The category name confused me. I never played Sonic 3 & Knuckles, so I have 0 idea about the game's mechanics, but from reading "minimum left/right inputs" I thought it meant holding both left and right at the same time. Holding both opposite directions can have unexpected effects in games, so I did not expect it to mean "no pressing left button, and no pressing right button." instead.
"minimum left/right inputs"
There was a run of mmx3 any% psx glitched(Submission #5175S) submitted a number of years ago, but was rejected on the grounds that the password intentionally triggers Bit/Byte/Vile's death flags before failing to unflag them after rejecting the password as a whole, so it's essentially just (partially) using a password.
Hello FractalFusion, Clean new strat that you used here! Out of curiosity, Approximatively how much time would this strategy save if we were going for the fastest completion time ?
Let y and z be as shown in the image above based on the subdivisions of the bottom edge. Then:
(2x)2+y2=145 (A)
(5x)2+(y+z)2=689 (B)
(4x)2+(z+3)2=377 (C)
Now let's declare that x,y,z are natural numbers. (Just do this. Don't think about it.)
Equation (A) says that 145 = 29*5 is a sum of two squares. Checking shows that 145 = 82+92 = 122+12 (you can also use 29 = 32+22 and 5 = 22+12, and the Brahmagupta-Fibonacci identity to get the above sums of two squares). The first gives x=4, y=9, and then in (C) z=8, and (B) checks out. The second gives x=6, but then (C) has no solution for z, so reject that. Therefore x=4.
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If you are unsatisfied with the above argument, specifically the declaration that x,y,z are natural numbers, note that no other argument I've found is interesting. You are free to solve it in real numbers by squaring equation(s) a few times to get rid of radicals. WolframAlpha will happily tell you that the only solutions are x=±4.
I foresee a judging problem here. I now consider regions 18 and 20 to be adjacent, but I didn't last time. Does that invalidate the old run...? Luckily, this is the favorable arrangement of regions 20 and 21, so even under the old interpretation, this coloring is still optimal.
There might be a problem in considering the yellow and gray regions adjacent, considering the purple and pink regions at the same "corner" are equally as close! (The regions to which the four-color theorem apply shouldn't be considered touching at "corners".)
- It's possible switching to Fast speed text could be faster:
- Upsides: Still investigating it, but I've heard that Mid speed text loses 1 frame per textbox. As to how many frames I actually lose over this run, I'm not sure.
- Downside: Complex battle manips are more difficult and costly. In this run there's a bunch of places I want to manip the enemy's move and my own separately by slowing down or speeding up text; I can't do that with Fast text. Help menu costs about 18 frames to open and close for that, so I'm not sure which is faster overall