A bag contains 300 marbles, each of the same size, but in six different colours. Suppose there are 50 marbles of each colour. What is the smallest number of marbles I must pick to be absolutely sure that there are 4 marbles of the same colour among the marbles I have picked blindfolded?

suppose you pick 18 marbles -- 3 of each color. The 19th marble must be the same color as one of the 18.

To find the smallest number of marbles you must pick to be absolutely sure that you have picked 4 marbles of the same color, you can use the Pigeonhole Principle.

The Pigeonhole Principle states that if you have more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon.

In this case, think of each color of the marble as a pigeonhole, and the number of marbles of each color as pigeons.

Since there are 6 different colors and 50 marbles of each color, you have a total of 6 pigeonholes and 6 x 50 = 300 pigeons.

To guarantee 4 marbles of the same color, you need to ensure that you have picked at least 3 marbles from each of the 6 pigeonholes (colors).

So the total number of marbles you need to pick would be: 6 x 3 + 1 = 19.

Therefore, the smallest number of marbles you must pick to be absolutely sure that there are 4 marbles of the same color is 19.