TL; DR
Fold a triangle for $n$ time. It will form a simple polygon, maybe with some holes. We define its number of sides as the sum of that in the polygon and the holes. What's its maximum number of sides for: 1. any polygon; 2. polygon without holes?
The problem
Assume we have a triangle and fold it for $n$ times. We can choose the initial shape of the triangle and which line to fold along each time. After the folds, it will form a polygon. What's the maximum number of sides of it after n folds of the triangle?
Note that the formed polygon may be concave, and even not simple. It can be proved that it must be a simple raw polygon with 0, 1 or some hole(s). In this case, we can define the number of sides as the sum of that in the raw one and the holes. The following polygon is a raw quadrilateral with two triangle and one quadrilateral holes, just for example. It has $4 + 3 + 3 + 4 = 14$ edges.
Let $f(n)$ and $g(n)$ be the maximum number of sides of the polygon, where the latter doesn't accept polygons with hole(s). (Is the definition of $g$ equivalent to this one: the maximum number of sides of the polygon, where we ignore the holes and just count the sides of the raw polygon?) It is trivial that $f(0) = g(0) = 3$.
My solution for n=1, and n=2 (partial)
$f(1) = g(1) = 7$ for folding an angle to its opposite side. Consider folding an equilateral triangle. Divide its left and right sides into three parts. Then chose the bottom ones and connect them. Fold the top angle down along the line. It can be proved that the answer is exactly $7$.
We can also find that $f(2) \ge 17, g(2) \ge 14$. The following shows the former, and the latter can be proved by a slight change to the triangle to remove the red hole. I can't prove if there exists an answer beyond that.
An important property of the heptagon generated in the first fold is that it can be generated from a quadrilateral like this:
And we can see if the points A, B, C and D are in the same side of E, respectively. We can use the notation here to express it:
- 0: A and B are in the same side of E.
- 1: B is, and A isn't.
- 2: A is, and B isn't.
- 3: Neither of them is.
And the same notation can be used with D and C. After excepting trivial (0-0), impossible (1-2, 2-2 and 2-3) and symmetric cases, we need to check these ones (I have calculated some):
| AB | DC | $f$ | $g$ |
|---|---|---|---|
| 0 | 1 | ? | ? |
| 0 | 2 | 11 | 14 |
| 0 | 3 | ? | ? |
| 1 | 1 | 11 | 11 |
| 1 | 3 | 11 | 11 |
| 3 | 3 | ? | ? |
I hope someone can help me complete this table!
it's much more difficult to find out the number with more folds. I have searched on the Internet and the OEIS, but none of them gives the answer.
I'd appreciate it if you would give the formula or help me compute more terms of it, for example, calculate the answer for $n = 2$.