1996 AHSME Problems/Problem 10
Problem
How many line segments have both their endpoints located at the vertices of a given cube?
Contents
Solution 1
There are choices for the first endpoint of the line segment, and choices for the second endpoint, giving a total of segments. However, both and were counted, while they really are the same line segment. Every segment got double counted in a similar manner, so there are really line segments, and the answer is .
In shorthand notation, we're choosing endpoints from a set of endpoints, and the answer is .
Solution 2
Each segment is either an edge, a facial diagonal, or a long/main/spacial diagonal.
A cube has edges: Four on the top face, four on the bottom face, and four that connect the top face to the bottom face.
A cube has square faces, each of which has facial diagonals, for a total of .
A cube has spacial diagonals: each diagonal goes from one of the bottom vertices to the "opposite" top vertex.
Thus, there are segments, and the answer is .
See also
1996 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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