Problem 17 · 1997 AJHSME
Hard
Geometry & Measurement
careful-counting
Show answer
Answer: E — 16.
Show hints
Hint 1 of 2
Diagonals come in two flavors: ones lying flat ON a face, and ones boring through the INSIDE of the cube. Count the two types separately so you don't miss the inside ones (like segment y).
Still stuck? Show hint 2 →
Hint 2 of 2
Sort by type to avoid double-counting or omissions: face diagonals (2 per face) plus interior space diagonals (one from each vertex to its far corner).
Show solution
Approach: split into face diagonals and space diagonals
- Face diagonals: each of the 6 faces is a square with 2 diagonals, so 6 × 2 = 12 (segment x is one of these).
- Space diagonals: each vertex has exactly one opposite vertex through the interior, and 8 vertices pair up into 8 ÷ 2 = 4 such diagonals (segment y is one).
- Total = 12 + 4 = 16.
Another way — all vertex pairs minus the edges:
- Connect every pair of the 8 vertices: that's C(8,2) = (8 × 7)/2 = 28 segments in all.
- Of those, 12 are edges (not diagonals). A diagonal is any vertex-pair that isn't an edge: 28 − 12 = 16.
- Lesson: 'count everything, subtract what doesn't qualify' is often faster than itemizing each kind.
Mark:
· log in to save