Track / Stage 1 / 145 of 240 #145 of 2444

Problem 145

Combinatorics Difficulty 1.4 Multiple choice CEMC Pascal · Canada · 2026

In the diagram, six $\$4\times$
4$ squares have a percentage of their area shaded.

Hide/Reveal Description of Diagram for Question 10

Six squares drawn on grid paper, arranged into two rows of three.

Top-left square: Divided into four regions of equal area. One of these regions is shaded.
Top-middle square: Divided into eight regions of equal area. Two of these regions are shaded.
Top-right square: Divided into two larger regions of equal area and two smaller regions of equal area. One of the larger regions is shaded and one of the smaller regions is shaded.
Bottom-left square: Divided into sixteen regions of equal area. Five of these regions are shaded.
Bottom-middle square: Divided into four larger regions of equal area and four smaller regions of equal area. Two of the larger regions are shaded and two of the smaller regions are shaded.
Bottom-right square: Divided into two smaller regions of equal area and one larger region twice the size of one of the smaller regions. One of the smaller regions is shaded.

How many of
the $4\times 4$ squares have exactly
$25\%$ of their area shaded?

Pick one

Official solution

The top-left square is divided into $4$ regions having equal area. Since $1$ of these $4$ regions is shaded, then $25\%$ of the square is shaded.

The top-middle square is divided into $8$ regions having equal area. Since $2$ of these $8$ regions are shaded, then $25\%$ of the square is shaded.

The top-right square is divided into $2$ larger regions having equal areas, and
$2$ smaller regions having equal
areas. Since $1$ of the larger
regions is shaded, then more than $25\%$ of the square is shaded.
Alternatively, the shaded region consists of three complete $1\times1$ squares together with $6$ half $1\times1$ squares, and thus the shaded
area is $6$ . The entire $4\times4$ square has area $16$ , and $\frac{6}{16}$ is not equivalent to $25\%$ .

The bottom-left square is divided into $16$ regions having equal area. Since $5$ of these $16$ regions are shaded, then more than
$25\%$ of the square is shaded.

The bottom-middle square is divided into $4$ larger regions having equal areas, and
$4$ smaller regions having equal
areas. Since $2$ of the $4$ larger regions are shaded, and $2$ of the $4$ smaller regions are shaded, then $50\%$ of the square is shaded.

In the bottom-right square, consider constructing a vertical line
segment between the midpoint of the top side of the square and the
midpoint of the bottom side of the square. The square would then be
divided into $4$ regions having equal
areas. Since $1$ of these regions is
shaded, then $25\%$ of the square is
shaded.

Thus, the top-left, top-middle, and bottom-right squares have exactly
$25\%$ of their area shaded and so
there are $3$ such squares.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.