This puzzle requires a little bit of thought, but has a satisfyingly neat solution.
Whilst sorting through some old jewellery a colleague of ours discovered seven pieces of a chain in different lengths. The seven pieces were two, three, four, five, six, seven and eight links long, making a total of 35 links.
It was a lovely chain, and she decided to have the pieces joined together so that she could wear it as a piece of jewellery. And so she sought the services of a jeweller to do the work.
She was disappointed to discover that the jeweller would charge £17.50 to join up the pieces, and asked for a breakdown of the charges. The jeweller said that it cost 50p to cut a link and £2.00 to weld the link back together. There were seven pieces, and so it required seven cuts and seven welds to make a continuous loop of chain. Seven lots of £2.50 comes to £17.50, the price the jeweller was quoting.
Is this the cheapest solution? And if not, what is the cheapest way to make the chain complete?
As we are posing this puzzle, you can reasonably expect that this is not the cheapest solution.
The best way to complete the chain is the take the two-link piece and the three-link piece, and make five cuts to get five individual links. You now have the five links and five other pieces of chain. Use the five individual links to join the five pieces of chain into a wearable piece of jewellery.
As this requires only five cuts and five welds, the price drops to £12.50.