All combination between 0.01 to 100$ in 4 notes

Asked Sep 13 '16 at 09:14

Active Sep 13 '16 at 10:01

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I found this interesting problem: http://mathfactor.uark.edu/2012/07/yoak-denominations-of-money/#comments

Devise an alternate set of denominations for coins and bank notes requiring a minimum number of denominations and such that any amount from \$0.01 to \$100 could be paid with four units of currency.

I found a smaller upper bound of the one proposed, however I am not sure it is right and I would like someone to check it.

There are 10.000 (from 10000 cent to 1 cent) possible combination of change that we need to express with 4 symbols.

If it was a positional system we would only need $log_4(10.000) <= 7$ symbols.

However we are talking about coins which is not a positional system.

At this point we could just expand the 7 symbols on the 4 bases to get $7 \times 4 = 28$ bank notes.

Which is still an upper bound since $4 ^ 7 = 16384 > 10000$

To make things clearer let me enumerate all the coin I think I need expressed in cent: $$ {0, 1, 2, 3, 4, 5, 6} \\ {7, 14, 21, 28, 35, 42, 49 } \\ {98, 147, 196, 245, 294, 343, 392} \\ {1029, 1372, 1715, 2058, 2401, 2744, 3087} \\ $$

I am missing something?

Cheers

edited Sep 13 '16 at 10:01

asked Sep 13 '16 at 09:14

Siscia

Erm, what are the denominations you propose? – Hagen von Eitzen Sep 13 '16 at 09:26
Sorry I thought that was the trivial part, since it is a base system you need 7 for the first digit (0 to 6 each times $7^0=1$), another 7 for the second digit, so (0 to 6 each times $7^1$=7) and so on... Am I wrong? – Siscia Sep 13 '16 at 09:36
Something like $\frac{1}{10}$th of a cent? – Sarvesh Ravichandran Iyer Sep 13 '16 at 09:50
@Siscia Won't you arrive at $7^4$ instead of $4^7$? – Hagen von Eitzen Sep 13 '16 at 09:51
@HagenvonEitzen humm... There is something I am missing too, but please take a look at the table of the coin... The maximun you can reach is 10290 which is enough... – Siscia Sep 13 '16 at 10:04
@астонвіллаолофмэллбэрг Sorry, I didn't get your comment, what you mean? – Siscia Sep 13 '16 at 10:04
@Siscia I was just joking, saying that with a currency of $\frac{1}{10}$th of a cent we could easily have paid everything without change. – Sarvesh Ravichandran Iyer Sep 13 '16 at 10:08
@HagenvonEitzen, yes, You are right, it would give me $7^4$... – Siscia Sep 13 '16 at 10:16

All combination between 0.01 to 100$ in 4 notes

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