Two can tango

[bonanova]

Two pennies can be placed on a table in such a way that every penny on the table touches (tangos with) exactly one other penny.

Three pennies can be placed on a table in such a way that every penny on the table touches exactly two other pennies.

What is the smallest number of pennies that can be placed on a table in such a way that every penny on the table touches exactly three other pennies?

(All pennies lie flat on the table and tango with each other only at their edges.)

[araver]

Spoiler

I think 16. Don't have a proof though.

The theoretical minimum is 4 (at least 1 touching at least 3). But you can't make opposite pairs touch themselves on a flat surface.

Start by arranging 4 pennies in a symmetrical way: North & South touch each other (place one on top of the other) then they each touch East & West (place two pennies left and right). We get a 4 penny rhomboidal structure where each of the E/W pennies need exactly 1 more connection (like an ion in chemistry), while the N/S pennies are full. Easy to see that this structure chains naturally with similar structures left & right.

To end the chain of similar 4-penny structures one would need a loop/circle of such structures while avoiding the N/S pennies touching another structure. A minimal arrangement of 4-rhomboidal seems to be when 4 of such structures are aligned in a circle.

Each N touches E, S, W from same structure. Each S touches E, N, W from same structure. Each W touches S, N from same structure and E from adjacent structure, etc. Can't properly draw this, but it was an interesting mini-game to turn the matter.js "Circle Stack" demo online to such an arrangement. I call it a game because that is a real physics environment where stuff bounces and it's very easy to disrupt your structure near the end

Edited January 6, 2018 by araver

[CaptainEd]

Nice job, araver! Nice puzzle, bonanova