New puzzles are as follows. A person owns a gold chain (not
circular) with 7 links. Each link is worth $1.
They wish to
open some of the links in the chain in such a way that they can
"make change" for any dollar
amount from $1 to $7. One way to do
this would be to open every other link to completely disconnect
the
chain. This would require opening 3 links. A more efficient
way would be to open the second and fifth links.
This produces
segments of lengths 1, 1, 2, 1, and 2 (including the opened
links). You can verify that any
dollar amount from 1 to 7 can be
obtained as a combination of these lengths. Several puzzles can
be based
on this setup. The first question is whether two links
are the fewest that must be opened on the seven link chain.
Suppose now that the chain has 20 links. What is the fewest
number of links on this longer chain that must be
opened in order
to make change for every dollar amount from 1 to 20? Suppose now
that three links are to be
opened on a chain of some length.
What is the longest chain for which it is possible to open three
links and make
change for every dollar amount?