A USSR Olympiad problem
A traveler having no money, but owning a gold chain having seven links, is accepted at an inn on the condition that he pay one link per day for his stay. If the traveler is to pay daily, but may take change in the form of links previously paid, and if he remains seven days, what is the least number of links that must be cut out of the chain?
The answer is 1.
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1 solution
if the third link is disengaged from the chain,then there are three pieces :the single link,a two-link piece ,and a foru link piece.On the first day the traveler gives the single link to the inn-keeper .On the second day he gives the innkeeper the two-link piece receiving the single link back in change.On the third day the travekker pays the single link,on the fourth day he pay the four link piece ,receiving as change the single link and the two link piece.On the fifth day he gives the single link ,ands on the sixth day the single link is returned to him as change for the two link piece.On the 7 day the single link pays for the seventh day lodging.Hence,only on link need be disengaged from the chain.