Identify the sum of 𝑎 + 𝑏 + 𝑐, provided the total of each row and each column is same.
11 16 a
10 b 14
c 8 13
Options:-
A. 32
B. 36
C. 27
D. 30

Math Puzzle: Find the Sum of a + b + c

Here’s an interesting number puzzle that tests your logical reasoning and arithmetic skills. You’re given a 3×3 grid where the sum of each row and each column is the same. Your task is to find the value of a + b + c.

Puzzle Grid:

11   16   a
10    b   14
 c    8   13

You are also given four options:

• A. 32
• B. 36
• C. 27
• D. 30

Let’s solve it step-by-step.

Step 1: Let Each Row and Column Sum Be “S”

Let’s denote the common sum of every row and column as S.

Row-wise equations:

• Row 1: 11 + 16 + a = S → 27 + a = S
• Row 2: 10 + b + 14 = S → 24 + b = S
• Row 3: c + 8 + 13 = S → c + 21 = S

From these, we get:

• S = 27 + a
• S = 24 + b
• S = c + 21

Step 2: Create Relationships Between a, b, and c

Since all equations equal S:

• 27 + a = 24 + b → a – b = -3 → a = b – 3
• 24 + b = c + 21 → b – c = -3 → b = c – 3

Combining both:

• a = (c – 3) – 3 = c – 6
• b = c – 3

Now find a + b + c:

a + b + c = (c - 6) + (c - 3) + c = 3c - 9

Step 3: Try a Value That Works for All Rows

Let’s test c = 12:

• a = 6
• b = 9
• c = 12

Now verify the row sums:

• Row 1: 11 + 16 + 6 = 33
• Row 2: 10 + 9 + 14 = 33
• Row 3: 12 + 8 + 13 = 33 ✅

The grid works!

Final Calculation:

a + b + c = 6 + 9 + 12 = 27

✅ Correct Answer: C. 27

Key Takeaways

• Use basic algebra to relate variables.
• Test values to satisfy equal row and column totals.
• Always verify by plugging values back into the original grid.