HMMT 二月 2013 · 冲刺赛 · 第 27 题
HMMT February 2013 — Guts Round — Problem 27
题目详情
英文原题
- [ 17 ] Let W be the hypercube { ( x , x , x , x ) | 0 ≤ x , x , x , x ≤ 1 } . The intersection of W and a
1 2 3 4 1 2 3 4
hyperplane parallel to x + x + x + x = 0 is a non-degenerate 3-dimensional polyhedron. What is
1 2 3 4
the maximum number of faces of this polyhedron? 1
解析
英文解析
- [ 17 ] Let W be the hypercube { ( x , x , x , x ) | 0 ≤ x , x , x , x ≤ 1 } . The intersection of W and a
1 2 3 4 1 2 3 4
hyperplane parallel to x + x + x + x = 0 is a non-degenerate 3-dimensional polyhedron. What is
1 2 3 4
the maximum number of faces of this polyhedron?
Answer: 8 The number of faces in the polyhedron is equal to the number of distinct cells (3 dimensional faces) of the hypercube whose interior the hyperplane intersects. However, it is possibleto arrange the hyperplane such that it intersects all 8 cells. Namely, x + x + x + x = intersects 3
1 2 3 4
1 1 1 1 1 12
all 8 cells because it passes through (0 , , , ) (which is on the cell x = 0), (1 , , , ) (which is on
2 2 2 6 6 61
the cell x = 1), and the points of intersection with the other 6 cells can be found by permuting thesequadruples.1 1