Leaky roof
Car talk Puzzler,
March 12-13, 2005
Twelve people were in a three-story house (no basement), some on each of three floors. It started to rain, and the roof started to leak.
So the people on the third floor went to the second floor. But the second floor people said, “Not all of you can stay here. We'll let only as many of you stay as we have here in our own group.” So some went back to the third floor.
It continued to rain, and the second floor ceiling began to leak.
So the people on the second floor went to the first floor. But the first floor people said, “Not all of you can stay here. We’ll let only as many of you stay as we have here in our own group.” So some went back to the second floor.
Now there were 4 people on each floor.
How many were on each floor to begin with?
There are several ways to solve this problem. Here is an algebraic solution.
Let a = the number of people originally on the third floor.
Let b = the number of people originally on the second floor.
Let c = the number of people originally on the first floor.
a + b + c = 12
| No. of people on | Time 1 | Time 2 | Time 3 |
| third floor | a | a - b | a - b = 4 |
| second floor | b | b + b | b + b - c = 4 |
| first floor | c | c | c + c = 4 |
| total | 12 | 12 | 12 |
2c = 4
c = 2
2b-c = 4
2b-2 = 4
2b = 6
b = 3
a = 12 - b - c = 12 - 3 - 2 = 7
So to begin with, there were 7 people on the third floor, 3 on the second, and 2 on the first.
A solution without algebra.
At the end, there were four people on each floor. But in the last move, the
people on the first floor let only two (half of four) people stay with
them. So originally there were two people on the first floor.
In the next-to-last move, then, there were six (two plus four) people on the
second floor. But the people on the second floor let only three
(half of six) people stay with the. So originally there were three
people on the second floor.
So originally on the third floor there were three plus four equals seven
people.