[Guest]

I have a few rolls of the most common US coins. I am laying them out on a table. Let's define the fill ratio as the area of the table covered by coins/the whole area (for a unit cell).

What is the maximum fill ratio if I am using:

a) Only Quarters

b) Only Pennies

c) Quarters and Dimes

d) Pennies and Dimes

e) Pennies, Nickels, Dimes, and Quarters

Diameters:

Penny: 19mm

Nickel: 21 mm

Dime: 18mm

Quarter: 24 mm

note: ignore edge effects. Only calculate the ratio for a representative sample somewhere in the middle of the pattern (unit cell)

Edited February 20, 2009 by voltage

[HoustonHokie]

The first two are the same: 0.785, assuming a "unit" table is square. If it were instead a regular hexagon, the ratio could increase to 0.906, but that sounds too much like another brain teaser I looked at recently... The 0.785 comes from the ratio of the area of a circle with diameter d to a square with sides d, which is (pi*d²/4) / d².

[HoustonHokie]

The first two are the same: 0.785, assuming a "unit" table is square. If it were instead a regular hexagon, the ratio could increase to 0.906, but that sounds too much like another brain teaser I looked at recently... The 0.785 comes from the ratio of the area of a circle with diameter d to a square with sides d, which is (pi*d²/4) / d².

Actually, the answer is the same no matter what you're using. The problem is that you haven't defined the ratio of quarters to dimes (for example). So, to maximize the ratio, I would use quarters to cover the majority of the table and place one dime in place of a quarter somewhere. My ratio would be very close to 0.785. Now, if I had to use the same number of each coin, that would be a different story. But I don't see that requirement, yet

Incidentally, my answer would change if a dime had a diameter smaller than 9.9 mm, because then I could fit it in the spaces between quarters.

Edited February 20, 2009 by HoustonHokie

[HoustonHokie]

I've been wrong before, and I guess I'll be wrong again. I actually get a ratio of about 0.80 with dimes and quarters if I place them in a rectangular grid. Guess I should have drawn it out first

A similar arrangement of pennies and dimes yields 0.786 (small improvement over just pennies or just dimes).

[Guest]

there two ways to line up the coins

one that four adjacent center form a square

one that those centers form a parallelogram

the square way is easier

[Guest]

Ok, I think I need to clarify. Assume an infinite table. Your pattern will repeat. Calculate the ratio for each repeating section.