r/mathpuzzles

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The factory you work for has a problem: they only ordered 1/2 splitters, and they need to divide the material input on one of their lines into 2 outputs of 1/10 and 9/10 the original amount. Being the smart one around, they ask you if there’s a way to use their half splitters to achieve this. And the area they use is only 3 metres wide, although it can be as long as you need. You’ve broken it down into the following mathematical problem:

EDIT: see comment for rewrite of problem

You have a grid of size n*3 composed of 1x1 squares. the grid sits on an xy axis, such that each square of the grid has whole number xy coordinate on which it sits. The objective is to be able to feed an infinite amount (Z) of materials through this grid starting at 1,2 and have 1/10 of Z end up on tile XaYb and 9/10 of z end up on XcYd (a can equal b, c can equal d. XaYb cannot equal XcYd.)

Each square in the grid can individually be assigned exactly ONE function that alters the position of the “material” occupying it. A grid can either be a CONVEYOR, or a SPLITTER.

- A splitter will take an input from any one of the axis, but will only take inputs from the first axis it recieved from.

- A splitter has a binary state and exists as either a 1 or a 0. This state is flipped whenever it passes an input. All splitters start in state 1. In state 1 it will “send” its input along the axis it was inputted, adding/subtracting 1 to its coordinate value in the direction of travel, and in state 0 it will add/subtract 1 to its perpendicular axis of the same +/-.
Example 1: a splitter on square 2,4 recieves 1 input from square 1,4. In state 1 it transforms the coordinate of the material to 3,4. in state 0, it transforms it to 2,5.
Example 2: a splitter on square 2,4 receives 1 input from square 3,4. In state 1 it transforms the coordinate of the material to 1,4. In state 0, it transforms it to 2,3.

- A square set to conveyor is assigned a value between 1 and 4. A conveyor set to 1 will add +1 to x to the individual coordinate of a material entering it; set to 2 will add +1 to y; set to 3 will add -1 to x, set to 4 will add -1 to y. This value cannot be changed once assigned.

RULES:
- The grid squares begin at 1,1.
- x and y for each square must be positive.
-Once a square has been assigned a function, it cannot be changed.
- The coordinates of the materials will change at the same rate of 1 “tick”, and is not continuous.
- A square can only have one material on it at a time.
- “materials” entering squares XaYb and XcYd are removed. Neither square can be square 1,2.
- The first input starts on square 1,2, and all other inputs enter the grid individually once the previous material has moved from this square

1. Find the minimum size of N to satisfy all criteria. It must be a positive number.
2. Find the minimum number of splitters required.
3. Find the function of each square within this grid and its coordinates on the axis. Find the value of each conveyor unit, if any are present.
4. Find the values of a, b, c, and d.

So I'm playing a game with my family, and I accidentally caused my brother to lose 10 points instead of gaining 10. I then commented that he lost out on 20 points because of my actions. They said "no, he only lost 10 points". I then tried to explain how he lost OUT on 20 points. They then started to get angry at me, saying (NO, that's not how it works. You start at zero points, lose 10 points, leaving you at-10. So you only lost out on 10 points). Normally I would just think that they don't understand logic and ignore it, but all 5 of them thought I was going crazy. I just need to know if I AM going crazy, or if all 5 of them had laps in logic at the same time.

Sorry for the paragraph.

Symbols include numbers, arrows, multiplication signs, etc. You can't put a variable representing a large number like TREE(3), or the infinity symbol. You CAN come up with new symbols to represent more complex mathematical operations.

I am a college undergrad who plays a lot of incremental games, many of which use their own notations for representing cartoonishly large numbers. I've also done some surface level research on things like Knuth's Up Arrow Notation, which inspired the question - what is the most "efficient" way to write large numbers?

This is really not my field so I'm not sure exactly what the limits of my question are, or even what type of math this is, but feel free to satisfy my curiosity!

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