2024 AMC 8 Problems/Problem 7

Problem

A x rectangle is covered without overlap by 3 shapes of tiles: x, x, and x, shown below. What is the minimum possible number of x tiles used?

Solution 1

We can eliminate B, C, and D, because they are not 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 tiles. There are squares and each or 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 and shape takes up an equal number of blue and red squares, so there must be more 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.

~arfekete

Solution 3

Suppose there are different tiles, different tiles and different tiles. Since the areas of these tiles must total up to (area of the whole grid), we have Reducing modulo gives , or or .

If , then . After some testing, there is no valid pair that works, so the answer must be , which can be constructed in many ways.

-Benedict T (countmath1)

Video Solution 1 (easy to digest) by Power Solve

https://youtu.be/16YYti_pDUg?si=KjRhUdCOAx10kgiW&t=59

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=L83DxusGkSY

See Also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.