2024 AMC 8 Problems/Problem 7

Problem

A $3$ x $7$ rectangle is covered without overlap by 3 shapes of tiles: $2$ x $2$ , $1$ x $4$ , and $1$ x $1$ , shown below. What is the minimum possible number of $1$ x $1$ tiles used?

$\textbf{(A) } 1\qquad\textbf{(B)} 2\qquad\textbf{(C) } 3\qquad\textbf{(D) } 4\qquad\textbf{(E) } 5$

Solution 1

We can eliminate B, C, and D, because they are not $21-$ any multiple of $4$ . Finally, we see that there is no way to have A, so the solution is $\boxed{\textbf{(E)} 5}$ .

Solution 2

Let $x$ be the number of $1x1$ tiles. There are $21$ squares and each $2x2$ or $1x4$ tile takes up 4 squares, so $x \equiv 1 \pmod{4}$ , so it is either $1$ or $5$ . Color the columns, starting with red, then blue, and alternating colors, ending with a red column. There are $12$ red squares and $9$ blue squares, but each $2x2$ and $1x4$ shape takes up an equal number of blue and red squares, so there must be $3$ more $1x1$ tiles on red squares than on blue squares, which is impossible if there is just one, so the answer is $\boxed{\textbf{(E)\ 5}}$ , which can easily be confirmed to work