Problem 2 in the same issue of the MSRI newsletter is the muffin problem that Jim Propp first brought to our attention in 2009. Veit On Nov 29, 2011, at 8:24 PM, Michael Reid wrote:
Thank you for the enjoyable puzzle. For now I will hold off on posting my solution unless there is some interest in seeing it.
I do not know about higher bases, but it seems reasonable to believe that there is a bound -- depending upon the radix, but not on the initial number -- for the number of steps needed.
My solution for the base 2 case does not seem to adapt easily, if it all, to other bases.
In the lastest issue of Emissary (the newsletter from MSRI) there's the following challenging puzzle:
For every positive integer n, write it in binary, and allow the following possible set of transformations:
You can insert "+" signs at arbitrary points within the binary expansion, and interpret that as a sum of binary numbers. For example
110101_2 --> 11_2+01_2+01_2 = 3 + 1 + 1 = 5 = 101_2.
Show that there is an absolute constant C such that for any positive integer n there is a sequence of at most C such transformations that results in 1.
Note: The minimal value of C is a real shocker
This made me wonder about the obvious generalization to other bases, where now we ask for a sequence of such transformation that results in a single digit base b number.
Victor
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