Four 1-inch by 1-inch squares are arranged so that each square touches at
least one other square. Any two squares touch each other according to
these rules: They can touch on the corners or they can touch entirely
across one side but not partially across a side. What are the possible
perimeters of the resulting figure?
Notice that the problem does not ask you to find all possible
arrangements of the squares, just all the different perimeters that
can arise. Draw pictures, and look for patterns!
-
Since the perimeter of each square is 4 inches, the perimeter of
the resulting figure can't be more than 16 inches. Also, when you place
a side of one square against a side of another, you lose 2 inches of
perimeter. Since you lose perimeter 2 inches at a time, the perimeter
of the resulting figure must be an even number. Now we have to
figure
out which even numbers can occur.
- By drawing figures as shown below, you can get perimeters of 16, 14,
12, 10 and 8. The perimeter cannot be less than 8, because the minimum
possible perimeter for a given area occurs for the square (given the
constraints of the problem -- in fact, the most efficient figure for
getting minimum perimeter for a given area is a circle - and the
circle that has area = 4 square inches has circumference = about 7.09
inches. So our perimeter must be more than that.)
