Zorg Has Socks

If Zorg took 9 socks, he could have 3 of each colour so no 2 pairs of one color. However, as soon as he takes the 10th sock, no matter which colour it is, he will have 4 socks of this colour. So 10 is enough to guarantee that.

Let's use algebra for this problem.

If we use $a$ , $b$ and $c$ as our sock colours, however we don't know which colour $a$ , $b$ or $c$ correspond to (in fact, it doesn't matter), this'll let us solve the problem easily.

Pick 1: $a$

Pick 2: $a + b$

Pick 3: $a + b + c$

Pick 4: $2a + b + c$

Pick 5: $2a + 2b + c$

Pick 6: $2a + 2b + 2c$

Pick 7: $3a + 2b + 2c$

Pick 8: $3a + 3b + 2c$

Pick 9: $3a + 3b + 3c$

Pick 10: $4a + 3b + 3c$

This way, Zorg is guaranteed to get 2 pairs of the same colour sock.