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This is somehow related to binary…and yet it’s also related to the Fibonacci Sequence. I’ll have to think about it.
Well, there goes my afternoon. ðŸ˜‰
In keeping with the idea of responding without words:
Neat! Looking at the second image I noticed things I hadn’t thought of when I made mine. =)
It’s almost a cellular automaton with a shift: add a column of green down the right hand side and you get a trace of Wolfram’s Rule # 106/120/169/225 (depending on which color you interpret to be 0 or 1, and on the order in which the cells are traced) with the output shifted one cell to the side at each step.
Very interesting observation! You say it’s very close to a trace of those rules but not exactly. Which makes me wonder whether some sort of cellular automaton can be devised which would give exactly this as its trace — or more likely, would include this as a subtrace, possibly with some extra intermediate states in between.
Edit to the comment I just made: add that column down the right hand side and you get something /very close to/ a trace of one of those rules I mentioned.
I think I have figured out the pattern and I can make pictures like xander has made. But I do not understand the math behind the pattern. Help? Also, why is is tagged “partition”?
If I recall correctly from undergraduate analysis, a partition of a set A is a collection of subsets such that the union of all of the subsets is A, and the intersection of any two subsets is empty. That is, the subsets completely cover the original set, but they do not overlap anywhere.
In the original image, imagine that you are laying stones for a garden path. You have blue stones, which are 1 foot square; and green stones, which are 1 foot by 2 feet. The path is 7 feet long. One question that you might ask is “How many ways can I lay stones to build my path?”
Each possible arrangement of stones represents one partition of the path—the stones completely cover the path, but do not overlap. What I find interesting is that the number of possible partitions of the type pictured is a Fibonacci number (specifically, the nth Fibonacci number, where n is the length of the garden path).
This can be thought of inductively (or recursively). Let us suppose that there are F_n ways to lay a path that is n feet long. First, note that there is only one way to pave a 1 foot long path, and that there are two ways to pave a 2 foot long path. Thus
F_1 = 1
F_2 = 2.
How many ways are there to lay stones for a path that is k feet long? That is, what is F_k? You can either lay a path that is k1 feet long, and put a 1 foot stone at the end; or you can lay a path that is k2 feet long, and put a 2 foot stone at the end. This gives us the relation
F_k = F_{k1} + F_{k+2}
which is the Fibonacci sequence.
As an added bonus, it seems that if you add a third type of stone that is 1 foot by 3 feet, you get a sequence defined by
S_1 = 1
S_2 = 1
S_3 = 2
S_n = S_{n1} + S_{n2} + S_{n3}
where S_n is the number of possible partitions for a path of length n.
xander
The solution was good. The explanation was great. Thanks. ðŸ™‚
Instant recognition. I use this problem with students (from middle school through teacher training)
Ghost the Bunny hops up a flight of 7 steps. Ghost can hop up one or two steps at a time, and never hops down. In how many ways can Ghost reach the top step? (3 double hops then a single is not the same as two doubles, a single, and another double)
Yeah, I also instantly thought of the same bijection. I call it the “stairstep”problem.
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