The Logical Puzzle - 28 Mar 2008

Solution to March’s Puzzle: Too Clever by Half

A chicken and a half lay an egg and a half in a day and a half. How many eggs would one chicken lay in three days? A builder and a half build a house and a half in a year and a half using a tool and a half. How many houses would one builder build in nine years? Can you generalize your calculation to solve both equations?

The intuitive yet incorrect answer to the chicken and eggs puzzle is three eggs. Our brain plays a trick on us and makes us think that if a chicken and a half lay an egg and a half in a day and a half, one chicken lays one egg in one day. But if you express the relationship between chickens, days, and eggs mathematically, you get the equation

3/2 chickens × 3/2 days = 3/2 eggs

Reducing the number of chickens from 3/2 to 1 is achieved by dividing the original number by 3/2. For the equation to be true, you also need to divide the number of eggs (3/2) by 3/2, giving you the equation 1 chicken × 3/2 days = 1 egg

Reducing the number of days from 3/2 to 1 has a similar effect on the number of eggs; namely, you need to divide 1 (egg) by 3/2, giving you the equation

1 chicken × 1 day = 2/3 egg

If you increase the number of days from 1 to 3, the effect on the number of eggs is a factor of 3 as well:

1 chicken × 3 days = 2 eggs

In a very similar manner you can express the relationships between builders, houses, years, and tools with the following equation:

3/2 builders × 3/2 years × 3/2 tools = 3/2 house

To reduce the number of builders, years, and tools to one each, you need to divide the number of houses by 3/2 three times; in other words, by (3/2)^3:

1 builder × 1 year × 1 tool = 3/2 ÷ 3/2 ÷ 3/2 ÷ 3/2 houses

Giving you the equation 1 builder × 1 year × 1 tool = 4/9 house

Thus, one builder with one tool will build four houses in nine years.

To generalization the equation, you need to divide the right side of the equation by 3/2 n times for n elements in the left side of the equation. Or, if you want to express the calculation as a multiplication instead of division, multiply by (2/3)^n. For example, in our last equation:

3/2 builders × 3/2 years × 3/2 tools = 3/2 houses

The left side of the equation contains three elements; therefore

1 builder × 1 year × 1 tool = 3/2 × (2/3)^3 houses

which is equal to

1 builder × 1 year × 1 tool = 4/9 house

April’s Puzzle: Suicidal Mosquito

Two trains drive toward each other on the same rail. Both trains drive at a speed of 100MPH. When the trains are 100 miles apart, a mosquito starts flying back and forth from the front of one train to the other at a speed of 200MPH. What total distance will the mosquito cover before the two trains crash?