2023 AMC 8 Problems/Problem 23
Each square in a grid is randomly filled with one of the
gray and white tiles shown below on the right.
What is the probability that the tiling will contain a large gray diamond in one of the smaller
grids? Below is an example of such tiling.
Contents
Video Solution 1 by OmegaLearn (Using Cool Probability Technique)
Solution 1
Probability is total favorable outcomes over total outcomes, so we can find these separately to determine the answer.
There are ways to choose the big diamond location from our
square grid. From our given problem there are
different arrangements of triangles for every square. This implies that from having
diamond we are going to have
distinct patterns outside of the diamond. This gives a total of
favorable cases.
There are 9 squares and 4 possible designs for each square, giving total outcomes. Thus, our desired probability is
.
-apex304, SohumUttamchandani, wuwang2002, TaeKim. Cxrupptedpat
Solution 2 (Linearity of Expectation)
Let , and
denote the
smaller
squares within the
square in some order. For each
, let
if it contains a large gray diamond tiling and
otherwise. This means that
is the probability that square
has a large gray diamond, so
is our desired probability. However, since there is only one possible way to arrange the squares within every
square to form such a tiling, we have
for all
(as each of the smallest tiles has
possible arrangements), and from the linearity of expectation we get
~eibc
Remark: This method might be too advanced for the AMC 8, and is probably unnecessary (refer to the other solutions for simpler techniques).
Solution 3
Note that the middle tile can be any of the four tiles. The white part of the middle tile points towards one of the corners, and for the white diamond to appear the three adjacent tiles must all be perfect. Thus, the solution is
~aayr
Animated Video Solution
~Star League (https://starleague.us)