Henry Ernest Dudeney/Puzzles and Curious Problems/234 - Pile Driving/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $234$

Pile Driving

During some bridge-building operations a pile was being driven into the bed of the river.

A foreman remarked that at high water a quarter of the pile was embedded in the mud,

one-third was under water,

and $17$ feet $6$ inches above water.

What was the length of the pile?

Solution

$26$ feet $3$ inches.

Proof

Let $L$ feet be the length of the pile.

You may be tempted to do this:

\(\ds L\)

\(=\)

\(\ds \dfrac L 4 + \dfrac L 3 + 17 \tfrac 1 2\)

\(\ds \leadsto \ \ \)

\(\ds 12 L\)

\(=\)

\(\ds 3 L + 4 L + 12 \times 17 \tfrac 1 2\)

\(\ds \leadsto \ \ \)

\(\ds 5 L\)

\(=\)

\(\ds 210\)

\(\ds \leadsto \ \ \)

\(\ds 5 L\)

\(=\)

\(\ds 42\)

No no no!

The part of the pile buried in the mud is also under water.

So:

\(\ds L\)

\(=\)

\(\ds \dfrac L 3 + 17 \tfrac 1 2\)

\(\ds \leadsto \ \ \)

\(\ds 3 L\)

\(=\)

\(\ds L + 3 \times 17 \tfrac 1 2\)

\(\ds \leadsto \ \ \)

\(\ds 2 L\)

\(=\)

\(\ds 52 \tfrac 1 2\)

\(\ds \leadsto \ \ \)

\(\ds L\)

\(=\)

\(\ds 26 \tfrac 1 4\)

and it is seen that the proportion of the pile which is buried in mud is no more than a distraction.

$\blacksquare$

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $234$. -- Pile Driving