MHB Smallest N for Ensuring 3 Colors in 100 Marble Draw

[anemone]

There are 111 marbles in a box, each being green, yellow, purple and blue. It's known that if 100 marbles are drawn, we can ensure getting marbles of all four colors.

Find the smallest integer $N$ such that if $N$ marbles are drawn, we can ensure getting marbles of at least three different colors._______________________________________________________________________________________________________

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[anemone]

No one answered last week's problem. :(

You can find the proposed solution below:

Spoiler

First off, note that $N>87$. This is because if there are $12$ blue, $12$ purple, $12$ yellow and $75$ green marbles in the box, we don't necessarily get marbles of at least three different colors when at most $87$ marbles are drawn (consider we many only get 75 green and 10 yellow marbles), and this combination satisfies the given condition if $100$ marbles are drawn, we can be sure to get marbles of all four colors because when $100$ marbles are drawn, only $11$ marbles are missing and hence no color can be missing from the marbles drawn as there are at least $12$ marbles of each color.

Next, we show that $N=88$ will meet the need. When $88$ marbles are drawn, $111-88=23$ marbles are left and it is entirely impossible to have 2 colors missing (in which case that requires at least $24$ marbles to be left over). It follows that $N=88$ is the answer.