Creating a Gold Chain

Here is a problem by J.A.H. Hunter.

'I've just remembered a teaser from when I was a kid,' said Fred. 'A long time ago!' commented his partner. 'Let's have it then.' 'Well, this man handed a jeweller five pieces of gold chain, each with only three links, saying he wanted them made into one chain. 'I'll charge a dime to cut a link and a dime to weld a link,' the jeweller told him.' 'That's easy,' laughed Joe. 'Four cuts and four welds, so it cost him eighty cents.' 'But you're wrong,' chuckled Fred. 'It didn't need so many cuts or welds, although they were welded links and the final chain had to be that way.' Yes, this is a really ancient teaser! But how could the job be done most economically?

Please give the answer and show it is really the most economical solution. Also as there is a hint that J.A.H. Hunter took the problem from an older source, if you can, give a source before 1959.

Peter Heichelheim

SPOILER

Take one of the chains and cut each link. Use each cut link to connect one of the other four chains to another one. Re-weld the links. That's three cuts and three welds.

three cuts, three welds, 60 cents

'sobvious

Amusements in Mathematics, Henry Ernest Dudeney, 1917:

'This is a puzzle based on a pretty little idea first dealt with by the late Mr. Sam Loyd. A man had nine pieces of chain, as shown in the illustration. He wanted to join these fifty links into one endless chain. It will cost a penny to open any link and twopence to weld a link together again, but he could buy a new endless chain of the same character and quality for 2s. 2d. What was the cheapest source for him to adopt? Unless the reader is cunning he may find himself a good way out in his answer.'

The picture shows nine chain segments, of lengths 8,7,6,6,6,5,5,4, 3.

I've seen a variation on this in H. E. Dudeney's 'Amusements In Mathematics' from 1917, and he mentions that it's based on an earlier puzzle by Sam Lloyd. Dudeney's variation goes like this:-

A Chain Puzzle

This is a puzzle based on a pretty little idea first dealt with by the late Mr. San Lloyd. A man had nine pieces of chain, as shown in the illustration. He wanted to join these fifty links into one endless chain. It will cost a penny to open any link and twopence to weld a link together, but he could buy a new endless chain for 2s. 2d. What was the cheapest course for him to adopt? Unless the reader is cunning he may find himself a good way out in his answer.

The illustration shows 9 pieces of chain with lengths 3, 4, 5, 5, 6, 6, 6, 7, 8.

(In those days, there were 12 pence in 1 shilling, so '2s. 2d.' is 26

'It's obvious' is rarely a satisfying answer.

There are, at the start, five part chains. That would have been four joins of an open-then-close type.

However, the cutting of all the links in one part reduces the number of closed parts by one, leaving four closed parts and only three joins to make, and three open links to do it.

If you don't get rid of a part, that is, don't cut all the links in one part, you will need at least four joins, so that can't lead to less than four open-then-close joins.

If you cut any links in the remaining four parts, you will eventually have to close them. So that would only increase the number of open-then-close joins.

So no other way of doing it can lead to less open-then-close

J.A.H. Hunter's answer was 'Cut each link of one piece: use one at each joint. Cost being 60 cents.' But I think if one piece were cut into its links there would be two cuts to get the three links separated and three welds to get the pieces into one chain. That would be 50 cents. Thank you for your reference.

Peter

<snip>

You can't weld an uncut link though, so you still have to cut all links of one piece.

Similar solution, of course: Cut all the links of the 4 and 3 segments, and use those seven links to join the remaining segments (8,7,6,6,6,5,5) into an endless chain.

you need to make Three cuts... two to release each link and the third to open the last link

wrote on 6/12/01 9:31 PM:

SPOILER

Cut all the links in one chain, and use them to weld the other four chains together.

Cutting and welding a link at the end of a chain can reduce the number of chains by at most one; cutting and welding a link in the middle of a chain does not reduce the number of chains at all; cutting and welding a link standing alone can reduce the number of chains by at most two. It follows that we must cut at least three links, of which at least one must be standing alone when cut; and if one of the links cut is in the middle of a chain, both others must be standing alone when cut. And the only way to do this is as above.