$$ \newcommand{\bsth}{{\boldsymbol\theta}} \newcommand{\va}{\textbf{a}} \newcommand{\vb}{\textbf{b}} \newcommand{\vc}{\textbf{c}} \newcommand{\vd}{\textbf{d}} \newcommand{\ve}{\textbf{e}} \newcommand{\vf}{\textbf{f}} \newcommand{\vg}{\textbf{g}} \newcommand{\vh}{\textbf{h}} \newcommand{\vi}{\textbf{i}} \newcommand{\vj}{\textbf{j}} \newcommand{\vk}{\textbf{k}} \newcommand{\vl}{\textbf{l}} \newcommand{\vm}{\textbf{m}} \newcommand{\vn}{\textbf{n}} \newcommand{\vo}{\textbf{o}} \newcommand{\vp}{\textbf{p}} \newcommand{\vq}{\textbf{q}} \newcommand{\vr}{\textbf{r}} \newcommand{\vs}{\textbf{s}} \newcommand{\vt}{\textbf{t}} \newcommand{\vu}{\textbf{u}} \newcommand{\vv}{\textbf{v}} \newcommand{\vw}{\textbf{w}} \newcommand{\vx}{\textbf{x}} \newcommand{\vy}{\textbf{y}} \newcommand{\vz}{\textbf{z}} \DeclareMathOperator*{\argmin}{argmin} \DeclareMathOperator\mathProb{\mathbb{P}} \renewcommand{\P}{\mathProb} % need to overwrite stupid paragraph symbol \DeclareMathOperator\mathExp{\mathbb{E}} \newcommand{\E}{\mathExp} \DeclareMathOperator\Uniform{Uniform} \DeclareMathOperator\poly{poly} \DeclareMathOperator\diag{diag} \newcommand{\pa}[1]{ \left({#1}\right) } \newcommand{\ha}[1]{ \left[{#1}\right] } \newcommand{\ca}[1]{ \left\{{#1}\right\} } \newcommand{\norm}[1]{\left\| #1 \right\|} \newcommand{\nptime}{\textsf{NP}} \newcommand{\ptime}{\textsf{P}} \newcommand{\R}{\mathbb{R}} \newcommand{\card}[1]{\left\lvert{#1}\right\rvert} \newcommand{\abs}[1]{\card{#1}} \newcommand{\sg}{\mathop{\mathrm{SG}}} \newcommand{\se}{\mathop{\mathrm{SE}}} \newcommand{\mat}[1]{\begin{pmatrix} #1 \end{pmatrix}} \DeclareMathOperator{\var}{var} \DeclareMathOperator{\cov}{cov} \newcommand\independent{\perp\kern-5pt\perp} \newcommand{\CE}[2]{ \mathExp\left[ #1 \,\middle|\, #2 \right] } \newcommand{\disteq}{\overset{d}{=}} $$

# The Triple Staple

When reading, I prefer paper to electronic media. Unfortunately, a lot of my reading involves manuscripts from 8 to 100 pages in length, with the original document being an electronic PDF.

Double-sided printing works really well to resolve this issue partway. It lets me convert PDFs into paper documents, which I can focus on. This works great up to 15 pages. I print the page out and staple it. I’ve tried not-stapling the printed pages before, but then the individual papers frequently get out of order or generally all over the place.

**However**, for larger manuscripts I frequently found myself in a pickle:

- I don’t want to manage loose leaf pages individually.
- Staplers that can handle stapling over 15 pages don’t occur naturally, at least near the printers I’m around.

Attempting to use a stapler beyond its capacity does not end successfully.

For a good deal of my life I’ve resigned myself to dealing with a reality of mediocre staplers and even more mediocre workarounds, e.g., a packet on a single topic now needs be represented by 3 independent, separately-stapled documents, which is 2 too many.

I’m confident many others also have this problem. To wit, I’d like to introduce a life hack, for all situations where you have documents of up to \(2X\) pages and staplers with penetration power rated at \(X\) pages.

## The Problem

I want to staple this thick paper stack.

*Optimality criteria*.

(A) Grip strength of resulting staple.

(B) Non-obstruction of reading material.

## Solution

- Staple pages \(1\) to \(X\).
- Staple pages \(X+1\) to \(2X\).
- Peel back the corner of pages \(1\) to \(\lfloor X/2\rfloor\) over the staple. Repeat for \(\lfloor 3X/2\rfloor\) to \(2X\)
- Insert the exposed corner of pages \(\lfloor X/2\rfloor +1\) to \(\lfloor 3X/2\rfloor - 1\) into the stapler, making sure the folded-away corners of the outer pages are out of the stapler’s line of fire.
- Apply the stapler to the middle pages, then fold the outer pages’ corners back up.

## Results

Step 1 and 2.

Step 3.

Step 4.

Step 5.

Additional results (skew angle, front, and back views).

## Discussion and Related Work

(A) is met due to each staple holding together at least \(X\) pages. Contrast this with related work which only staples two pages \(X,X+1\) with an intermediate staple, resulting in a single point of failure at page \(X\).

(B) UX is equivalent to a single-stapled page, as opposed to binder-clip methodology which frequently requires clipping past the margin.

## Future Work

There exists a straightforward alternating iteration of our method that can be shown, by induction to apply to documents of length up to \(n X\) for any \(n\in\mathbb{N}\). We leave evaluation to future work.