Penny Problem!

Math

I subbed in for the Calc teacher again. One student showed me a neat puzzle involving pennies! Seriously, try this out using actual pennies/etc.

I totally recommend finding four pennies and trying this out before looking at the solution.

If you can't see the animation above, here's an explanation:

Objective:

Move the pennies into a straight line, any direction.

Rules:

• Move only one penny at a time;
  
• Pennies may only be moved to a spot where it will touch at least two other pennies;
  
• Pennies may not be picked up (you may only slide them around);
  
• Use as many moves as possible, then try to minimize the number of moves.
  

To start, four pennies are arranged so that each penny touches two others; a parallelogram.

If you found that easy, try the next step--use five pennies:



Seeing the Solution

I think the difficulty comes in seeing the line we're trying to build. We're so accustomed to perpendicular lines that we can't see the 60-degree lines until we've physically moved the pennies around for a few minutes to get used to it.


The Solution

Really, the trick to creating a line in the least amount of moves is to make a gap large enough for a single penny. The last move will be to fill in the gap--no other move can be the last move.

At least four pennies must be present for this; but because of the initial arrangement (a "diamond" shape), this is not necessarily intuitive. To build the gap, you must first build a column of three. Then, you must be able to see the line.

With five set up in one column of three and one column of two, the process becomes more obvious--simply remove the middle penny in the column of three.

From there on, it's the same game: make a gap, and stagger the pennies to build the rest of the line. It's interesting to note, though, that the minimum number of moves is equal to the number of pennies! I found that pretty exciting!

I'll figure out a proof for it, some day... I've still got another proof to figure out; it's for a card trick, and I get the feeling it uses Perms and Combs, which sucks for me.

--Charlie!

PS: Happy Mothers' Day!