Problem
of the Week
Problem
B and Solution
Not
a Tetris^{TM} Game

Problem

On June 6, 2024, the puzzle game Tetris^{TM} will be 40 years
old! The game of Tetris^{TM} uses pieces called “tetrominoes”,
which are shapes composed of four identical squares, like the ones given
at the bottom of this page. This problem is inspired by
Tetris^{TM}.

In this problem, tetromino pieces are to be placed in a grid
according to the following rules:

Pieces may be rotated or reflected (flipped over).

Pieces may not overlap each other and each
square in a piece must be placed directly on top of a square in the
grid.

Only the given pieces may be used, but you do not need to use all
of them.

The goal is to cover as many squares in the grid as possible with the
pieces. Is it possible to cover all the squares in the given grid?
Explain why or why not.

When answering this question, you may find it helpful to cut out the
given tetrominoes and place them on the grid.

There are five
different types of tetrominoes. Each type is made by arranging four
identical squares in a different way.

In the first type, the four squares are arranged into a column.
There are three pieces of this type.

In the second type, three squares are arranged into a column and
the fourth square is placed along the right side of the middle square in
the column. There are four pieces of this type.

In the third type, three squares are arranged into a column and
the fourth square is placed along the right side of the top square in
the column. There are four pieces of this type.

In the fourth type, two squares are arranged into a column, the
third square is placed along the left side of the top square in the
column, and the fourth square is placed along the right side of the
bottom square in the column. There are five pieces of this
type.

In the fifth type, the four squares are arranged into two rows
and two columns forming a larger square. There are five pieces of this
type.

Solution

The grid has a total of \(7 \times
10=70\) squares and each piece has \(4\) squares. Since \(70 \div 4 = 17.5\), which is not a whole
number, that tells us that \(70\) is
not a multiple of \(4\). So it is not
possible to cover all the squares in the grid. At most, we would be able
to cover \(17 \times 4 = 68\) of the
squares. One such possibility is shown.