A leprechaun places a magic penny under a girl’s pillow. The next night there are 2 magic pennies under her pillow. The following morning she finds four pennies. Apparently, while she sleeps, each penny turns into two magic pennies. The total number of pennies seen under the pillow each day is the grand total; that is, the pennies from each of the previous days are not being stored away until more pennies magically appear. How many days would elapse before she has a total of more than $21 billion? (Proceed by trial and error.)
The Correct Answer and Explanation is:
We are given a scenario where each night, every magic penny doubles in number. This is an example of exponential growth. The total number of pennies under the pillow each morning follows this pattern:
- Day 1: 1 penny
- Day 2: 2 pennies
- Day 3: 4 pennies
- Day 4: 8 pennies
- And so on…
This follows the pattern:
Total pennies on day n = 2^(n – 1)
We want to find the smallest day n such that the total number of pennies is more than 21 billion, or:
2n−1>21, 000, 000, 0002^{n – 1} > 21,\!000,\!000,\!0002n−1>21,000,000,000
Let’s try a few values by trial and error:
- 2¹⁰ = 1,024
- 2²⁰ = 1,048,576
- 2³⁰ ≈ 1.07 billion
- 2³¹ ≈ 2.15 billion
- 2³² ≈ 4.29 billion
- 2³³ ≈ 8.59 billion
- 2³⁴ ≈ 17.18 billion
- 2³⁵ ≈ 34.36 billion
So, 2³⁵ is the first power that exceeds 21 billion.
Since 2^(n – 1) = 2³⁵, then
n – 1 = 35 → n = 36
✅ Final Answer: 36 days
Explanation
This problem involves exponential growth where the number of pennies doubles each night. On the first night, the girl has 1 magic penny. The next night, each penny turns into 2 pennies. Therefore, the growth follows powers of 2. The total number of pennies after n days is given by the expression 2^(n – 1), since we start with 1 penny on the first day (2⁰).
To find when the total exceeds 21 billion, we need to find the smallest power of 2 greater than 21,000,000,000. By testing successive values of 2 raised to increasing powers, we find that 2³⁴ equals approximately 17.18 billion, which is still under 21 billion. However, 2³⁵ equals around 34.36 billion, which is more than 21 billion. That means on the 35th day, the total number of pennies is still under the limit, but on the 36th day, the count exceeds 21 billion.
This trial and error approach is effective for problems with exponential growth where a formula is known but the goal is to determine when a threshold is crossed. It is important to track the powers carefully and check values systematically. The key is understanding that doubling every day leads to rapid growth — in just over a month, the number can go from a single penny to billions.
