Step 1: Identify the pattern in the table (figure 1). The table has 3 rows and 6 columns: Row 1: 1, 2, 3, 2, 10, 12 Row 2: 2, 5, 12, 10, 16, 13 Row 3: 1, 2, 1, ?, 10, 24 Add up the three entries in each column: Column 1 = \(1+2+1=4\), Column 2 = \(2+5+2=9\), Column 3 = \(3+12+1=16\), Column 5 = \(10+16+10=36\), Column 6 = \(12+13+24=49\).
Step 2: Spot the pattern among the column sums. The sums 4, 9, 16, 36, 49 are perfect squares: \(2^2, 3^2, 4^2, 6^2, 7^2\). Column 1 gives \(2^2\), column 2 gives \(3^2\), column 3 gives \(4^2\), so the columns follow \(n^2\) where \(n\) increases by 1 for each column. That means column 4 should equal \(5^2 = 25\), column 5 should equal \(6^2 = 36\) (which matches), and column 6 should equal \(7^2 = 49\) (which also matches).
Step 3: Solve for the missing number. Column 4's entries are 2, 10 and the unknown value, and their sum must equal 25: \(2 + 10 + x = 25\), so \(x = 13\).
Final Answer: The missing number is 13, so option C is correct. \[ \boxed{13} \]
