hi, please help me on this problem, I have no idea how to start. thank you
A 64 x 64 grid is filled with 2 x 2 matrices. The four entries in each matrix are chosen without replacement from the set {a,b,c,d,e,f}. What is the least number of identical matrices guaranteed to appear in the grid? Justify your answer.
Let n be the number of empty matrices on the grid, and let m be the number of all possible distinct matrices filled with elements of {a,b,c,d,e,f}. Find n and m.
This is a problem on the generalized pigeonhole principle. Here containers are distinct filled matrices, pigeons are initially empty matrices on the grid, and a pigeon (empty matrix) is in a container (filled matrix) if it is going to be filled that way. The link gives a formula for the minimum number of pigeons in a container.
An intuitive way to think about this is to consider the worst case, when as many matrices as possible are different. We fill the grid with filled matrices #1, #2, ..., #m, then again #1, #2, ..., #m and so on. How many such complete series of m matrices will be there and what about the remaining matrices?