2022 AMC 8 Problem 23
Attempt Problem 23 of the 2022 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 2022 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
23.
A or is placed in each of the nine squares in a grid. Shown below is a sample configuration with three ’s in a line.
How many configurations will have three ’s in a line and three ’s in a line?
Small hint:
Split by how many triangles are in the grid
Big hint:
With exactly triangles, diagonal triangle lines do not work
Solution:
Let be the number of triangles. To have both a triangle line and a circle line, we need .
If , the three triangles must form a line. A row or column works, because the remaining six circles contain a full circle row or column. A diagonal does not work, because its complement contains no full line of circles. Thus there are configurations for . By symmetry, there are also configurations for .
If , choose the triangle line and then the extra triangle. If the line is a row or column, all possible extra positions work, because one parallel row or column remains all circles. This gives configurations. If the triangle line is a diagonal, no extra position leaves a full circle line. By symmetry, also gives configurations.
The total number of configurations is .
Thus, the correct answer is D.
Problem 23 in other years
- 1985 AMC 8
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