AMC12 2008 B

AMC12 2008 B · Q2

AMC12 2008 B · Q2. It mainly tests Arithmetic misc.

A $4\times4$ block of calendar dates is shown. \[ \begin{array}{|c|c|c|c|} \hline 1 & 2 & 3 & 4\\ \hline 8 & 9 & 10 & 11\\ \hline 15 & 16 & 17 & 18\\ \hline 22 & 23 & 24 & 25\\ \hline \end{array} \] 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?

给出一个日历日期组成的 $4\times4$ 方块： \[ \begin{array}{|c|c|c|c|} \hline 1 & 2 & 3 & 4\\ \hline 8 & 9 & 10 & 11\\ \hline 15 & 16 & 17 & 18\\ \hline 22 & 23 & 24 & 25\\ \hline \end{array} \] 将第二行的数字顺序反转；然后将第四行的数字顺序反转。最后，把两条对角线上的数字分别相加。两条对角线和的正差是多少？

(A) 2 2

(B) 4 4

(D) 8 8

(E) 10 10

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): The two sums are $1+10+17+22=50$ and $4+9+16+25=54$, so the positive difference between the sums is $54-50=4$. Query: If a different $4\times 4$ block of dates had been chosen, the answer would be unchanged. Why? \[ \begin{array}{|c|c|c|c|} \hline 1 & 2 & 3 & 4\\ \hline 11 & 10 & 9 & 8\\ \hline 15 & 16 & 17 & 18\\ \hline 25 & 24 & 23 & 22\\ \hline \end{array} \]

答案（B）：两组和分别是 $1+10+17+22=50$ 和 $4+9+16+25=54$，因此两者的正差为 $54-50=4$。问题：如果选择了不同的 $4\times 4$ 日期方块，答案将保持不变。为什么？ \[ \begin{array}{|c|c|c|c|} \hline 1 & 2 & 3 & 4\\ \hline 11 & 10 & 9 & 8\\ \hline 15 & 16 & 17 & 18\\ \hline 25 & 24 & 23 & 22\\ \hline \end{array} \]

Topics

AR.015 Arithmetic misc