2024 AMC 8
Complete problem set with solutions and individual problem pages
A rectangle is covered without overlap by 3 shapes of tiles: , , and , shown below. What is the minimum possible number of tiles used?
- A.
- B.
- C.
- D.
- E.
Solution 1
We can eliminate B, C, and D, because they are not subtracted by any multiple of . Finally, we see that there is no way to have A, so the solution is .
Solution 2
Let be the number of by tiles. There are squares and each by or by tile takes up 4 squares, so , so it is either or . Color the columns, starting with red, then blue, and alternating colors, ending with a red column. There are red squares and blue squares, but each by and by shape takes up an equal number of blue and red squares, so there must be more by tiles on red squares than on blue squares, which is impossible if there is just one, so the answer is , which can easily be confirmed to work.
