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2024-02-20
Back in November, I wrote about making 2n-page zines.
Thanks to some conversations I had at Big MathsJam
in later November, I've been able to work out how many 128-page zines there are: 315434.
The insight
At Big MathsJam, Colin Beveridge pointed out
something he'd noticed about the possible zines: when drawing the line connecting the pages
in order, there were some line segments that were always included. For example, here are all of the possible
64-page zines:
![](javascript:showlimage("64zines.png"))
Every single one of these includes these line segments:
![](javascript:showlimage("64zines-always.png"))
Colin conjectured that for a zine of any size, a pattern like this of alternative horizontal segments
must always be included. He was close to justifying this, and since MathsJam I've been able to fill
in the full justificication.
The justificiation
First, consider the left-most column of pages. They must be connected like this:
![](javascript:showlimage("64zines-leftcol.png"))
If they were connected in any other way, there would be two vertical connections in a row,
which would create a page that is impossible to open (as every other connection must be a horizontal
that ends up in the spine). Additionally, the horizontal lines in this diagram must all be in the
spine (as otherwise we again get pages that cannot be opened).
Next, consider a horizontal line that's in the spine (shown in red below), and we can look
at all the possible ways to draw the line through the highlighted page, paying particular
attention to the dashed blue line:
![](javascript:showlimage("64zines-highlighted.png"))
The six possible ways in which the line could travel through the highlighted page are:
![](javascript:showlimage("64zines-options.png"))
The three options in the top row do not give a valid zine: the leftmost diagram has two vertical
connections in a row (leading to pages that do not open). The other two diagrams in the top row
have the horizontal line that we know is in the spine, followed by a horizontal line not in the
spine, then a vertial line: this vertical line should be in the spine, but as it is vertical
it cannot be (without making a page that doesn't open).
In each of the diagrams in the bottom row, the connection shown in dashed blue
is included and must be in the spine: in the leftmost diagram, the horizontal line that we know is in the spine
is followed by a horizontal not in the spine, then the horizinal in the dashed blue position
that must therefore be in the spine. The othe other two diagrams in the bottom row,
the dashed blue position is connected to a vertical line: this means that the dashed blue connection
must be in the spine (as otherwise the vertical would cause a page that doesn't open).
Overall, we've now shown that the leftmost column of lines must always be included and
must all be in the spine; and for each horizontal line in the spine, the line to the right of it
after a single gap must also be included and in the spine. From this, it follows that all the horizontal
lines in Colin's pattern must always be included.
Calculating the number of 128-page zines
Now that I knew that all these horizonal lines are always included, I was able to update
the code I was using to find all the possible zines
to use this. After a few hours, it had found all 315434 possibilites. I was very happy to get this
total, as it was the same as the number that
Luna (another attendee of Big MathsJam) had calculated but wasn't certain was correct.
The sequence of the number of 2n-page zines,
including the newly calculated number,
is now published on the OEIS.
I think calculating number of 256-page zines is still beyond my code though...
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