Log: Conjecture

## Audiolog Code: Conjecture

I am the dilletantMathematician, and I am here to tell you about some of the math behind the Land of Rods and Screens. JR came to me for help designing a puzzle going into the Land of Rods and Screens. Part of the requirements were that making any modification would alter multiple on-screen elements. I dug up an old mathematical idea called the Collatz conjecture. I am going to read to you from its Wikipedia article. "The **Collatz conjecture** is a conjecture in mathematics that concerns a sequence defined as follows: start with any positive integer *n*. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that no matter what value of *n*, the sequence will always reach 1." The Collatz conjecture has not yet been proven but no-one has yet discovered a number that fails to reach one when you apply the rules. I figured we could encode the state as an integer and make it so that the target's state is one, then give the player the option of dividing by two or calculating three x plus one. If the user picks the correct option we perform the math. If the user picks the wrong option then we donk out the state a little. A player who follows the procedure outlined in the Collatz conjecture should eventually get to one and win the game. Of course, the players are somewhat handicapped by the fact that we did not tell them the rules or even how to interpret the state of the board. The saving grace is that there is something of a flaw in the punishment logic. If the user chooses to divide by two [REDACTED] the Javascript throws a random number into the state and returns x over two. But that state gets recalculated from x over two, making the punishment null and void. So it's pretty easy to win by spamming [REDACTED] for all three state groups until you get to the target image. If you dig into the code you might also notice that the [REDACTED] option has no punishment. This is because tripling a number you should be reducing to one is punishment enough. There are also 4 dials. Their combined state encodes the target image which effects how the screen states encode into the integer that is fed into the Collatz math. Four dials with four positions each have two hundred and fifty six combinations. I also came up with an idea that never went into LoRaS. I called it Sodemodularity. Regular base ten number use a modulus of ten for every digit, base two uses a modulus of two for every digit. I had recently studied arithmetic encoding used for compression of HD64 video and envisioned an encoding of numbers where each digits' position could have a different modulus. Each screen in the game would choose its image by mixing a couple of digits from the states encoding different than the mix of digits from the adjacent screen's encoding. JR thought that was too much of a hassle and she was right, so you currently have the stripes that are three wide. I hope this insight into the math of LoRaS was illuminating and if it was not take solace in the fact that I am piping for Azathoth.