[tasty_freeze]

This problem reminds me of another problem.

There is a round table and two players, A and B. The players take turns placing a coin on the table in any location they desire, but coins may not overlap. The first person who is unable to place a coin loses. What is the winning first move?

Answer: the winning move is for player A to place the first coin in the center of the table. After that, no matter which location player B chooses, player A can always play the 180 degree rotationally symmetric location.

[stevesimmons]

I solved this in a hedge fund interview with "Consider the limiting case of a coin the same size as the table... You can only place it at the centre, and you win... Now make the coin smaller. How does the strategy change?... It can't, because of symmetry."

They didn't consider it a valid solution :|

[fenomas]

This solution doesn't hold - just because a game is symmetrical, doesn't guarantee that the optimal move in some limiting case works in all cases.

For example, consider if the game was played with N>2 players, or if the win condition was to make the first move modulo N, for some N>2. In those cases everything would still be symmetrical, and playing in the center would still be a winning move for huge coins, but not (necessarily) for smaller coins.

[petters]

It is not clear to me how to play following your strategy. There are things missing. You need to describe what the second move should look like.

[toxik]

Well, that explanation is a total cop out, so I can see why they would think that.

[Y_Y]

I disagree. Arguments to symmetry like this are found all over the place, especially in the kind of geometry and physics I'm familiar with. In fact the solution I came up with when I read the problem was very similar.

[quietbritishjim]

The original comment (by tasty_freeze) was an argument to symmetry. Just saying "because of symmetry" is not a proper explanation.

[Y_Y]

You're right that it's not a proper explanation. This feels just like a case of "details are left as an exercise". This is most frustrating when you're learning a subject, and you want to be able to check the details, or you are skeptical of a claim. Inevitably though people elide things that they think the reader (who they model as similar to themselves) will immediately be able to guess based on their experience. I'm sure you can think of an equivalent situation in your field of expertise.

[quietbritishjim]

I can think of things where I would leave out details that a non-expert would need filled in. Except, I wouldn't do that in an interview situation where someone is specifically asking me to explain that one thing. (Yes they started with the giant coin thing, but "symmetry" is still the substantial majority of their argument.)

The sibling to my first comment says it a lot better than I did: in a similar but different situation there's symmetry but no analogous strategy, so further explanation really is needed here.

In a sense, that makes it even more damning. Eliding details that you don't realise are important, but actually are, shows a real lack of understanding of your own argument. And that's something I have definitely witnessed in my own field!

[aqme28]

Working through this myself...

If you place a coin in the center, then due to symmetry there must be an even number of remaining places for a coin (or you didn't put it in the center).

You can work up from remaining=2 to see that player A always wins with an even remainder.

[tzs]

It is not fully correct to think in terms of even numbers or remainders, because there are an uncountably infinite number of places that a coin can be placed.

It is the right general idea, but don't try to express it in numbers. Just think that for each non-center point P there is a unique corresponding point P' such that the center point is exactly midway between P and P'. That gets you your pairs of places to work with without having to bring numbers into it.

For a coin placed at the center it covers any given non-center point Q if and only if at covers Q's corresponding point Q'. After the first player plays the center we then have that for every point that is not yet covered that point's corresponding point is not yet covered.

Then you have to show that once the center point has been covered by a coin any subsequent coin played that covers a point R cannot also cover R's corresponding point R'.

Then you can proceed with your pairing argument that whenever player 2 covers a set of points player 1 can cover the corresponding points on the next turn, and that after this we have restored the condition that for every point still available its corresponding point is also still available.

[aqme28]

Even though it's uncountable and potentially infinite, it's still even, because of exactly the symmetry relation you're saying.

I just thought that the intuition for an even number of places is a lot easier (for me) to grok.

[porphyra]

Reminds me of mirror go: https://en.wikipedia.org/wiki/Mirror_Go

[hoten]

And if they don't, then what?

[GolDDranks]

Then they are not guaranteed to win.