Money Equilibrium? Maximizing Entropy w/ Given Denominations

[alxm]

Just a musing I had.. how far from thermodynamic equilibrium is the change in my pocket?

Obviously having any given amount in the smallest denomination only, maximizes a degree of freedom in the sense that I can form the largest number of sums with it, meaning higher entropy. But on the other hand, I have lower entropy in the sense that I then only have one kind of coin.

So the question is.. for a given distribution of denominations, e.g. {1, 5, 10, 25, 50, 100} and a given amount of money (e.g. 100 cents), which distribution of coins will maximize the entropy?

Any combinatorics fans want to take a crack at this?

 

[davee123]

I guess the question is how you measure entropy given the two possibilities. My instinct answer is that with 5 pennies, 2 nickels, 1 dime, 1 quarter, and 1 half dollar, you can create ALL integer values between 0-100, and have the widest variety of coins possible that still allows you to create all values. But that's far from a scientific answer.

I think the interesting question would be somewhere along the lines of:

Given T coins totaling 100 cents in value, N coins are chosen randomly where 1 <= N <= T. The value of the N coins is V. What distribution of coins should be chosen to maximize the probable number of distinct values for V?

In other words, if you had 100 pennies, then V = N for all values of N. But with a different distribution, V is variable. And obviously, with 95 pennies and 1 nickel, V you have potential for V = N or V = N+4. But also, N+4 is relatively unlikely when choosing only a few coins.

So, how would you measure that? I'm pretty sure I could write a program to simulate this action, and get a good guess at the best distribution. But how do you measure the randomness of a given sample set of V for a given value N? Do you take the standard deviation? The number of distinct values? Some combination of the two? I honestly don't recall how you assign a specific number for a value of entropy...

DaveE