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 marbles and boxes

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nick
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PostSubject: 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.html

I 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.
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Bruno
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PostSubject: 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.

study
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nick
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PostSubject: Re: marbles and boxes   Thu Oct 01, 2009 1:02 am

Smile good point,

Smile 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.
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