This problem actually has nothing to do with evolution!

You have six lengths of chain, each containing five links, and from these you wish to construct a single 30-link ‘endless chain’ – that is, a closed chain loop.

Now it costs 80 cents to cut a link, and $1.80 to weld a link together again.

Also a new endless chain can be bought for $15.

What is the least expensive way of obtaining such an endless chain?

Hint 1

Is there anything significant about the number of links in each given length?

Hint 2

Experiment with the different possible ways of obtaining the endless chain, and work out a costing for each.

Solution

Let us consider the various possible ways of obtaining the endless chain.

The first way is to buy a new one, and the cost is $15.00.

Another option is work in a clockwise direction around the lengths of chain as shown in the diagram, cutting the link at one end of each length, looping it around the end link of the adjacent length, and welding it together again. This means cutting and welding six links at a cost of 6 x (0.80 + 1.80) = $16.80. This is more expensive than buying a new one.

An alternative is to operate on the two end links of each alternate length of chain, but the cost for cutting and welding is again $16.80.

The third possibility is to cut the five links in a single length of chain, and use them to combine the remaining five lengths. This involves cutting and welding five links at a cost of 5 x (0.80 + 1.80) = $13.00.

The fact that each length contains five links, which is just the required number, gives us assurance that this is the expected solution to the problem!

Extensions

1. Experiment with different variations of this problem. Suppose you had five equal lengths of chain. How many links would be a natural choice for the problem?

2. Suppose now that we have eight lengths of chain, each containing four links? What is the best solution this time? Is there some general theory emerging here?

Hint 1

Hint 2

Solution

Extensions