Hi, I'm doing some recreational maths involving layers of paper, and I've got a problem that I have no idea how to tackle.
I start with a base sheet of paper, which I number 0. I then add or remove strips of paper until a sort of heightmap is built up, so that a cross-section would reveal a 'staircase' of layers of paper from left to right. Each step is numbered based on it's height. Putting one strip on the base sheet would be 1, another strip on top of that would be 2, etc. Doing that, I can reduce the staircase to a series of numbers. So a normal-looking staircase would look like this: 0,1,2,3,4. Or i could have a stack that was thickest at the middle, like this: 0,1,2,1,0.
Each step must be +1 or -1 from the last step, so nothing like 0,0,0,0,4. Also, each stack must be unique. What I mean by this is that if you added 1 to every number in a stack, you'd get another one, but it wouldn't be unique. So 1,2,3,4,5 is equivalent to 0,1,2,3,4. What this means practically is that every arrangement must have a 0 in it somewhere.
The amount of steps in the staircase I'll call n. I'm trying to work out a formula for finding how many configurations there are for a general case of n. From just trying it out, the number of arrangements is like this:
n=1 - 1 arrangement
n=2 - 2 arrangements
n=3 - 4 arrangements
n=4 - 7 arrangements
n=5 - 14 arrangements
You can probably see thaf I have no real experience with maths beyond GCSEs, so I'm not sure where to even start with this. Does anyone have areas to research, or ways to try and solve it? I'll happily provide more info if this isn't clear.
Thanks.
-Ben
I start with a base sheet of paper, which I number 0. I then add or remove strips of paper until a sort of heightmap is built up, so that a cross-section would reveal a 'staircase' of layers of paper from left to right. Each step is numbered based on it's height. Putting one strip on the base sheet would be 1, another strip on top of that would be 2, etc. Doing that, I can reduce the staircase to a series of numbers. So a normal-looking staircase would look like this: 0,1,2,3,4. Or i could have a stack that was thickest at the middle, like this: 0,1,2,1,0.
Each step must be +1 or -1 from the last step, so nothing like 0,0,0,0,4. Also, each stack must be unique. What I mean by this is that if you added 1 to every number in a stack, you'd get another one, but it wouldn't be unique. So 1,2,3,4,5 is equivalent to 0,1,2,3,4. What this means practically is that every arrangement must have a 0 in it somewhere.
The amount of steps in the staircase I'll call n. I'm trying to work out a formula for finding how many configurations there are for a general case of n. From just trying it out, the number of arrangements is like this:
n=1 - 1 arrangement
n=2 - 2 arrangements
n=3 - 4 arrangements
n=4 - 7 arrangements
n=5 - 14 arrangements
You can probably see thaf I have no real experience with maths beyond GCSEs, so I'm not sure where to even start with this. Does anyone have areas to research, or ways to try and solve it? I'll happily provide more info if this isn't clear.
Thanks.
-Ben
