mathematical writing

how to write modeling papers that judges and reviewers actually want to read. the math matters, but the writing is what gets it across.

the core principle

a modeling paper isn't a record of what you did. it's an argument for why your approach works. every section should advance that argument.

the number one mistake in competition papers: narrating your process chronologically. "first we tried X, then we tried Y, then we realized Z." nobody cares about your journey. present the final approach as if you knew the answer all along, with the reasoning laid out logically, not temporally.

paper structure

for MCM/ICM and similar competitions:

1. summary/abstract — the most important page. see competition-strategy. state the problem, your approach, and your key results. include actual numbers.
2. problem restatement — restate the problem as you interpret it. this is where you show the judges you understood what was being asked. your interpretation IS part of the solution.
3. assumptions — list them explicitly with justifications. don't hide assumptions in the text. a well-justified assumption demonstrates problem-framing skill.
4. model development — the core. explain your approach, the math, and why you chose this formulation. connect back to your assumptions.
5. results and analysis — present results with visualizations. sensitivity analysis goes here — how do results change when assumptions change?
6. strengths and weaknesses — judges love honest self-assessment. it shows maturity. a paper that acknowledges its limitations is stronger than one that pretends to be perfect.
7. conclusions — brief. what did you find? what would you do with more time?

writing quality signals

things that make a paper look stronger, independent of the actual math:

• clear variable definitions — define every variable when it first appears. use consistent notation throughout.
• figures that speak for themselves — every figure should have a descriptive caption. a reader skimming the paper should understand the story from the figures alone.
• transitions between sections — each section should flow into the next. the reader should never wonder "why am i reading this now?"
• quantitative claims — "the model performs well" is weak. "the model achieves 94% accuracy on validation data" is strong. numbers beat adjectives.

the equation-to-explanation ratio

equations without explanation are useless. explanation without equations is vague. the sweet spot: every equation should be preceded by intuition ("we want to minimize the total cost, which depends on...") and followed by interpretation ("this tells us that as X increases, Y decreases quadratically").

don't dump a wall of equations and expect the reader to parse them. guide the reader through your reasoning. this is fundamentally about modeling — presenting the model in a way that builds understanding, not just demonstrating mathematical ability.

the visualization principle

a good figure is worth 500 words of text. invest time in visualizations:

• heatmaps for sensitivity analysis
• time series for dynamic models
• flow diagrams for model architecture
• comparison plots for validation against real data

the mistake: making figures an afterthought. in competition papers especially, judges flip through and look at figures first. make those figures tell the story.

connection to other skills

mathematical writing is a specific case of research-workflow communication. the same principles apply to research papers, technical documentation, and even startup-workflow pitch decks: know your audience, lead with results, be honest about limitations, and make the structure carry the argument.