Fox, chicken and grain problem
This is a solution to simplify the fox, chicken and grain problem and other similar puzzles.

Imagine you have to transport a fox, chicken and a bag of grain from one side of the river to the other. But you can only take one at a time and you can’t leave the chicken with the grain and you can’t leave the fox with the chicken.

This can be worked out by trial and error. Similar puzzles with more items to take over and more rules can be more tedious to work out so here’s a solution to simplify these problems. If an item is on the first side of the river lets say it is in position 0 and lets say when it is on the other side of the river it is in position 1. If we want to represent the position of all three items we could do this.

FOX    CHICKEN   GRAIN

  0             0               0

  
Discrete space state

Here’s a 3-dimensional map of all possible moves when there are no rules.

1. Whenever we go from left to right on the yellow lines it represents the fox going from side 0 to side 1.
2. Whenever we go from the bottom to the top on the blue lines it represents the chicken going from side 0 to side 1.
3. Whenever we go from the front to the back on the green lines it represents the grain going from side 0 to side 1.

1. When you go from right to left on the yellow lines it represents the fox going back to the first side.
2. When you go from top to bottom on the blue lines it represents the chicken going back to the first side.
3. When you go from back to front on the green lines it represents the grain going back to the first side.

Adding rules Now we need to manually remove disallowed paths.The rules in our original game were

1. We were not allowed to take the fox leaving the chicken and grain together
2. We were not allowed to take the grain leaving the fox and chicken together

If Fox 0/1 = Chicken 0/1 on the same vertex remove the line grain from that vertex. If Chicken 0/1 = Grain 0/1 on the same vertex remove the line Fox from that vertex. Now the solutions becomes much easier to see, by taking the steps by following the line from 0,0,0 to 1,1,1.

if (fox = chicken ){remove lines that are not chicken and fox}

How to know what points connects to other points

If we have a single point in 0 dimensions, that will be point 1 of dimension
If we create another point representing point 1 in its set

The first number represents their number in their own 0 dimensional set, the second number number represents it's number in the 1 dimensional set and so on.

Ever point on the same dimension where the number is the same on the dimention below will connect