Free Printable Sum Matrix With Faces - Free Printable

Let’s solve the first puzzle step by step.

We have a 4x4 grid. Each face = a different number.
Rows and columns add up to the numbers shown — except one row or column is missing its sum (marked with ?).

In Puzzle 1:

Faces:
- Brown hair girl → let’s call her B
- Blue crown boy → C
- Yellow hair boy → Y
- Sun → S

Grid:

Row 1: B, C, C, Y → sum = 30
Row 2: C, B, C, Y → sum = 30
Row 3: C, S, S, S → sum = 30
Row 4: Y, B, S, Y → sum = ?

Columns:
Col 1: B, C, C, Y → sum = 30
Col 2: C, B, S, B → sum = 20
Col 3: C, C, S, S → sum = 32
Col 4: Y, Y, S, Y → sum = 37

Wait — notice Row 1 and Row 2 are almost the same!

Row 1: B + C + C + Y = 30 → B + 2C + Y = 30
Row 2: C + B + C + Y = 30 → same as above → also B + 2C + Y = 30 → consistent.

Now look at Column 1: B + C + C + Y = 30 → same as Row 1 → so that checks out.

Column 2: C + B + S + B = 20 → C + 2B + S = 20 → Equation (1)

Column 3: C + C + S + S = 32 → 2C + 2S = 32 → divide by 2 → C + S = 16 → Equation (2)

Column 4: Y + Y + S + Y = 37 → 3Y + S = 37 → Equation (3)

Row 3: C + S + S + S = 30 → C + 3S = 30 → Equation (4)

Now we can use Equations (2) and (4):

From (2): C = 16 - S
Plug into (4): (16 - S) + 3S = 30 → 16 + 2S = 30 → 2S = 14 → S = 7

Then from (2): C = 16 - 7 = 9

Now plug S=7 into (3): 3Y + 7 = 37 → 3Y = 30 → Y = 10

Now plug C=9, Y=10 into Row 1 equation: B + 2C + Y = 30 → B + 18 + 10 = 30 → B + 28 = 30 → B = 2

Check Column 2: C + 2B + S = 9 + 4 + 7 = 20 → correct ✔

Now find Row 4: Y + B + S + Y = 10 + 2 + 7 + 10 = 29

So the missing sum in Puzzle 1 is 29

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Now Puzzle 2:

Same faces? No — problem says “The number represented by a particular face is not the same in the two different puzzles.” So we must solve independently.

Puzzle 2 Grid:

Row 1: C, C, S, B → sum = 29
Row 2: Y, B, Y, S → sum = 33
Row 3: C, S, Y, C → sum = 22
Row 4: Y, B, C, S → sum = ?

Columns:
Col 1: C, Y, C, Y → sum = 20
Col 2: C, B, S, B → sum = 38
Col 3: S, Y, Y, C → sum = 24
Col 4: B, S, C, S → sum = 33

Let’s write equations.

Row 1: 2C + S + B = 29 → Eq (A)
Row 2: 2Y + B + S = 33 → Eq (B)
Row 3: 2C + S + Y = 22 → Eq (C)
Row 4: Y + B + C + S = ? → this is what we need

Column 1: 2C + 2Y = 20 → divide by 2 → C + Y = 10 → Eq (D)
Column 2: C + 2B + S = 38 → Eq (E)
Column 3: S + 2Y + C = 24 → Eq (F)
Column 4: B + 2S + C = 33 → Eq (G)

Use Eq (D): C = 10 - Y

Plug into Eq (C): 2(10 - Y) + S + Y = 22 → 20 - 2Y + S + Y = 22 → 20 - Y + S = 22 → S - Y = 2 → S = Y + 2 → Eq (H)

Plug C = 10 - Y and S = Y + 2 into Eq (A):

2(10 - Y) + (Y + 2) + B = 29 → 20 - 2Y + Y + 2 + B = 29 → 22 - Y + B = 29 → B - Y = 7 → B = Y + 7 → Eq (I)

Now plug into Eq (B): 2Y + B + S = 33
Substitute B and S: 2Y + (Y+7) + (Y+2) = 33 → 2Y + Y + 7 + Y + 2 = 33 → 4Y + 9 = 33 → 4Y = 24 → Y = 6

Then:
S = Y + 2 = 8
B = Y + 7 = 13
C = 10 - Y = 4

Check Row 1: 2C + S + B = 8 + 8 + 13 = 29 ✔
Row 2: 2Y + B + S = 12 + 13 + 8 = 33 ✔
Row 3: 2C + S + Y = 8 + 8 + 6 = 22 ✔
Column 1: 2C + 2Y = 8 + 12 = 20 ✔
Column 2: C + 2B + S = 4 + 26 + 8 = 38 ✔
Column 3: S + 2Y + C = 8 + 12 + 4 = 24 ✔
Column 4: B + 2S + C = 13 + 16 + 4 = 33 ✔

Now Row 4: Y + B + C + S = 6 + 13 + 4 + 8 = 31

So missing sum in Puzzle 2 is 31

Final Answer:
29, 31