The unit of money is the finbar. The transaction must involve at most two coins including the change. Show that at least six denominations are needed to enable payment for all purchases from 1-29 finbars.

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The unit of money is the finbar. The transaction must involve at most two coins including the change. Show that at least six denominations are needed to enable payment for all purchases from 1-29 finbars.

Mathematics
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This is only some of the information from the question...In vortzegia, the unit of money is the finbar.The Vortzegians like to keep things simple, so they passed a law long ago stating that when you want to buy an item,the transaction must involve at most two coins including the change. In Western vortzegia before 2011 there were only 1- finbar and 8- finbar coins,.With these coins only goods with the following values could be purchased: 1 finbar,2 finbars (1+1),7 finbars (8-1, pay and get 1 finbar change),8 finbars,9 finbars (8+1) or 16 finbars (8+8) In Eastern vortzegia before 2011 there were only three coin denominations in use, but they could transact all amounts from 1 to 10 finbars. In 2011 Eastern and Western vortzegia merged to form one powerful empire under Prime Minister Otto von O chsenkopf.It was announced that all old coins would be withdrawn from circulation and a new set of coins would be released for the entire country with the face of the Prime Minister on them. The vortzegian Mint seeks to implement a system that would make it possible to transact all amounts from 1 to 22 finbars and still obey the two-coins-only-per-transaction law.The mint says this must be done without using any coins with a value of more than 11 finbars. Question: Show that at least six denominations are needed to enable payment for all purchases from 1 to 29 finbars.
I hope this makes sense :) Thank you
It is not possible with 5 or less, because then you would only have 25 possible transactions and not 29: Give only 1 coin -> 5 Give two coins -> 4+3+2+1 = 10, because only the order in which you give the coins is not important Give one coin and get one back -> 4+3+2+1 = 10 because what you'll get back will be less than what you give.

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Thank you so much
In fact if you try out all possibilities (I did it with a program) you'll see, that you need 8 denominations!
Sorry, but there is a flaw in your reasoning nowhereman, I'm not saying you can do it, but if you give two coins, there are 15 possibilities, 5+4+3+2+1=15, say the coins are a,b,c,d and e. you left out a+a, b+b, c+c, d+d and e+e.

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