bonanova 84 Posted January 6, 2018 Report Share Posted January 6, 2018 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.) Quote Link to post Share on other sites

0 Solution araver 10 Posted January 6, 2018 Solution Report Share Posted January 6, 2018 (edited) 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 Quote Link to post Share on other sites

0 CaptainEd 26 Posted January 6, 2018 Report Share Posted January 6, 2018 Nice job, araver! Nice puzzle, bonanova Quote Link to post Share on other sites

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## bonanova 84

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.)

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