Question

The Dog Lovers Society has 49152 members nationwide. The president, vice-president, and secretary contact four members (round 1), each of whom contact four more members (round 2), each of whom contact four more members (round 3), etc.
How many rounds of communications are needed before the entire organization is notified?

Answer 1

To solve the problem, we need to analyze how many members can be contacted after each round of communication.

Let's break it down as follows:

1. **Initial Contacts**: In round 1, the president, vice-president, and secretary contact 4 members. Hence, the total number of members contacted after round 1 is:
\[
4
\]

2. **Subsequent Rounds**: Each of the contacted members continues this pattern. In round 2, these 4 members each contact 4 more members. Therefore, the number of new members contacted in this round is:
\[
4 \times 4 = 16
\]
The cumulative total after round 2 is:
\[
4 + 16 = 20
\]

3. In round 3, each of the newly contacted 16 members also contacts 4 more members, resulting in:
\[
16 \times 4 = 64
\]
The cumulative total after round 3 is:
\[
20 + 64 = 84
\]

4. In round 4, each of the 64 members contacts 4 more members:
\[
64 \times 4 = 256
\]
The cumulative total after round 4 is:
\[
84 + 256 = 340
\]

5. Continuing this process, in round 5:
\[
256 \times 4 = 1024
\]
The cumulative total after round 5 is:
\[
340 + 1024 = 1364
\]

6. In round 6:
\[
1024 \times 4 = 4096
\]
The cumulative total after round 6 is:
\[
1364 + 4096 = 5460
\]

7. In round 7:
\[
4096 \times 4 = 16384
\]
The cumulative total after round 7 is:
\[
5460 + 16384 = 21844
\]

8. In round 8:
\[
16384 \times 4 = 65536
\]
The cumulative total after round 8 is:
\[
21844 + 65536 = 87380
\]

Now, let’s summarize the members contacted through each round:

- Round 1: \(4\)
- Round 2: \(16\)
- Round 3: \(64\)
- Round 4: \(256\)
- Round 5: \(1024\)
- Round 6: \(4096\)
- Round 7: \(16384\)
- Round 8: \(65536\)

Now we notice that by round 8 we have exceeded 49,152 members. We will now check how many rounds we actually needed to reach at least 49152 members.

### Calculating in Exponential Terms

Instead of iteratively summing, we notice we can calculate the total directly:
- In round \(n\), the contactable members can be represented as \(4^n\).
- The sum of members contacted cumulatively through \(n\) rounds can be represented as:
\[
4^1 + 4^2 + 4^3 + ... + 4^n = 4 \left( \frac{4^n - 1}{3} \right) \text{ (using the formula for the sum of a geometric series)}
\]

We need:
\[
4 \frac{4^n - 1}{3} \geq 49152
\]
This simplifies to:
\[
4^n - 1 \geq 3 \cdot 49152 / 4
\]
Calculating:
\[
3 \cdot 49152 / 4 = 36864 \implies 4^n \geq 36865
\]
Calculating powers of 4:
- \(4^1 = 4\)
- \(4^2 = 16\)
- \(4^3 = 64\)
- \(4^4 = 256\)
- \(4^5 = 1024\)
- \(4^6 = 4096\)
- \(4^7 = 16384\)
- \(4^8 = 65536\)

Thus, we find \(4^8 = 65536\) is the first power that is greater than 36865.

### Conclusion
This means that the entire organization is notified by the end of round 8. Therefore, the number of rounds of communications needed is:

\[
\boxed{8}
\]