Henry Ernest Dudeney/Puzzles and Curious Problems/356 - A Rail Problem/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $356$

A Rail Problem

There is a garden railing similar to our design.

In each division between two uprights there is an equal number of ornamental rails,

and a rail is divided in halves and a portion stuck on each side of every upright,

except that the uprights at the end have not been given half rails.

Idly counting the rails from one end to another, we found that there were $1223$ rails, counting two halves as one rail.

We also noticed that the number of those divisions was five more than twice the number of whole rails in a division.

How many rails were there in each division?

Solution

There are $51$ divisions of $23$ whole rails each.

Proof

Let $n$ be the number of whole rails in a division.

Let $d$ be the total number of divisions.

Each of the outer divisions has $n + \dfrac 1 2$ rails.

Each of the inner divisions has $n + 1$ rails, that is, $n$ plus the two half rails at either end.

Hence we have:

\(\text {(1)}: \quad\)

\(\ds 1223\)

\(=\)

\(\ds \paren {d - 2} \paren {n + 1} + 2 \paren {n + \dfrac 1 2}\)

\(\ds \)

\(=\)

\(\ds d n + d - 1\)

\(\text {(2)}: \quad\)

\(\ds d\)

\(=\)

\(\ds 2 n + 5\)

... the number of those divisions was five more than twice the number of whole rails in a division.

\(\ds \leadsto \ \ \)

\(\ds 1223\)

\(=\)

\(\ds \paren {2 n + 5} n + \paren {2 n + 5} - 1\)

substituting for $d$ from $(2)$ into $(1)$

\(\ds \leadsto \ \ \)

\(\ds 2 n^2 + 7^2 - 1219\)

\(=\)

\(\ds 0\)

simplifying

\(\ds \leadsto \ \ \)

\(\ds n\)

\(=\)

\(\ds \dfrac {-7 \pm \sqrt {7^2 + 4 \times 2 \times 1219} } {2 \times 2}\)

Quadratic Formula

\(\ds \)

\(=\)

\(\ds \dfrac {-7 \pm 99} 4\)

simplifying

\(\ds \)

\(=\)

\(\ds \dfrac {-7 + 99} 4\)

obviously it is the positive root we need here

\(\ds \)

\(=\)

\(\ds 23\)

simplifying

\(\ds \leadsto \ \ \)

\(\ds d\)

\(=\)

\(\ds 2 \times 23 + 5\)

substituting for $n$ in $(2)$

\(\ds \)

\(=\)

\(\ds 51\)

simplifying

$\blacksquare$

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $356$. -- A Rail Problem
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $256$. A Rail Problem