HMMT 二月 2015 · 冲刺赛 · 第 16 题

英文解析

16. [ 8 ] Determine the number of unordered triples of distinct points in the 4 × 4 × 4 lattice grid
    3 3
    { 0 , 1 , 2 , 3 } that are collinear in R (i.e. there exists a line passing through the three points).
    Answer: 376 Define a main plane to be one of the xy, yz, zx planes. Define a space diagonal tobe a set of collinear points not parallel to a main plane. We classify the lines as follows:
    (a) Lines parallel to two axes (i.e. orthogonal to a main plane). Notice that given a planeof the form v = k , where v ∈ { x, y, z } , k ∈ { 0 , 1 , 2 , 3 } , there are 8 such lines, four in one directionand four in a perpendicular direction. There are 4 × 3 = 12 such planes. However, each line lies
    8 × 4 × 3
    in two of these ( v, k ) planes, so there are = 48 such lines. Each of these lines has 4 points,
    so there are 4 possible ways to choose 3 collinear points, giving 4 × 48 = 192 triplets.2
    Guts
    (b) Diagonal lines containing four points parallel to some main plane. Consider a plane ofthe form ( v, k ), as defined above. These each have 2 diagonals that contain 4 collinear points.
    Each of these diagonals uniquely determines v, k so these diagonals are each counted once. Thereare 12 possible ( v, k ) pairs, yielding 12 × 2 × 4 = 96 triplets.
    (c) Diagonal lines containing three points parallel to some main plane. Again, consider aplane ( v, k ). By inspection, there are four such lines and one way to choose the triplet of pointsfor each of these lines. This yields 4 × 12 = 48 triplets.
    (d) Main diagonals. There are four main diagonals, each with 4 collinear points, yielding 4 × 4 = 16
    triplets.
    (e) Space diagonals containing three points. Choose one of the points in the set { 1 , 2 } to be 3
    the midpoint of the line. Since these 8 possibilities are symmetric, say we take the point (1 , 1 , 1).
    There are four space diagonals passing through this point, but one is a main diagonal. So each ofthe 8 points has 3 such diagonals with 3 points each, yielding 8 × 3 = 24 ways.
    Adding all these yields 192 + 96 + 48 + 16 + 24 = 376.