Each of the points $A$, $B$, $C$, $D$, $E$, and $F$ in the figure below represents a different digit from 1 to 6. Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is 47. What is the digit represented by $B$?

A. 1

B. 2

C. 3

D. 4

E. 5

1. 2020 AMC 8 官方解析

The total of the five sums is
$(A + B + C) + (A + F + E) + (B + D) + (B + F) + (C + D + E)$
$= 2A + 3B + 2C + 2D + 2E + 2F$
$= 2(A + B + C + D + E + F) + B$,
which equals 47. The expression $A + B + C + D + E + F$ equals 21, the sum of the digits from 1 to 6, so $2 \cdot 21 + B = 47$. Therefore $B = 47 - 42 = 5$.

这五个和的总和是
$(A + B + C) + (A + F + E) + (B + D) + (B + F) + (C + D + E)$
$= 2A + 3B + 2C + 2D + 2E + 2F$
$= 2(A + B + C + D + E + F) + B$,
等于47．表达式$A + B + C + D + E + F$等于从1到6的数字总和21，代入上面的式子可得
$2 \cdot 21 + B = 47$，因此$B = 47 - 42 = 5$

Note that every point lies on exactly two lines except for $B$, which lies on three lines. Because the sum of the digits from 1 to 6 is 21, this gives a total sum of $2 \cdot 21 + B = 47$. Therefore $B = 47 - 42 = 5$.

除了$B$之外，每个点都位于两条直线上，而$B$则位于三条直线上．因为从1到6的数字总和为21，所以总和为$2 \cdot 21 + B = 47$．因此$B = 47 - 42 = 5$．