John has arranged 100 empty piggy bank dolls and numbered them serially from 1 to 100. In the first turn he puts one coin in each doll. In the second turn he puts one coin in each of the 2nd, 4th, 6th, …, 98th, and 100th doll. In the third turn he puts one coin in each of the 3rd, 6th, 9th, …, 96th, and 99th doll. He continues this process and at last puts a coin in the 100th doll. How many dolls have an odd number of coins?

25

20

15

10

To solve this problem, we need to analyze the pattern of coins being added to each piggy bank doll.

1. **Understanding the process**:
   - On the first turn, John puts one coin in each of the 100 dolls.
   - On the second turn, he adds one coin to every doll with an even number.
   - On the third turn, he adds one coin to every doll whose number is a multiple of 3.
   - This pattern continues, where on the \( n \)th turn, he adds a coin to every doll whose number is a multiple of \( n \).

2. **Counting the number of coins in each doll**:
   - For each doll \( k \), a coin is added to it on each turn \( n \) where \( k \) is a multiple of \( n \).
   - Therefore, the number of coins in doll \( k \) equals the number of divisors of \( k \).

3. **Odd number of coins**:
   - A doll will have an odd number of coins if the number of its divisors is odd.
   - The number of divisors of a number is odd if and only if the number is a perfect square. This is because divisors usually come in pairs, but a perfect square has one unpaired divisor (the square root).

4. **Finding the dolls with an odd number of coins**:
   - We need to count how many numbers between 1 and 100 are perfect squares.
   - The perfect squares between 1 and 100 are \( 1^2, 2^2, 3^2, \dots, 10^2 \), which are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

There are 10 such perfect squares.

**Answer**: The number of dolls that have an odd number of coins is **10**.

Would you like more details or have any questions about this? Here are some related questions you might find interesting:

1. What is the general formula to calculate the number of divisors of a number?
2. Can this pattern be generalized to more than 100 dolls?
3. How many dolls would have an even number of coins in this scenario?
4. What happens if we only consider dolls numbered with prime numbers?
5. How can we determine if a number is a perfect square without calculating its square root?

**Tip**: Whenever you encounter a problem that involves divisors or factors, remember that the number of divisors can give you important information, like whether the total count is odd or even, which can indicate something special about the number.

Explore how to solve John's piggy bank problem involving dolls and coins. Learn about the distribution of coins across dolls and identify how many dolls have an odd number of coins. Suitable for high school students interested in number theory and problem-solving.

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