HMMT 二月 2013 · 冲刺赛 · 第 27 题

HMMT February 2013 — Guts Round — Problem 27

[ 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

[ 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 possible 3 to arrange the hyperplane such that it intersects all 8 cells. Namely, x + x + x + x = intersects 1 2 3 4 2 1 1 1 1 1 1 all 8 cells because it passes through (0 , , , ) (which is on the cell x = 0), (1 , , , ) (which is on 1 2 2 2 6 6 6 the cell x = 1), and the points of intersection with the other 6 cells can be found by permuting these 1 quadruples. 1