HMMT 二月 2020 · 团队赛 · 第 2 题

HMMT February 2020 — Team Round — Problem 2

[25] Let n be a fixed positive integer. An n -staircase is a polyomino with cells arranged in the 2 shape of a staircase, with arbitrary size. Here are two examples of 5-staircases: Prove that an n -staircase can be dissected into strictly smaller n -staircases.

解析

[25] Let n be a fixed positive integer. An n -staircase is a polyomino with cells arranged in the 2 shape of a staircase, with arbitrary size. Here are two examples of 5-staircases: Prove that an n -staircase can be dissected into strictly smaller n -staircases. Proposed by: James Lin Solution 1: Viewing the problem in reverse, it is equivalent to show that we can use multiple n - staircases to make a single, larger n -staircase, because that larger n -staircase is made up of strictly smaller n -staircases, and is the example we need. For the construction, we first attach two n -staircases of the same size together to make an n × ( n + 1) rectangle. Then, we arrange n ( n +1) of these rectangles in a ( n +1) × n grid, giving an n ( n +1) × n ( n +1) n ( n +1) 2 2 size square. Finally, we can use of these squares to create a larger n -staircase of n ( n + 1) 2 smaller staircases, so we are done. Solution 2: An alternative construction using only 2 n + 2 staircases was submitted by team Yeah Knights A. We provide a diagram for n = 5 and allow the interested reader to fill in the details.