Solve the Halloween Picture Puzzle by assigning values to each object based on row totals.

This is a Halloween-themed picture puzzle that requires solving a system of equations. Each Halloween object (witch, cat, pumpkin, and witch hat) has a unique integer value between 1 and 6. The goal is to find the value of each object using the row totals provided.

I will solve this step-by-step by setting up equations for each row and then solving them systematically.

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Step 1: Assign Variables

Let’s assign variables to each object:
- Let W = value of the Witch
- Let C = value of the Cat
- Let P = value of the Pumpkin
- Let H = value of the Witch Hat

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Step 2: Write Equations from Rows

From the image, we can write the following equations based on the row totals:

- Row 1: C + P + W + C = 12 → `2C + P + W = 12` ...(1)
- Row 2: P + H + H + H = 20 → `P + 3H = 20` ...(2)
- Row 3: W + C + C + W = 14 → `2W + 2C = 14` → Simplify: `W + C = 7` ...(3)
- Row 4: H + W + P + W = 16 → `H + 2W + P = 16` ...(4)

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Step 3: Solve the System of Equations

We now have 4 equations with 4 variables. We can solve them step by step.

#### From Equation (3): `W + C = 7`
→ We can express W as: `W = 7 - C` ...(5)

#### Substitute (5) into Equation (1):
Equation (1): `2C + P + W = 12`
Substitute W:
→ `2C + P + (7 - C) = 12`
→ `C + P + 7 = 12`
→ `C + P = 5` ...(6)

#### Now use Equation (2): `P + 3H = 20`
→ `P = 20 - 3H` ...(7)

#### Substitute (7) into Equation (6):
Equation (6): `C + P = 5`
→ `C + (20 - 3H) = 5`
→ `C = 5 - 20 + 3H`
→ `C = 3H - 15` ...(8)

#### Now substitute (5) and (7) into Equation (4):
Equation (4): `H + 2W + P = 16`
Substitute W = 7 - C and P = 20 - 3H:
→ `H + 2(7 - C) + (20 - 3H) = 16`
→ `H + 14 - 2C + 20 - 3H = 16`
→ `-2H - 2C + 34 = 16`
→ `-2H - 2C = -18`
→ Divide by -2: `H + C = 9` ...(9)

#### Now use Equation (8) in Equation (9):
Equation (9): `H + C = 9`
Substitute C = 3H - 15:
→ `H + (3H - 15) = 9`
→ `4H - 15 = 9`
→ `4H = 24`
→ `H = 6`

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Step 4: Find Other Values

Now that we have H = 6:

- From Equation (7): `P = 20 - 3H = 20 - 18 = 2`
- From Equation (8): `C = 3H - 15 = 18 - 15 = 3`
- From Equation (5): `W = 7 - C = 7 - 3 = 4`

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Step 5: Verify All Values

Let’s check if these values satisfy all original equations:

- Row 1: 2C + P + W = 2(3) + 2 + 4 = 6 + 2 + 4 = 12 ✔
- Row 2: P + 3H = 2 + 3(6) = 2 + 18 = 20 ✔
- Row 3: 2W + 2C = 2(4) + 2(3) = 8 + 6 = 14 ✔
- Row 4: H + 2W + P = 6 + 2(4) + 2 = 6 + 8 + 2 = 16 ✔

All values are correct and within the range 1 to 6.

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Final Answer

- Witch Hat (H) = 6
- Cat (C) = 3
- Pumpkin (P) = 2
- Witch (W) = 4

These are the values you should record in the boxes below the grid.