Please solve the following problem: When the base-16 number $66666_{16}$ is written in base 2, how many base-2 digits (bits) does it have?

Amanita, let's solve the problem step by step:

Step 1: Understand the Problem We are given a number in base-16 (hexadecimal): $66666_{16}$. We need to find out how many digits (bits) this number will have when written in base-2 (binary).

Step 2: Convert the Base-16 Number to Base-10 First, let's convert $66666_{16}$ to base-10 (decimal). This will help us work with the number more easily. The formula for converting from base-16 to base-10 is:

$66666_{16} = 6 \times 16^4 + 6 \times 16^3 + 6 \times 16^2 + 6 \times 16^1 + 6 \times 16^0$

Step 3: Find the Number of Binary Digits Once we have the decimal value, the number of binary digits (bits) in a number $n$ is given by:

$\text{Number of bits} = \lfloor \log_2 n \rfloor + 1$

where $\lfloor x \rfloor$ means the greatest integer less than or equal to $x$.

Let's execute these steps.

Step 2: Convert $66666_{16}$ to decimal

Let's calculate the decimal value. Amanita, here’s how we solved the problem:

1. We first converted the base-16 number $66666_{16}$ to decimal. Using the formula for base conversion, we found:

$66666_{16} = 6 \times 16^4 + 6 \times 16^3 + 6 \times 16^2 + 6 \times 16^1 + 6 \times 16^0 = 419430$

2. To find out how many binary digits (bits) this number has, we used the formula:

$\text{Number of bits} = \lfloor \log_2{419430} \rfloor + 1$

Calculating this, we get:

$\lfloor \log_2{419430} \rfloor + 1 = 19$

Final Answer:
When the base-16 number $66666_{16}$ is written in base 2, it has $\boxed{19}$ binary digits (bits).