Touching pennies

[ceptimus]

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]

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]

Is this a stable configuration that can exist freestanding?

[Grammatron]

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]

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?

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]

may I heat the pennies? really, really, really hot? :D

[roger]

okay, I have 5 in an unstable configuration.

[Grammatron]

okay, I have 5 in an unstable configuration.

I think 6 is the limit.

[Tanja]

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]

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?

[Tanja]

Well, you did not misunderstand me, I just assumed that pennies that touch in one point actually do touch.

[ceptimus]

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]

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]

pennies aren't square

[ceptimus]

If you look at them edge-on they are 'square'. More accurately, they have a rectangular cross-section (approximately).

[exarch]

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.