This guy has come up with a negative one cent.

Again, applies to US system only.

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For US currency: my preference would be dime and dollar, and round off transactions to the nearest 10¢. Maximum number of coins would be needed for change amounts between $4.85 and $4.94 - 13 coins.

Or how about a 10 and negative-1 cent piece? So to make 49c in change, it's 5 dimes and 1 anti-penny. If we think change amounts would be biased towards terminal digits 5-9 (implying transaction amounts ending in 0-4), this reduces expected number of coins. It's like a "leave-a-penny" dish.

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I found this reply interesting too; never noticed that those denominations follow this logarithmic pattern!

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Right now I have only hand-waving arguments, so here they are FWIW.

I think the answer depends on the number system we use and the typical cost of the smallest non-electronic transaction we normally make. Any coin less than a quarter is pretty much useless in today's America, where coffee is around $1.25 /cup and a candy bar can go for $1. So the question is about the smallest denomination we feasibly want to live with, and for this we address the number system.

Since we have a decimal currency system the U.S., traditional currency multiples have been 1, 2, 5, 10, 20, ,50, 100, 500, 1000. What is interesting about this sequence is it is pretty much evenly spaced over a logarithmic scale:

number logarithm (base 10)

1 0

2 0.3

5 0.7

10 1

20 1.3

50 1.7

100 2

200 2.3

500 2.7

1000 3

The denominations of the Euro -- both for coin and banknote -- follow this pattern exactly. The underlying assumption seems to be that factors of 5 in this currency are easy to keep in one's head, since 2 fives make 10 and 10's are easy to keep track of . Assuming that 5 is as good number for a decimal system I would choose a 2 unit and a 5 unit currency note.

If we were to restrict ourselves to US coinage, I wold stick to dimes and quarters.

Other currencies (octal,hex, sexigesimal, etc.) need to be looked at also.

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Aditya