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hey math people

Salix BabylonikaSalix Babylonika
in General
me and my younger brother thought of like a weird game thing today and i want to know if there's a name for this

so you've got a 4x4 grid of circles, any two of which can be filled in.

image

Now, you take an imaginary marker (an X below) and start at any circle. Your goal is to touch every empty circle exactly once, you can't touch filled-in circles and you can't touch the same circle more than once. You can only move vertically or horizontally, not diagonally.

So for the grid above one solution is this:

image

So that's the principle of the thing, and I've got a couple questions. Bare with me, I'm real bad at math.

1. someone has presumably thought of this before, is there a name for it?

2. are there any configurations where this is unsolvable?

3. is there some kind of rule you could write to figure out what configurations would be unsolvable without actually doing them?

and anything else you want to bring up is fine too.

idk I thought it was interesting, maybe not, but hopefully so.
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Comments

  • CalicaCalica
    edited 2015-07-22 20:49:36
    Sup bitches, witches, Haters, and trolls.
    The general case is called a Hamiltonian path, and not only is there no rule to figure out if a configuration is solvable without actually doing it, it's strongly believed that it's impossible to do so efficiently.
  • Salix BabylonikaSalix Babylonika
    https://i.imgur.com/IoGc2jA.jpg
    Oh neat.
  • CalicaCalica
    Sup bitches, witches, Haters, and trolls.
    As for #2, the answer is yes: consider the configuration 

    OOOO
    XOOO
    OOOO
    OOXO

    and note that any path will have to start at one corner and end at another, and cross the cell between the two marked out exactly once, but the two corners are on the same side of said middle cell so it will have to cross an even number of times, which is a contradiction.
  • Salix BabylonikaSalix Babylonika
    https://i.imgur.com/IoGc2jA.jpg
    cool!
  • Princess AlicePrincess Alice
    You are the end result of a “would you push the button” prompt where the prompt was “you have unlimited godlike powers but you appear to all and sundry to be an impetuous child” – Zero, 2022
    This would make a cool mobile phone game, I think. Drag your finger between the circles and touch each one only once.

    The inability to easily determine whether or not a particular grid is solvable would be an issue, though.
  • CalicaCalica
    Sup bitches, witches, Haters, and trolls.
    Central Avenue said:

    This would make a cool mobile phone game, I think. Drag your finger between the circles and touch each one only once.


    The inability to easily determine whether or not a particular grid is solvable would be an issue, though.
    there are a couple puzzles like that.  numberlink is pretty close and has a pretty nice android version called flow free
  • KlinotaxisKlinotaxis
    kill living beings
    another one is flow free, which has multiple "entrance/exit" pairs and is really easy after a while, possibly due to a stereotyped puzzle generator, dunno
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