## Touching pennies

ceptimus
Posts: 1464
Joined: Wed Jun 02, 2004 11:04 pm
Location: UK

### Touching pennies

How many pennies can be arranged so that each penny touches every other penny. Don't just give a number; a description and/or diagram is required.
Grammatron
Posts: 37447
Joined: Tue Jun 08, 2004 1:21 am
Location: Los Angeles, CA

### Re: Touching pennies

Abdul Alhazred wrote:
ceptimus wrote:How many pennies can be arranged so that each penny touches every other penny. Don't just give a number; a description and/or diagram is required.
Four.

Three pennies flat on the table tangent to one another and another penny laid on top.

Do I win?
In that case two would work as well.
roger
Posts: 389
Joined: Wed Jun 09, 2004 6:15 pm
Location: USA
Is this a stable configuration that can exist freestanding?
Grammatron
Posts: 37447
Joined: Tue Jun 08, 2004 1:21 am
Location: Los Angeles, CA
Also, wouldn't three work as well. Two pennies touch each other with one on top of the two touching both? I think that there are far more possibilities that are based on the given criteria.
ceptimus
Posts: 1464
Joined: Wed Jun 02, 2004 11:04 pm
Location: UK
Grammatron wrote:Also, wouldn't three work as well. Two pennies touch each other with one on top of the two touching both? I think that there are far more possibilities that are based on the given criteria.
Yes. Two, three and four are all possible, as Abdul and you have described. But can you get a group of five or more pennies to each touch all the others in the group?
roger wrote:Is this a stable configuration that can exist freestanding?
Not necessarily. To beat Abdul's four (assuming that is possible), you may have to use some supports. You can embed the pennies in clay, or a clear plastic resin, to support them in the desired configuration.
roger
Posts: 389
Joined: Wed Jun 09, 2004 6:15 pm
Location: USA
may I heat the pennies? really, really, really hot? :D
roger
Posts: 389
Joined: Wed Jun 09, 2004 6:15 pm
Location: USA
okay, I have 5 in an unstable configuration.
Grammatron
Posts: 37447
Joined: Tue Jun 08, 2004 1:21 am
Location: Los Angeles, CA
roger wrote:okay, I have 5 in an unstable configuration.
I think 6 is the limit.
Tanja
Posts: 53
Joined: Tue Jun 08, 2004 8:02 am
Location: London
If we have three pennies lying flat on the table touching each other, than surely we can put another three on top of them and have six?
ceptimus
Posts: 1464
Joined: Wed Jun 02, 2004 11:04 pm
Location: UK
Tanja wrote:If we have three pennies lying flat on the table touching each other, than surely we can put another three on top of them and have six?
Where four objects all touch at a single point, it's generally accepted that they don't all touch each other. Here is a poor ASCII diagram showing a vertical cross-section of four pennies, A and B on the top layer, C and D below:

Code: Select all

-------+-------
A | B
-------+-------
C | D
-------+-------
Now everyone accepts that the pairs AB, AC, BD and CD touch. The problem is with AD and BC - do either or both of those pairs touch? Obviously one of the pairs can be made to touch easily, by slightly staggering the arrangement:

Code: Select all

--------+------
A | B
-------++------
C | D
-------+-------
So now AD is also a touching pair, but BC clearly isn't. I would argue that in the top diagram, either the pair AD is touching or the pair BC is, but not both at the same time. Does that make it clearer? Perhaps I misunderstood what you meant?
Last edited by ceptimus on Wed Sep 15, 2004 1:35 pm, edited 1 time in total.
Tanja
Posts: 53
Joined: Tue Jun 08, 2004 8:02 am
Location: London
Well, you did not misunderstand me, I just assumed that pennies that touch in one point actually do touch.
ceptimus
Posts: 1464
Joined: Wed Jun 02, 2004 11:04 pm
Location: UK
If A and D touch, wouldn't you agree that this forms a barrier, preventing B from touching C? Obviously, if we consider the third dimension, they might touch elsewhere., but in a two-dimensionsal cross-section, I don't think it's possible.
Tanja
Posts: 53
Joined: Tue Jun 08, 2004 8:02 am
Location: London
I suppose you are right strictly speaking. I suppose I took the stance of "they are close enough".

If you think at the cross section as where four countries meet, would you say that all countries border each other, or would you say that countries AD and BC don't border each other, or that AD do have a border, but they are separated by one milimetre of countries B and C? Or would those countries end up with one square milimetre of disputed territory?

Oops, late for work....might continue thinking about it later
ManfredVonRichthoffen
Posts: 285
Joined: Sun Jun 13, 2004 3:24 pm
pennies aren't square
ceptimus
Posts: 1464
Joined: Wed Jun 02, 2004 11:04 pm
Location: UK
If you look at them edge-on they are 'square'. More accurately, they have a rectangular cross-section (approximately).
exarch
Posts: 897
Joined: Wed Jun 02, 2004 10:51 pm
Location: Beyond redemption
Grammatron wrote:
roger wrote:okay, I have 5 in an unstable configuration.
I think 6 is the limit.
I think 5 is the limit ...

Without bending them that is.