Cigarette pairings

[BMAD]

You have an infinite cache of cigarettes. What is the maximum number of cigarettes you can place on a table so that every cigarette touches every other cigarette? (You can't bend the cigarettes.)

[bonanova]

six is straightforward, and I think seven is possible. I'll work on a picture.

They don't all touch the table, though. It's not clear whether OP requires that.

[TimeSpaceLightForce]

I cant figure out the seven

Edited October 27, 2013 by TimeSpaceLightForce

[bonanova]

Here's one

If the pencils have ideal pointed tips, it's trivial.

Arrange radially as spokes with each tip touching the origin, and thus each other.

This only works for a mathematical model of a pencil, not real objects that are quantized by molecules.

Stand one pencil vertically on its eraser.

Lay three pencils radially spaced 120 degrees that are tangent to the vertical pencil,

and go beyond it until they touch one of the other two (and thus are touched by the other of the other two.)

Crudely, it looks like this:

\ /
\ /
\ /
\ /
\ /|
\O|
\|
|
|
|

Now you can lay three more pencils, lying on, and offset at a slight angle from, the first three.

Each of the seven touch the other six.

Edit: now I see that the solution for seven is given in the above link.

Edited October 29, 2013 by bonanova
Link to the seven solution