2023 AMC 8 Problem 23
Attempt Problem 23 of the 2023 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
23.
Each square in a grid is randomly filled with one of the shaded-and-unshaded tiles shown below on the right.
What is the probability that the tiling will contain a large shaded diamond in one of the smaller grids? Below is an example of such a tiling.
Answer: C
Small hint:
Choose which grid contains the large diamond
Big hint:
Once a grid is chosen, its four tile orientations are forced
Video solution:
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Written solution:
There are possible tilings. There are possible grids where a large shaded diamond could appear.
After one of these grids is chosen, the four tile orientations inside it are forced, and the other squares can be filled in any way. This gives tilings for each chosen grid.
Two different grids cannot both contain a large shaded diamond, because their overlapping squares would require incompatible tile orientations. Therefore the number of favorable tilings is
The desired probability is then
Thus, C is the correct answer.
Problem 23 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
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- 1993 AMC 8
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