P(l)aying with Euro's

Paying a small amount of money in Euro's can always be done with the following set of coins and (bank)notes:
0.01 , 0.02 , 0.02 , 0.05 , 0.10 , 0.20 , 0.20 , 0.50 , 1.00 , 2.00 , 2.00 , 5.00 , 10.00 , 20.00 , 20.00 , 50.00 . Some multiple of this is what people commonly have in their wallet when they go to the local supermarket, here in the Netherlands.
Paying an amount of money now goes like here. € 99.99 = 50.00 + 20.00 + 20.00 + 5.00 + 2.00 + 2.00 + 0.50 + 0.20 + 0.20 + 0.05 + 0.02 + 0.02 (though 50.00 + 50.00 and receiving back 0.001 would have been much simpler).
It is remarked that paying the exact money involves a series of binary decisions, twelve (12) in our case. This is due to the above standarized subdivision in coins and notes. Herewith, counting all the way up to 99.99 (like paying with single cents 0.01 ) is replaced by the much more efficient divide and conquer approach which is common in our cash money economy.
Some redundancy is involved, too, with most divide and conquer approaches. If we restrict attention to cents only, 3 yes or no - 1 or 0 - decisions about 0.01 , 0.02 , 0.02 , 0.05 - in that order, then we find:
0.00 = free of charge (0000) , 0.01 = 0.01 (0001) , 0.02 = 0.02 (0010 or 0100) , 0.03 = 0.02 + 0.01 (0011 or 0101) , 0.04 = 0.02 + 0.02 (0110) , 0.05 = 0.05 (1000) or 0.02 + 0.02 + 0.01 (0111) .
It is seen that all of the 23 = 8 possible combinations of yes (1) or no (0) decisions are present here. But some of these denote the same transaction of money and the maximum amount involves only 5 cents (while 7 cents could have been possible with a binary decision tree instead of one matching base 10). There is a redundancy with the cashing of Euro's.
Some of the redundancy could be removed by allowing multiple decisions. With ( 0.01 , 0.02 , 0.05 ) it becomes:
0.02 = 0.02 (010) , 0.03 = 0.02 + 0.01 (011) , 0.04 = 2 x 0.02 = (020) , 0.05 = 2 x 0.02 + 0.01 (021) or 0.05 (100) .
Thus 0.02 and 0.03 now become unique, but such is still not the case with 0.05 and, moreover, the whole Boolean Objects character of our decision support system is destroyed, with the advent of more than one decision at a time.

In retrospect, the above is only a very primitive 'Ansatz' to a far more sophisticated (and far less trivial) mathematical theory, which is known as "The Coin-Exchange Problem of Frobenius". I've discovered this only recently on the Internet.