The problem can be solve with Pigeon Hole Principle. It states that
If you put n + 1 pigeons in n holes then at least one hole would have more than one pigeon.
Consider 4 different colors as 4 holes, now we want to find out the minimum number of pens such that at least one whole contains more than 4 pens.
If we start with 4 holes- black, blue, green, red.
Start putting a black , blue, green and red pen in respective hole. Now each hole has exactly one pen.
If we repeat this again, then each hole will contain exactly 2 pens.
Repeat the same third times to have exactly 3 pens in each hole.
Now if we select a random pen and put in a hole, at least one hole will hold 4 pens, which is what we need .
The total number of pens selected = 4 + 4 + 4 + 1 = 13
