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# Choices and Coins (June 2022 challenge question)

Posted May 31, 2022 12:00 AM
Pathfinder Tags: challenge question

You and a friend decide to play a novel game, wherein each of you has one giant bucket of quarters and a large, rectangular table. The rules of this game are as follows.

1. Each player takes a turn placing one quarter on the table. You cannot skip or cede a turn.

2. Quarters can touch sides, but can never overlap. Quarters cannot stand on their edge, or overhang the table edge.

3. The loser of this game is the first person who cannot place a quarter without a creating an overlap. The winner is the last person to successfully place a quarter, per the rules.

What strategy would you use to ensure you win the game?

(In theory, at least, the type of coin and size of table doesn't matter.)

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Guru

Join Date: Mar 2007
Location: by the beach in Florida
Posts: 31852
#1

### Re: Choices and Coins (June 2022 challenge question)

05/31/2022 1:11 AM

There is no way to assure a win, the variability of the spacing between coins negates any plan you might have, like placing your coin in the center and mirroring your opponents moves...the table may be symmetrical, and the coins may be symmetrical individually, but the placement of the coins creates a non-determinant variable...

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Guru

Join Date: Apr 2010
Location: About 4000 miles from the center of the earth (+/-100 mi)
Posts: 9254
#3

### Re: Choices and Coins (June 2022 challenge question)

05/31/2022 8:01 AM

If you move first, place the coin in the exact center of the table. Wherever the opponent places a quarter, place yours the exact distance opposite the center quarter. This assures a win for the player who moves first.

Guru

Join Date: Aug 2005
Posts: 5492
#4

### Re: Choices and Coins (June 2022 challenge question)

05/31/2022 8:11 AM

If you go second you have to hope that your opponent will at some point place a coin which partially covers the centre position.

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Anonymous Poster #1
#9

### Re: Choices and Coins (June 2022 challenge question)

06/20/2022 11:23 AM

This is the admin's answer as well. There is only on position that cannot be countered with an equivalent move.

Go first and take the very center of the table. Thereafter, mirror your opponent's move on the other side of the table, if axes are applied across the center length and center width of the table. Your opponent will be the first one to run out of unique spaces to place a coin.

Guru

Join Date: Aug 2005
Posts: 5492
#2

### Re: Choices and Coins (June 2022 challenge question)

05/31/2022 5:12 AM

I don't believe you can guarantee a win, but here's a strategy that would probably work against most opponents.

Try to create an even number of "holes" big enough for an odd number of coins.

Your "holes" need to be one coin "wide", n coins long and "oblong". What do I mean by oblong: I mean that you could slide a coin from one end to the other: if a "hole" is shaped so that you can't slide a coin past two intrusions, then it is two holes. For example it's easy to see how you could create a hole big enough for 3 coins, but that you cannot fit one so that there is only enough space for one more: this space is really just three one coin holes.

As the table fills look for opportunities to create pairs of odd sized holes.

At the end of the game it's easy to see how this works for an even number of single holes. For triple holes: if he places a coin in one slot, you place another in the same hole; if he places a coin over two slots, you do the same in another hole.

Etc. Etc.

If he figures out your strategy, or, is working on a similar one, before the shapes on the table become "definable" then it's anyones game. The trouble is that it will get to point where placements will become critical to within tiny fractions of a millimetre: the only way to reasonably play would be on a digital table placing coins by the coordinates of their centres.

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Guru

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#5

### Re: Choices and Coins (June 2022 challenge question)

06/01/2022 7:53 AM

I cant explain why mathematically,but intuitively I think it must always be an even number of coins,so going 2nd is the winning strategy.Even going down to the molecular size should not matter.Of course,going up in size to fill the sides would limit it to 1,but I am excluding this from my guess.

Just a SWAG on my part without actually proving it with math.

Shippers never ship or store an odd number of round articles without a spacer somewhere,and I am sure they have done the math on this.
Example:6 pack,12 pack,etc.

Oil used to come in round containers till someone realized how much space they were losing.Eventually,all foods may come in square-sided containers with rounded corners,like gallon milk or water bottles.The radius and thickness of the corners will be the minimum possible to endure the stresses involved. I leave it to industry to figure the best way,and I am sure they have.

A wild thought is:Tilt the table,and vibrate it by tapping on the bottom.This will arrange the quarters into the tightest possible arrangement.No math needed.

(Presuming there is an edge to prevent the coins from slipping off of the table.)

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Guru

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#6

### Re: Choices and Coins (June 2022 challenge question)

06/01/2022 8:31 PM

Shippers never ship or store an odd number of round articles without a spacer somewhere,and I am sure they have done the math on this.

24 beers in a case. 24 hours in a day. Coincidence? I think not!

The number of quarters on the table could be even or odd. If it is odd, the first player wins, and if it's even the second player wins.

The first player forces it to be odd by placing the first quarter exactly in the center and by countering each quarter placed by his opponent with one directly opposite, thus maintaining symmetry.

This works for a rectangular table, a square table, a regular hexagon table, a round table, etc.

This strategy doesn't work for an equilateral-triangle-shaped table or any regular polygon with an odd number of sides.

My guess is that it is necessary for there to exist 2 perpendicular axes of symmetry to the table for it to work.

Power-User

Join Date: Apr 2007
Posts: 158
#7

### Re: Choices and Coins (June 2022 challenge question)

06/02/2022 8:08 AM

In this case, you certainly could have an odd number. The square of an odd number equals an odd number. You could have 9, 25, 49, 81, 121, 169, 225, 289, etc.

Guru

Join Date: Oct 2006
Location: 44.56024"N 15.307971E
Posts: 6944
#8

### Re: Choices and Coins (June 2022 challenge question)

06/02/2022 8:17 AM

I stand corrected.Sometimes the gut feeling is not right ,(except when you are hungry).

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"A man never stands so tall as when he stoops to help a child." "Never argue with a stupid person.They will drag you down to their level and beat you with experience"
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