*How much of a dollar bill can you cover up using change for a dollar? The coins must not overlap one another.*

(See previous post and the one before that.)

I made a few attempts at this, see pictures below.

**Version 1, wide-open**. Here I covered the dollar bill with 45 pennies, going for a hexagonal lattice. I'm guessing this covered a percentage of the dollar in the mid- to high-80's. (One easily shows that an infinite hexagonal lattice covers a fraction Pi*Sqrt[3]/6, or a little less than 91%, so I'm assuming the below is pushing up against that limit.)

**Version 2, intermediate**. I nudged the configuration above in order to make room for the additional coins to touch the dollar bill - that seemed to force a decision to remove five of the pennies and replace the nickel with a dime.

I did some quickie image analysis on the above awful picture to give me some inputs for a Monte Carlo calculation of the covered area:

The result was just under 85% coverage. I didn't do a sensitivity analysis. I'm guessing this configuration is low- to mid-80's.

**Version 3, tight**. Here, my attempt uses 8 dimes, 15 pennies and a nickel. According to the spreadsheet, this configuration covers just over 64% of the dollar.

It would be nice to be able to replace that nickel with 5 pennies - according to the spreadsheet, that would bump the area up to 74.6%. But I wasn't seeing quite enough room to do that, and truth be told I have put a lot more time into blogging this puzzle than I've spent trying to solve it. Feel free to blow these attempts out of the water!

***

"May I help you?"

"I'd like change for a dollar."

"Here you go!" (Hands the customer four quarters.)

"Why, you cheat! I had 16 square inches' worth of currency and you gave me back only 3 square inches' worth!"

"I'm sorry, sir. Here are 7 dimes and 30 pennies instead."

"OK, close enough...."

I thought it was amusing, this image of a person valuing an exchange along a totally irrelevant dimension. As if one were to share a long kiss with somebody, and then become upset because the other person took a different number of breaths than you did.

***

The above puzzle takes the form, "How many X can you cover with Y?" I'm sure there a lot of problems and solutions of this nature to be found in the literature on circle-packing. I don't know that literature, but a quick search just now turned up some neat pictures here.