2 dimensional co-ordinate game

Hythlodaeus

I'm not great at mathematics (not incompetent, but not great), so sorry if I got anyone to expect a problem that was actually challenging. I imagine this is ridiculously simple by the standards on this board. It's a question I've been thinking about since watching "School of Hard Sums" earlier in the week (Yay! Popular math programming seems to work on some people). They were dealing with the optimal meeting points on Cartesian co-ordinates. My question isn't exactly the same, though it's related.

You have a fairly standard Cartesian grid with different individuals located at different points. Two players (P1) and (P2) are tasked with attracting these individuals to a focal point (the winner is the one who attracts the most to his focal point). Each individual will ALWAYS choose the closest focal point to him or herself. if the two are equidistant, you can assume a 50% probability either way. P1 gets to go first and P2 gets to go second. The game then continues ad infinitum. Where are the focal points dropped during move 1 and move 2? Finally, where will the two focal points end up (you may include more steps and I would appreciate visual displays if people are feeling up to it).

The individuals are situated as follows :

There are 2 at (4,3)
There is 1 at (4,1)
There are 2 at (-1,3)
There are 3 at (-2,-2)
There is 1 at (1,-1)


Keep in mind that I am a layman who hopes to learn the method involved if possible. Consequently, I am looking for the explanations. If I can't grasp the mathematics, I would like to gain some sort of intuition for why the method works. I can read most LaTex symbols, but visual illustration would definitely be preferred where possible. Feel free to send private messages/files to my acccount.

Brussels Sprout

Is it a case that the two players pick their focal points in turn and then the people migrate or what?

i.e. 1. Player 1 picks (0,0)
2. Player 2 picks (-1.5,0.5)
3. The individuals go to their nearest focal point

(in the above Player 2 would win 5-4)


From your description it would appear that there is more to it than this but I'm having trouble figuring out exactly what you mean.

Hythlodaeus

Yes. Exactly as you described. I was unsure whether the game would continue or not, but I felt allowing an infinite number of turns would allow for all possibilities. If player 1 cannot pick a better spot after the second turn, then the game should stop there, as he will only choose the same spot and they will both stay there. I'm afraid I don't know how to be any clearer. There are probably some very precise terms I could use if I knew them, but I am somewhat hampered by my casual understanding of mathematics.

Could you explain how you arrived at the result?

Brussels Sprout

Those were just example numbers that I chose to make my example clearer.

I'm still confused by your explanation. If all of the individuals migrate to their closest focal point after both players have each taken 1 turn then how can there be any more turns taken by the players? If the individuals have migrated to a focal point then it will be impossible to come up with a focal point that is closer to them. Let me illustrate this using the previous example

Player 1 chooses (0,0)
Player 2 chooses (-1.5,0.5)

Now the migration occurs. The 2 individuals at (-1,3) and the 3 at (-2,-2) migrate to Player 2's focal point at (-1.5,0.5) as this is closer to them than (0,0). The other 4 migrate to (0,0) as this is closer to them.

Now how can player 1 move his focal point to draw the 5 individuals away from (-1.5,0.5)? Since they are standing right at it he can't pick a closer point. The only thing he could do is pick the exact same point.

So yeah I think I'm not understanding the rules correctly.

Hythlodaeus

Ah. I see how poorly I explained the rules.

The people scattered across the board return to their starting positions each turn (This is a really important rule that I totally neglected to explain properly). Let's just say they "respawn" at the same point.

*facepalm*

The players, however, will each maintain their position from the previous turn.

So, to put it as simply as I can;

If P1 is to the far north of the board after the first turn, P2 would beat him simply by being to the less extreme south of the board after the people have respawned.

In hindsight, people were a bad analogy for this.