Four. (25 points) There are several (more than enough) socks in red, yellow, blue, and white. If any two socks of the same color can make 1 pair, the question is: What is the minimum number of socks needed to ensure that 10 pairs of socks can be formed?
Solution 1 Since there are 4 colors, among 5 socks, there must be 1 pair.

After taking out 1 pair, 3 socks remain. By adding 2 more socks, another pair can be formed.

Following this logic, the number of pairs of socks $(x)$ and the number of socks needed $(y)$ have the following relationship:
$$
y=2 x+3 \text {. }
$$

Thus, to form 10 pairs of socks, only 23 socks are needed.

If 22 socks are taken out, 9 pairs can certainly be formed. If the remaining 4 socks are all different colors, then 22 socks cannot form 10 pairs.

Therefore, at least 23 socks are needed to ensure 10 pairs can be formed.

Solution 2 Since the maximum number of single-color socks that can remain is 4, 24 socks can certainly form 10 pairs.

When 23 socks are taken out, 9 pairs can certainly be formed, leaving 5 socks;
Among the 5 socks, 1 pair can be formed, so 23 socks can also form 10 pairs.
The rest is the same as Solution 1.