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# marbles and boxes

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Euclid

Posts : 95
Join date : 2009-09-15
Age : 57
Location : Alexandria

 Subject: marbles and boxes   Mon Sep 28, 2009 7:29 am ''Combinatorics is putting different-colored marbles in different-colored boxes" said teacher Rota.http://web.mit.edu/newsoffice/1998/rota-1028.htmlI took it as a problem at the time I read the interview, and I found it funny to solve this problem; also, I managed to get some formula. It was a one page solution using a drawing and the theory of species. I'd like to compare my solution to others, thank you.

Posts : 184
Join date : 2009-09-15
Age : 32
Location : the infinite, frictionless plane of uniform density

 Subject: Re: marbles and boxes   Mon Sep 28, 2009 11:19 pm Hi nick,thanks for the article! I read it but I didn't find the problem. If you'd like to state the problem maybe I could give it a try.

Euclid

Posts : 95
Join date : 2009-09-15
Age : 57
Location : Alexandria

 Subject: Re: marbles and boxes   Thu Oct 01, 2009 1:02 am good point, yes, let me take a look at solution.- it is about n definitively named marbles eg: marbel_a, marbel_b, .... - it is about no-named colors meaning that coloring induce a partition not a surjection- boxes are also no-named - there are no empty boxes- there could be k common colors among the box-colors and the marble-colors that make these k common colors to be special.Example : two marbles, "red" and"blue" in a "blue" box is a different situation to the same marbles in a "yellow" box. "Blue" is a special color, but it is still no-named (of course, if "Blue" is the only one special color, one could name it without any risk.)The formula required depends only on n and k, summing all distinct possibilities of:- coloring the named marbles with no-name colors (partitioning them)- placing them in the no-name boxes- coloring the boxes for all possible amounts of boxes and colors.=============In fact I had some difficulties with the common colors.Suppose there are four boxes and four marbles, and I am colorblind. Then I ask someone to put for me the marbles in the right boxes. I find that there are four different common colors and they are now one-to-one at their places.Of course, the next thing I do is to mark them with a permanent marker Does not this means that:- establishing a bijection between boxes colors and marbles colors is equivalent to- giving names to the no-named colors ?So, in my solution I considered that the common colors are named colors. That is why I asked here for another solution, to clarify this philosophy of colors.