Problem of the rods
Probability
Level
2
Shown below are three types of interlocking rods.There are a total of 12 rods, of which 6 are type A, 2 are type B and 4 are type C. What is the maximum length that can be achieved by connecting the rods properly?
80
66
70
76
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1 solution
Total number of male ends is 6 ⋅ 2 + 2 ⋅ 1 = 1 2 + 2 = 1 4
Total number of female ends is 2 ⋅ 1 + 4 ⋅ 2 = 2 + 8 = 1 0
Two of the male ends may be at the end of a rod, but that still leaves two with nothing to connect to. So one of the type A connectors will not be used.
One possible arrangement of connectors will be A C A C A C A C A B B . In any event, 5 A's, 2 B's, and 4 C's will make up the final shape.
The length of the rod will be 5 ⋅ 8 + 2 ⋅ 6 + 4 ⋅ 4 = 4 0 + 1 2 + 1 6 = 6 8 for the thick section, plus 2 for the ends to a total length of 7 0