My questions are:

Which section of math the problem falls under?

Is there a formula to find the number of possible solutions for either of the scenarios?

What is the most efficient way of solving this type of problem?

Thank you.

The problem

There are variables X1, X2, X3, and X4. Each variable is equidistant from the subsequent one (distance between variables is unrelated to their values). I need to find Y(max) under 2 scenarios. Scenario 1: Intervals cannot overlap, intervals have to be equidistant, and an interval can only be formed from right to left by subtraction. Scenario 2: Intervals cannot overlap, intervals don’t have to be equidistant, and an interval can only be formed from right to left by subtraction.

If I am not mistaken all possible solutions for scenario 1 are

Y1 = (X2-X1) + (X3-X2) + (X4-X3) |

Y1 = (X2-X1) + (X4-X3) |

Y1 = (X3-X2) + (X4-X3) |

Y1 = (X2-X1) + (X3-X2) |

Y1 = (X3-X2) |

Y1 = (X3-X1) |

Y1 = (X4-X3) |

Y1 = (X4-X2) |

Y1 = (X4-X1) |

Y1 = (X2-X1) |

Scenario 2 all possible solutions would be all of the above with addition of

Y1 = (X2 - X1) + (X4 - X2) | |||||

Y1 = (X3 - X1) + (X4 - X3) |

So my questions are which section of math this type of problem falls under? Is there a formula to find the number of all possible solutions for either scenario? What is the most efficient way to solve these type of problems? (Excel or program etc) Thank you.