PUMaC 2019 · 团队赛 · 第 3 题

PUMaC 2019 — Team Round — Problem 3

Julia is placing identical 1-by-1 tiles on the 2-by-2 grid pictured, one piece at a time, so that every piece she places after the first is adjacent to, but not on top of, some piece she’s already placed. Determine the number of ways that Julia can complete the grid. 4 3 1 2

解析

Julia is placing identical 1-by-1 tiles on the 2-by-2 grid pictured, one piece at a time, so that every piece she places after the first is adjacent to, but not on top of, some piece she’s already placed. Determine the number of ways that Julia can complete the grid. 4 3 1 2 Proposed by: Frank Lu Answer: 16 Julia can choose any of the 4 pieces to place first. Next, she can choose to place any 2 of the pieces adjacent to this first piece. From here, she can place the final two pieces in any order, since both are adjacent to one of the two pieces already placed. There are 2 choices for such an order, This gives us a total of 4 ∗ 2 ∗ 2 = 16 ways for Julia to fill the grid. Note: We also accepted the interpretation where the pieces cannot be placed vertically on top of another. The solution for this case is as follows: Note that in a given column, we must place the piece at the top before the piece at the bottom. We have 2 choices for the first piece, taking either the top left or top right corner. Then, if our 2nd piece fills up the top row, we have 2 more ways to fill in the grid to get 4 ways in this case. Else, our second piece fills up the column, and our last 2 pieces must be, in order, placing the top corner and bottom corner of the opposite corner. This gives us 4 + 2 = 6 cases, which we also accepted as an answer. 1