AMC 10 Step by Step
Log inSign up
Printable

2008 AMC 10B Problem 2

Problem onlyBack to the problem
2008 AMC 10B Problem 2AMC 10 Step by Step
Name:                             Date: September 13, 2026
Sets, Estimation & Miscellaneous · difficulty 1 of 5 · about 2 min · position P1-10

A 4×44\times 4 block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?

12348910111516171822232425\begin{array}{|c|c|c|c|} \hline 1&2&3&4\\\hline 8&9&10&11\\\hline 15&16&17&18\\\hline 22&23&24&25\\\hline \end{array}

  1. A)

    22

  2. B)

    44

  3. C)

    66

  4. D)

    88

  5. E)

    1010

Answer

B

Key insight

Just rewrite the block after the two reversals and add the diagonals: 1+10+17+22 = 50 versus 4+9+16+25 = 54.

Solution

Reverse the second row (8,9,10,11→11,10,9,88, 9, 10, 11 \to 11, 10, 9, 8) and the fourth row (22,23,24,25→25,24,23,2222, 23, 24, 25 \to 25, 24, 23, 22). The block becomes
12341110981516171825242322\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 11 & 10 & 9 & 8 \\ 15 & 16 & 17 & 18 \\ 25 & 24 & 23 & 22 \end{array}

Main diagonal (top-left to bottom-right): 1+10+17+22=501 + 10 + 17 + 22 = 50.

Other diagonal (top-right to bottom-left): 4+9+16+25=544 + 9 + 16 + 25 = 54.

The positive difference is 54−50=454 - 50 = 4.

The answer is (B) 4\boxed{\textbf{(B)}\ 4}.

Why this works

The problem is pure bookkeeping, and the safest method is to physically write the new block rather than track changes in your head. For a quicker check: in the original block both diagonals sum to 5252. Reversing a row swaps its two diagonal entries, so row two moves 11 from one diagonal to the other and row four moves 33; the sums shift by ±(3−1)=±2\pm(3 - 1) = \pm 2, giving a gap of 44.

The trap

Reversing the wrong rows (for example the first and third) or reading the diagonals off the original block.

Common mistakes

  • Reversing the wrong rows (for example the first and third) or reading the diagonals off the original block.
  • Reversing the columns instead of the rows, or forgetting that the difference is asked as a positive number.

Techniques

Set up the equation/formula and compute; no special trick needed