Henry Ernest Dudeney/Modern Puzzles/139 - A Crease Problem/Solution

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Modern Puzzles by Henry Ernest Dudeney: $139$

A Crease Problem
Fold a page, so that the bottom outside corner touches the inside edge and the crease is the shortest possible.


Solution

Dudeney-Modern-Puzzles-139-solution.png

Bisect $AB$ at $C$ and construct $CG$ parallel to $BH$.

Bisect $AC$ at $D$ and construct the semicircle between $B$ and $D$.

Let the semicircle $BD$ intersect $CG$ at $E$.

Let $DE$ be produced to intersect $BH$ at $F$.

Then $DF$ is the required crease.


Proof

First we assume that the length of the page is sufficiently long.

Otherwise simply folding the page in half could produce a shorter crease.


Suppose $t$ of the length of the bottom edge is folded over.

Then $1 - t$ of the length of the bottom edge is not folded over.

The angle adjacent to the bottom edge of the small triangle produced after folding is therefore:

$\cos^{-1} \paren {\dfrac {1 - t} t}$

The angle of the crease with respect to the bottom edge would be:

$\dfrac 1 2 \paren {180^\circ - \cos^{-1} \paren {\dfrac {1 - t} t} } = 90^\circ - \dfrac 1 2 \cos^{-1} \paren {\dfrac {1 - t} t}$

The length of the crease is therefore:

\(\ds \frac t {\cos \paren {90^\circ - \dfrac 1 2 \cos^{-1} \paren {\frac {1 - t} t} } }\) \(=\) \(\ds \frac t {\sin \paren {\frac 1 2 \cos^{-1} \paren {\frac {1 - t} t} } }\) Cosine of Complement equals Sine
\(\ds \) \(=\) \(\ds \frac t {\sqrt \frac {1 - \cos \paren {\cos^{-1} \paren {\frac {1 - t} t} } } 2}\) Half Angle Formula for Sine
\(\ds \) \(=\) \(\ds \frac t {\sqrt \frac {1 - \frac {1 - t} t} 2}\)
\(\ds \) \(=\) \(\ds \frac t {\sqrt {\frac {2 t - 1} {2 t} } }\)
\(\ds \) \(=\) \(\ds \sqrt {\frac {2 t^3} {2 t - 1} }\)

of the length of the bottom edge.


Let $\map f t = \dfrac {2 t^3} {2 t - 1}$.

Then $\map {f'} t = \dfrac {6 t^2 \paren {2 t - 1} - 2 \paren {2 t^3} } {\paren {2 t - 1}^2} = \dfrac {8 t^3 - 6 t^2} {\paren {2 t - 1}^2}$

$\map {f'} t = 0$ when $t = 0$ or $t = \dfrac 3 4$.

Hence the maximum length of the crease occurs when $t = \dfrac 3 4$, which is shown in the construction in the solution.

$\blacksquare$


Sources

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