HMMT 二月 2004 · GEN1 赛 · 第 10 题
HMMT February 2004 — GEN1 Round — Problem 10
题目详情
- A floor is tiled with equilateral triangles of side length 1, as shown. If you drop a needle of length 2 somewhere on the floor, what is the largest number of triangles it could end up intersecting? (Only count the triangles whose interiors are met by the needle — touching along edges or at corners doesn’t qualify.) 2
解析
- A floor is tiled with equilateral triangles of side length 1, as shown. If you drop a needle of length 2 somewhere on the floor, what is the largest number of triangles it could end up intersecting? (Only count the triangles whose interiors are met by the needle — touching along edges or at corners doesn’t qualify.) Solution: 8 Let L be the union of all the lines of the tiling. Imagine walking from one end of the needle to the other. We enter a new triangle precisely when we cross one of the lines of the tiling. Therefore, the problem is equivalent to maximizing the number of times the needle crosses L . Now, the lines of the tiling each run in one of three directions. It is clear that the needle cannot cross more than three lines in any given direction, √ since the lines are a distance 3 / 2 apart and the needle would therefore have to be of √ length greater than 3 3 / 2 > 2. Moreover, it cannot cross three lines in each of two different directions. To see this, notice that its endpoints would have to lie in either the two light-shaded regions or the two dark-shaded regions shown, but the closest two points of such opposite regions are at a distance of 2 (twice the length of a side of a triangle), so the needle cannot penetrate both regions. Therefore, the needle can cross at most three lines in one direction and two lines in each of the other two directions, making for a maximum of 3 + 2 + 2 = 7 crossings and 7 + 1 = 8 triangles intersected. The example shows that 8 is achievable, as long as the √ needle has length greater than 3 < 2. 4