Problem of the Week
Problem B and Solution
Scaryman
Problem
You are playing the very scary video game Scaryman. Randomly generated
rooms are rectangular, and each has a perimeter of 20 m. The length and width
of each room is a whole number.
a) What are the dimensions of the room with the least area?
b) What are the dimensions of the room with the greatest area?
c) There are five silent monsters randomly placed in the room with the greatest
area. Each monster takes up one square metre. If you step through the door
into the room of greatest area, what is the probability that you will stumble
into a monster?
Solution
The rooms are rectangular, each with a perimeter of 20 m. Since twice the sum
of the side lengths, say l m and w m, must equal the perimeter, we have
2 × (l + w) = 20, or l + w = 10 m. Thus the possible side lengths must sum to
10 m, and hence they are
l + w = 10 m=(1 + 9)m, (2 + 8)m, (3 + 7)m, (4 + 6)m, and (5 + 5)m.
The corresponding rectangles have areas of 1 × 9 = 9 m2 , 2 × 8 = 16 m2 ,
3 × 7 = 21 m2 , 4 × 6 = 24 m2 , and 5 × 5 = 25 m2 .
(Note that a rectangle can be a square.)
a) The room with least area has dimensions 1 m by 9 m.
b) The room with greatest area has dimensions 5 m by 5 m.
c) Since each of the 5 monsters takes up 1 m2 , and there are 25 squares of that
size in the 5 m by 5 m room, the chance that you will stumble into a
5
monster is 5 in 25, or 25
= 51 .