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# EventHorizon

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1. ## Convex Quadrilaterals

Was there an easier method? Concave
2. ## Lanterns of Eden.

Good. Yeah, I overthink everything and can get rather paranoid.
3. ## Lanterns of Eden.

And by "put the devil in his place", I didn't mean you @harey. I reread that and remembered you said "As I am a nice guy, I let you the role of the angel." So if you found that belittling, I apologize, as that was not my intent. I am sure you see things I don't and vice-versa.
4. ## Lanterns of Eden.

First, let's nail down the definition of configuration so we know exactly when the devil wins. It cannot simply be the state of the lanterns, since not changing a lantern on a given step would then repeat the configuration. So the location of the devil and angel need to be part of it. There are two ways I see of incorporating this. Include which lantern number the devil and angel are at and the string of lantern states. Which could be notated by a location marked in a string of lantern states (e.g., "01101;001") or a number and the string (6,01101001). The lantern numbers don't matter, just look forward from the lantern the angel and devil are at (e.g., ";00101101"). And then you don't need the location marker ";" because it is always at the front. Option 1 has more states, lanterns x 2^lanterns compared to just 2^lanterns. Let's choose option 2 since less states means more of a chance of repeat and gives the devil a better chance. That is the notation I used for my code. I notice that you use the word "turn" to sometimes mean one whole stroll around all the lanterns. That is something that threw me off a little. I'd call that a revolution or a full rotation. Alright, time to put the devil in his place.
5. ## Lanterns of Eden.

I couldn't follow that, harey.
6. ## 7-11 store, 4 items to buy

Just saw your post harey.
7. ## Lanterns of Eden.

recreating the results of the code...
8. ## Lanterns of Eden.

sorted configurations with 6 lanterns (and added devil best choices I omitted before) 5 lanterns 4 lanterns 3 lanterns Edit: Another thing of note. The devil and angel both choose between the same 2 options once each (except the choice between 000...00 and 000...01 for angel and 111...10 and 111...11 for devil, due to angel win conditions).
9. ## Lanterns of Eden.

On the "ugly strategy"
10. ## Lanterns of Eden.

Edit: Oops. I guess switch to 1 meaning off and 0 meaning on to match the story. Who wins? Ugly Strategy with not much intuition as to how/why it works. (Obviously, I don't consider this as the answer...) Yay code Code output for 4 lanterns 5 lanterns 6 lanterns
11. ## Relating sides of a quadrilateral

Which angles correspond to which sides? There are two sides that create a given angle and two away.
12. ## n^5 has the same last digit as n

Proof 3: Factors of x^5-x
13. ## n^5 has the same last digit as n

Proof 1: Modulus 10 Proof 2: Modulus 5 I have a third proof idea... I'll see if I can flush it out later. Are you a math professor/researcher? If not, where do you find / get inspiration for these?
14. ## A Gambler's Problem

Constant fraction of current money to maximize the median/mode: Code output (assumes infinitely divisible money): A little more math:
15. ## Find the Rat

It seems my initial guess and idea about how the probabilities would end up were... umm, bad. Yay code... My best approach so far... It seems we calculate the expected value for the day the rat is found differently.
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