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

Modern Puzzles by Henry Ernest Dudeney: $212$

A Chain Puzzle

A man has $80$ links of old chain in $13$ fragments, as shown in the diagram.

It will cost him $1 \oldpence$ to open a link and $2 \oldpence$ to weld one together again.

What is the lowest price it must cost him to join all the pieces together so as to form an endless chain?

A new chain will cost him $3 \shillings$ (that is, $36 \oldpence$)

What is the cheapest method of procedure?

Remember that the small and large links must run alternately.

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 $2 \shillings 6 \oldpence$

$\blacksquare$

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $212$. -- A Chain Puzzle
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $513$. A Chain Puzzle