1990 AMC 8 Problem 25
Attempt Problem 25 of the 1990 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1990 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
25.
How many different patterns can be made by shading exactly two of the nine unit squares in a grid? Patterns that can be matched by flips and/or turns are not considered different.
Small hint:
The nine squares come in three types: corners, edge-middles, and center; describe each pattern by which types its two shaded squares use and how far apart they are
Big hint:
Split into cases by whether a corner is shaded, being careful that flips and turns merge many placements into one pattern
Solution:
Classify the two shaded squares up to rotations and reflections. Patterns that include a corner: two adjacent corners (same edge), two diagonally opposite corners, corner with the center, corner with an adjacent edge-middle, and corner with a far edge-middle. That is patterns.
Patterns with no corner: two adjacent edge-middles, two opposite edge-middles, and an edge-middle with the center. That is more.
In total there are distinct patterns.
Thus, the correct answer is C .
Problem 25 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1991 AMC 8
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