Circular Quarters
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You are playing a game with your friend sitting across a perfectly round tabletop. You each have a supply of quarters which you will place on the table one at a time, without moving any other quarters and without overlapping. The last person who successfully puts a quarter on the table without running out of space on the table wins the game. You get the lucky break of going first.
Where should you put your quarter to ensure that you win, and why?
[Bonus Round] You don’t get to go first. Depending on where your partner puts his quarter first, where do you put your quarter second to best your odds? Can you guarantee a win or a stalemate?
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You have to put it right in the center. The other spots that will have left will be simetrical.
First one to the center, others symetrically to oponents quarters.
If the quarters will all be placed most efficiently (edge to edge in a hexagonal pattern), then you should place your quarter in the exact center of the table. No matter how large the table, your center quarter will always be surrounded by an even number of quarters (6 in the first tier, 12 in the 2nd, etc) and whomever places the 1st quarter will also be placing the last. If the quarters are placed haphazardly… well… uh… hmmmmmmmmm…
Mirror placement for all remaining quarters – of course! Good point!
This cannot be answered, because we don’t know the size of the table and the pattern in which should be quarters placed. There is no exact definition of the game. Can we place quarters on each other?
There’s a lot of approaches that I thought of, the simplest would be to:
1. Take a powersaw, cut the table down until there is only enough room for one quarter.
2. Place the quarter.
3. Win.
Note: Use a cheap table for this.
Bonus Round Answer: Based on the lexicon of the first part of the puzzle “to ensure that you win” we must assume that either you cannot guarantee either a win or a stalemate or that the entire puzzle is invalid.
To ‘best’ my odds, I’d put my second quarter in the first person’s face, which I would subsequently put on whichever spot on the table seemed the most appealing (except, of course, the center – as that’s already been taken).
If we take that part that center of the table is the only place that don’t have symmetric pair on the table, we can guarantee, that there is odd number of places where to put quarters. And the person that starts the game always puts an odd quarter on the table so he/she always wins, because there is always the remaining odd place where he can put his quarter.
the “correct answer” is invalid. The rules clearly state that you cannot move any other quarters on the table. Therefore there is no guarantee that the quarters will be symmetrical. You can purposely add white-space between quarters by spacing them out at variable lengths to prevent another quarter from fitting between.
The question is not answerable.
Point for steve:)
No points for steve, because despite your attempts at foiling the pattern, the symmetry remains by placing you quarter in the exact same spot on the opposite side of the table. Symmetry is guaranteed because your turn always follows theirs and all the space does not need to be filled to be symmetric.
Another big question is does the quarter have to fit totally on the table or can it be hanging off?
Answer: They can hang off.
The “correct” answer is wrong. Any answer relating to symmetry is wrong. Nowhere does the problem statement require subsequent players to place their quarters adjacently to each other. Just because you put a quarter in the middle, doesn’t mean I cannot place mine at the edge of the table, or anywhere else to break the symmetry. With Subsequent players placing quarter in random non-adjacent locations the whole problem becomes unsolvable.
@John
The “correct” answer is still correct, and so are the posts for symmetry.
You are also correct in that there are no “rules” requiring players to put quarters in any particular place.
However, with perfect play, the first move is to place a quarter in the center & mirror your opponent’s move 180 deg across the center in order to force a win for yourself.
Still pondering the bonus. I’m assuming it’s some other solution trying to set up triangles on the table (if you can) until the other player gives up their symmetrical move.
Thank you for another good job. Where else can a person such information in a simple to understand presentation.