Question 174
João has a single piece of chain with $n$ links, each weighing $1~\mathrm{g}$. Each time a link of some piece of chain is broken, we obtain 3 pieces. For example, if a piece has 9 links and we break the fourth one, we will then have pieces with the following numbers of links: 3, 1 and 5. A broken link still weighs $1~\mathrm{g}$.
a) If $n=8$, show an example of how to break only 2 links and obtain pieces that can be combined so as to make any weight from $1~\mathrm{g}$ to $8~\mathrm{g}$.
b) If $n=16$, show an example of how to break only 2 links and obtain pieces that can be combined so as to make any weight from $1~\mathrm{g}$ to $16~\mathrm{g}$.
c) If $n=63$, show an example of how to break only 3 links and obtain pieces that can be combined so as to make any weight from $1~\mathrm{g}$ to $63~\mathrm{g}$.
d) If $n=k \cdot 2^{k}-1$, show an example of how to break only $k-1$ links and obtain pieces that can be combined so as to make any weight from $1~\mathrm{g}$ to $n~\mathrm{g}$.
a) If $n=8$, show an example of how to break only 2 links and obtain pieces that can be combined so as to make any weight from $1~\mathrm{g}$ to $8~\mathrm{g}$.
b) If $n=16$, show an example of how to break only 2 links and obtain pieces that can be combined so as to make any weight from $1~\mathrm{g}$ to $16~\mathrm{g}$.
c) If $n=63$, show an example of how to break only 3 links and obtain pieces that can be combined so as to make any weight from $1~\mathrm{g}$ to $63~\mathrm{g}$.
d) If $n=k \cdot 2^{k}-1$, show an example of how to break only $k-1$ links and obtain pieces that can be combined so as to make any weight from $1~\mathrm{g}$ to $n~\mathrm{g}$.