## Comment Re:Not yet... (Score 3, Informative) 943

Sorry, this is complete crap.

I'm going to compare the USA 1,5,10,25,100 system against the very common 1,2,5,10,20,50,100 system (125 system, for short.)

By my calculation (done by hand, so may be a little out) to make all the quantities of change from 0 to 99, the USA system needs 470 coins, the 125 system needs 340 coins. The only quantities for which the USA system needs fewer coins are 25, 26, 35 and 36, for which it is one coin more efficient in each case.

If you took away the 2c coin to have 1,5,10,20,50 (i.e. a 20 and 50 instead of a 25 compared to the USA system) you'd still need only 420 coins (vs 470) for all change from 0 to 99c.

The 125 system works so well because the denominations are nearly equally spaced logarithmically. Starting with 1, you get the next denominations by multiplying by 2, 2.5, 2, 2, 2.5, 2. In the USA system, the multipliers are 5, 2, 2.5, 4. It is those large multipliers (5 and 4) which make the system so inefficient.

If you wanted to span the 1-100 range with just four coins, I believe 1, 3, 10, 30 would be optimal, as it is closest to logarithmic spacing. (Actually, 1,3,10,32 might be better, but that would be a real pain to make change with.) (Even more efficient would be a true logarithmic system: 1,2,4,8,16... or 1,3,9,27,81,... or 1,4,16,64,256... but none of these mesh well with our decimal system.)

On what evidence do you know "THE AMOUNTS MOST COMMONLY NEEDED"? Even accepting your unsupported claim "the very large majority of "change" needed is less than 50 cents", for each amount of change 0 to 49 cents, USA system needs 185 coins vs 145 for the 125 system, so you fail on this count too. If you remove the 2c from 125 and consider only change less than 50c (effectively removing the 50c from the 125 system as we'll never use it, so denominations are 1,5,10,20) then you *finally* get marginally ahead: 185 vs 190. Against this, the 125 system is about one coin better for every number from 50c to 99c, so your 'very large majority' will have to be greater than 90% to even get marginally ahead. (And remember, this is without the 2c, which the 125 system would normally have.)