This seems like such a simple problem but I am stuck.
I have 50 different pizzas. There are 30 humans that have to share them.
I want that each person eats from 6 different pizzas. We can have two different persons eating from the same pizza, but each individual has to eat from 6 different pizzas.
Also each pizza has to be proved, else the solution would be to cut 6 pizzas in 30 pieces and ignoring the rest.
How many slices do I need to cut the pizzas, at the minimun?
First I thought about doing this with combinations and permutations but I am doing something wrong. Is this a minimization problem? Part of the problem for me is it has been a long while since I tried to solve this kind of problems, and I am having trouble finding the "context" or branch of mathematics I should use.
I am sorry if this is not the right place/way to ask.
Think of the pizzas as an excel-sheet with 50 columns. Now in the first row start entering the first name, for six columns, then the next name, for the next six columns... 8x6=48, so you'll be able to fit eight names in the first row, with two pizzas not yet sampled.
As Matti P. said, 6 x 30 = 180 pizza-slices (or in our case excel-cells) need to be named. Three rows will get you 150 cells, four rows 200 cells, so depending on whether you have to cut all pizzas the same, you'll either have to cut them all into 4 pieces, or 30 into four pieces and 20 into three.
No cell (=piece) can be occupied by more than one name, because every tester eats one whole piece, therefore you need at least (Number_of_testers x number_of_samples) 180 servings. The minimal cutting depends on the rules for cutting (pizzas individually (30@4p 20@3), or stack the pizzas and cut all at once (50@4p, with some uneaten pieces).