Henry Ernest Dudeney/Modern Puzzles/212 - A Chain Puzzle/Solution

by : $212$

 * A Chain Puzzle

Solution
Open all the links of the piece of chain with $4$ links, on the right hand end of the middle row.

Open all the links of the piece of chain with $3$ links, in the middle of the middle row.

Open all the links of one of the remaining pieces of chain with $3$ links, for example from the middle of the bottom row.

We now have $10$ open links: $5$ large and $5$ small.

At the ends of the remaining $10$ pieces, there are $10$ large links and $10$ small links in total.

Each of the large open links can be used to connect $2$ of the small open links.

Similarly, each of the small open links can be used to connect $2$ of the large open links.

This costs $2 \shillings 6 \oldpence$

Proof
To open a link and rejoin it costs $3 \oldpence$

By opening one link at the end of each of the $13$ pieces will cost $39 \oldpence$, that is, $3 \shillings 3 \oldpence$

Using taht strategy it would be cheaper to buy a new chain.

The best that can be done is to open all the links of $3$ pieces containing $10$ links of the appropriate nature so as to use them to join the remaining $10$ pieces at a cost of $30 \oldpence$, that is $2 \shillings 6 \oldpence$