○ + △ + ☆ = y + x + z + y = 5 + 3 + 4 + 5 = 17 ✔

All match!

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Final Answer:

- ○ = 3
- ☆ = 5
- △ = 4

✔ All row and column sums are satisfied.

Let me know if you'd like a visual explanation or further breakdown!