Henry Ernest Dudeney/Puzzles and Curious Problems/347 - The Iron Chain/Solution

by : $347$

 * The Iron Chain

Solution
The two pieces of chain contained $9$ and $15$ links respectively.

Proof
The difficult bit here is getting the picture.

In order to visualise the situation, we present a diagram of a $3$-link chain:


 * Dudeney-Puzzles-and-Curious-Problems-347-solution.png

We see that the length of a chain is equal to:
 * the total lengths of the inner diameters of each link

plus:
 * the thickness of the link for both of the links at either end.

Let $d$ inches be the inner diameter of one link.

Let $n$ be the number of links in the shorter chain.

We are given that the links are made of metal half an inch thick.

So twice the thickness of the links is $1$ inch.

Recall there are $12$ inches to the foot, so a $3$ foot length is $36$ inches

So, we have that:

So the shorter chain has $9$ links, while the longer chain has $9 + 6 = 15$ links.