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Problem of the Week

Problem D and Solution

Another Path Using Math

A landscaper needs to fill a path measuring 2 feet by 8 feet with patio stones. The patio stones are each 1 foot by 2 feet, so the landscaper calculates that she will need 8 of them.
Before arranging the patio stones, the landscaper wants to look at all her options. She can not cut or overlap the stones, and they all must fit inside the path area without any gaps. Two possible arrangements of the stones are shown below. How many different arrangements are there in total?

One possible arrangement of eight, 1 by 2 rectangles filling a 2 by 8 rectangle. There is a pair of rectangles with a horizontal orientation one on top of the other, followed by one rectangle with a vertical orientation, followed by another pair of rectangles placed horizontally, then another vertical rectangle and finally another pair of horizontal rectangles.   Another possible arrangement consisting of two vertical rectangles side by side, followed by a pair of horizontal rectangles one on top of the other, then another pair of horizontal rectangles, and finally two more vertical rectangles side by side.


Let’s consider the ways that the patio stones can be arranged. We will imagine we are looking at the path from the side, just like in the images shown in the question. First, notice that there must always be an even number of patio stones that have a horizontal orientation, because they must be placed in pairs.

Therefore, the total number of different arrangements of the patio stones is \(1+7+(5+4+3+2+1) + (4+3+2+1)+1 = 34.\)