Question

\[\begin{array}{|c|c|c|} \hline 7 & 6 & 8 \\ \hline 5 & 4 & 9 \\ \hline 3 & 2 & 1 \\ \hline 83 & 56 & ? \\ \hline \end{array} \]

• A. 146
• B. 128
• C. 136
• D. 148

Hint:

When two columns fit a clean rule, apply the same operation to the third to find the missing number.

Answer 1

The Correct Option is A

Rule (column-wise): Each bottom entry equals the sum of the squares of the three numbers above it in that column. \[ \begin{aligned} \text{Col 1: } 7^2+5^2+3^2 &= 49+25+9 = 83 \quad \checkmark \\ \text{Col 2: } 6^2+4^2+2^2 &= 36+16+4 = 56 \quad \checkmark \\ \text{Col 3: } 8^2+9^2+1^2 &= 64+81+1 = 146 \end{aligned} \] \[\boxed{146}\]

Answer 2

Check each option by subtracting the squares of the first two column entries in the third column (\(8^2+9^2=64+81=145\)) and seeing whether the remainder equals the square of the third entry (\(1^2=1\)).

1. Option (a) 146: \(146-145=1=1^2\) — confirmed.
2. Option (b) 128: \(128-145=-17\), not a valid square — rejected.
3. Option (c) 136: \(136-145=-9\), not \(1^2\) — rejected.
4. Option (d) 148: \(148-145=3\), not \(1^2\) — rejected.

Only option (a) leaves exactly \(1^2\) after removing the first two squares.

the correct answer is 146.