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Combinatorics Difficulty 4.8 AIME Find the answer Canada

In the diagram, each row, each column, and each shape shown by
the thick lines must contain the letters $A$ , $B$ , $C$ , $D$ , and $E$ .

Hide/Reveal Description of Diagram for Question 24

A five by five grid with thick lines dividing the grid into five different shapes. Each shape is made up of five connected squares from the grid. Numbering the rows in the grid 1 through 5 from top to bottom and the columns 1 through 5 from left to right, the squares making up each of the five shapes are as follows:

Squares 1 through 4 in column 1 and square 3 in column 2.
Squares 2 through 5 in row 1 and square 4 in row 2.
Squares 2 and 3 in row 2, square 3 in row 3, and squares 3 and 4 in row 4.
Square 2 in row 4 and squares 1 through 4 in row 5.
Square 3 in column 4 and squares 2 through 5 in column 5.

There are letters in five of the squares as follows:

$A$ is in row 2, column
4.
$B$ is in row 5, column
3.
$C$ is in row 3, column
2.
$D$ is in row 2, column
3.
$E$ is in row 5, column
5.

The square in row 4, column 1 is shaded.

If each square contains exactly one letter, what letter must be
placed in the shaded square?

Pick one

Solution

We begin by naming shapes shown by the thick lines 1, 2, 3, 4,
and 5, as shown in the diagram.

[[IMAGE0]]

Hide/Reveal Description of Shapes 1 - 5

Shape 1: Squares 1 through 4 in column 1 and square 3 in column
2.
Shape 2: Squares 2 through 5 in row 1 and square 4 in row
2.
Shape 3: Squares 2 and 3 in row 2, square 3 in row 3, and squares
3 and 4 in row 4.
Shape 4: Square 2 in row 4 and squares 1 through 4 in row
5.
Shape 5: Square 3 in column 4 and squares 2 through 5 in column
5.

We also adopt the notation [row number, column number] to refer to
the contents of specific squares in the diagram.

For example, we are given that $[2,3]$ is $D$ (this is in shape 3), and $[2,4]$ is $A$ (this is in shape 2).

It is important to note that the solution that follows is one of many
different ways to arrive at the final answer.

Since $[3,2]$ is $C$ , then $[2,1]$ cannot be $C$ (shape 1 already has $C$ ), and $[2,2]$ cannot be $C$ (column 2 already has $C$ ), and thus the $C$ in row 2 must be $[2,5]$ .

Since row 5 already has an $E$ , then
the $E$ in shape 5 cannot be in row
5, and thus $[4,2]$ is $E$ .

Row 2 is missing $B$ and $E$ and since column 2 has an $E$ , then $[2,1]$ is $E$ and $[2,2]$ is $B$ .

The letters determined to this point are included in the diagram
below.

[[IMAGE1]]

Since shape 2 already contains $A$
and row 1 is missing $A$ , then $[1,1]$ is $A$ .

Also, column 4 already contains $A$
and so $[5,4]$ is not $A$ which means that $[5,2]$ is $A$ (since shape 5 is missing $A$ ).

The letters determined to this point are included in the diagram
below.

[[IMAGE2]]

Shape 1 must contain $D$ somewhere
in column 1 and so $[5,1]$ is not
$D$ . Since shape 5 is missing $D$ , then $[5,4]$ is $D$ .

Shape 4 is missing $A$ , $B$ and $D$ . Since column 4 already contains $A$ and $D$ , then $[3,4]$ cannot be $A$ or $D$ , and thus $[3,4]$ is $B$ .

Finally, shape 1 is missing $B$ and
$D$ . Since row 3 contains $B$ , then $[3,1]$ cannot be $B$ , and so the letter in the shaded
square, $[4,1]$ , is $B$ as shown.

[[IMAGE3]]

The completed diagram is included below. Given the initial 5 letters
and their locations, there is exactly one way to complete this
diagram.

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