Polyforge Levels Table

A table of the first 700 levels in the mobile game Polyforge

Polyforge is a mobile game where you, playing as an orbiting triangle, try to hit every side of a shape at the center of the screen. In each level, your goal is to hit every side exactly once; you win a level once you hit all of its sides, but if you hit even a single side you've already hit, you have to restart the level. It starts out simple: Level 1 is a triangle, Level 2 is a square, Level 3 is a pentagon, and so on, but over time the levels get more and more interesting:

Polyforge's levels are generated via an algorithm, meaning there's an infinite number of them... but unlike other games with infinitely many levels, Polyforge's levels actually have a difficulty curve to them, since they go by order of sides: for example, all the levels with 54 sides are grouped together, then all the levels with 55 sides, then all the levels with 56 sides, and so on (well, except for every tenth level, but I'll get to that later; for more information on that, go here). The fact that Polyforge's levels keep getting harder and more intricate/interesting as you go, without end, is the main reason I love this game so much, enough to have played up to Level 700 and written down every level's shape as I went.

So how does the level generation algorithm work? Well, I'm not the one who made the game, so I don't know for sure, but I have a theory. It seems to me that the levels are split into a bunch of different level types, different shape templates, and each shape template gets a shape with threefold symmetry, a shape with fourfold symmetry, a shape with fivefold symmetry, and so on, and from there the levels are sorted by number of sides. I was originally going to post a list of the levels, but when I realized this structure, I decided that a table would fit better. Therefore, this table's two dimensions are shape template and symmetry number. The rows correspond to different shapes, the columns correspond to different symmetry numbers. The shape templates also tell you how many sides per symmetry they have, so a shape template with [15s] listed below its picture has 45 sides in the threefold version, 60 sides in the fourfold version, and 15s sides in the s-fold version. There's a picture for each template, as well as some words describing the template: I've included a list of the terminology I used for those descriptions here. Without further ado, here is a table of all of the regular levels in Polyforge up until Level 700:

Here are some notable points about this table:

• Levels 506, 507, 508, 509, and 511 are repeats of Levels 501, 502, 503, 504, and 505 respectively. These are the only repeat levels I've seen so far, and I have no idea why they repeat. Notably, these are all the levels with 95 sides: all the 95-sided (regular) levels repeat, and none of the other levels repeat.
• The color scheme the game uses is random and changes every so often; I just chose to make my pictures of the level templates in a different color scheme for different templates, and the same color scheme for variations of the same template, for thematics.
• The three types of "varied incline trios" levels with 20 and with 27 sides per trio are where I'm least confident that I sorted them correctly; I'm pretty confident I got the rest of the parts right, but I may have misjudged the algorithm on those varied levels specifically.
• There are 101 different shape templates... but for all I know, there may be more templates I haven't discovered yet, given that a template was introduced at Level 607 after the previous newest template being introduced at Level 416. Even though I'm very far into the game, at Level 700, there's still time for another template to appear even further into the game.
• The table doesn't mention a notable property of higher Polyforge levels: Multishot. Starting at Level 101, you have the option of shooting multiple triangles at once instead of just one, making the level easier. The amount of triangles given by Multishot depends on the amount of sides the level has:
  • If the level's amount of sides is a multiple of 4 and is at least 76, Multishot gives 4 triangles.
  • Otherwise, if the level's amount of sides is a multiple of 3 and is at least 51, Multishot gives 3 triangles.
  • Otherwise, if the level's amount of sides is a multiple of 2 and is at least 40, Multishot gives 2 triangles.
  • Otherwise (as long as the level number is above 100; the first 100 levels do not have Multishot), treat the level as if it has 1 (or 2, or 3, as many as it takes to get down to a number that works) less side(s) than it actually does: for example, a level with 55 sides will have 3-triangle Multishot because 54 sides has 3-triangle Multishot.
• I transcribed the levels from written descriptions to a table in a drawing program to this table, so I may have made some errors along the way. If you also play this game and notice any level I got wrong, please let me know at mathcookie17@gmail.com.

Also, you may have noticed that Levels 20, 30, 40, and indeed all of the multiples of 10, other than Level 10 itself, are not included on this table. That's because those levels are special: rather than having a single shape, these levels consist of multiple shapes, and they don't obey the normal algorithm. Their algorithm is arguably simpler: there are nine multi-shape levels (Levels 20, 30, 40, 50, 60, 70, 80, 90, and 100), and after those nine, the same nine repeat over and over, but each multi-shape level gains additional sides on each of its shapes on each repeat, with each multi-shape level changing at its own constant rate between each repeat. Notably, the only shapes used in multi-shape levels are regular polygons and stars. Also, Multishot is not available in multi-shape levels.

Since these multi-shape levels don't fit into the symmetry template algorithm, I've given them their own table below. Note that whereas the s in [15s] refers to the symmetry number, the r in [8 + 8r] refers to the number of repeats. All the pictures in the following table are of the original levels, where r = 0.