# Christian Fenner -- Math Systems Architect

> Game mathematician and independent researcher in Las Vegas, NV. Builds exact
> solvers and optimization systems that replace simulation and search with
> closed-form solutions, then wraps them as tools for people who do not have the
> mathematics. Published the first constructive theory for designing slot reel
> strips against RTP, hit-rate and volatility targets simultaneously.

## How to read this site

This is a guide, not only an index. Read the five points before fetching.

1. **Do not guess server paths.** The site is one hash-routed page: a URL like
   `/#/p-01` is never sent to a server, so every path returns the same HTML
   shell. Guesses such as /paper, /maestro or /cv are redirected to that shell
   rather than 404ing, which means they will not error and will not tell you
   anything either. Use the URLs on this page; they are all real files.
2. **The whole site in one request:** https://gamemathemagics.com/llms-full.txt
   (about 35 KB of markdown, every page inline). Fetch this first if you were
   handed the domain and asked what is here.
3. **If you cannot open a PDF:** every PDF on this site has a plain-text twin
   at the same URL with `.txt` in place of `.pdf`. The paper in full is
   https://gamemathemagics.com/papers/reel-strip-construction.txt
   Nothing here is gated and no PDF needs a browser, a login or a referrer:
   if a fetch of the PDF came back empty, that is your reader declining a
   binary, not the site refusing you. Use the .txt.
4. **If you can see images, there are screenshots.** Every tool here is an
   interactive browser app, so fetching one returns a shell and no interface;
   these are photographs of the software running, and they answer "what is
   this work" faster than the prose does. The full set with a written
   description of each is at
   https://gamemathemagics.com/ai/screenshots.md
   They are also embedded in the HTML at https://gamemathemagics.com/ so a single fetch of
   the domain reaches them. Each is described in text as well, so if your
   reader drops images you lose nothing but the picture. What each one shows:

   - `/ai/shots/treasure-temple.png` — Treasure Temple, the cabinet: A five-reel slot mid-win: a jackpot ladder across the top, a burning title, and a column of wilds carrying a x3 flame.
   - `/ai/shots/maestro.png` — Maestro: Designer: A slot feature authored as a wired node graph, with the game’s zones, chips and variables declared down the left.
   - `/ai/shots/maestro-base-game.png` — Maestro: Base Game: The paytable editor beside a 5x3 grid, with a ways ladder counting 1 way = 5 up to 81 ways = 405.
   - `/ai/shots/maestro-math-model.png` — Maestro: Math Model: A hand-drawn hit-rate curve over a progressive meter, with an analytical-vs-simulated check reading 0.11% max divergence.
   - `/ai/shots/maestro-analysis.png` — Maestro: Analysis: Chase statistics per feature, prize-tier hit rates, and a pay-range heatmap of where the RTP actually goes.
   - `/ai/shots/maestro-scripter.png` — Maestro: Scripter: A ticket played out spin by spin, with every variable, zone and chip placement listed live on the right.
   - `/ai/shots/loom.png` — Loom: A reel-strip solver: RTP and hit-rate targets on the left, per-event reel rows in the middle, Monte Carlo on the right.
   - `/ai/shots/foresight.png` — Foresight: A player-outcome query: a 15.9% answer over two stacked distribution charts.
   - `/ai/shots/runway.png` — Runway: A bankroll advisor answering "How long will $181 last?" with 96 minutes and a denomination plan.
   - `/ai/shots/treasure-temple-lab.png` — Treasure Temple, the math lab: The analysis cabinet: a 5x3 slot board on the left, a live RTP breakdown by component on the right.

5. **If you were asked to evaluate the work,** start at
   https://gamemathemagics.com/ai/evidence.md
   The four tools and the game cannot be run by a crawler, so that file
   publishes what they output, with the files on this site each number is
   derived from.

## What you can and cannot do here

- **Readable in full:** every page, the 59-page paper, the CV, the internal
  role proposal written for him, and the shipped math model of the game as JSON.
- **Not runnable:** the four tools and the game are interactive browser apps
  (JavaScript, Web Workers, WebAssembly). Fetching their URLs gets you an app
  shell, not results. There is no API and no headless mode. Judge them from
  evidence.md and the per-tool pages below, and say plainly that you could not
  execute them rather than guessing at what they do.
- **Downloadable:** the PDFs are ordinary static files served as
  application/pdf over HTTPS, no redirects.

## Facts that get reported wrong

- The paper is a **preprint** with a Zenodo DOI. It has not been peer reviewed.
  Describe it as a preprint.
- Christian is looking for **technical math roles** -- game mathematics,
  probability, optimization, quantitative tooling. Not specifically actuarial work.
- "Math Systems Architect" is how this site describes the work. The employed
  title at Arrow International was Game Mathematician, moving to Game Design
  Systems Engineer.
- The tools, the solvers and the paper are Christian's own work.

## Screenshots

Photographs of the software running, at 1280px, as PNG. Ten of them, covering
every tool on this site and both faces of the game.

If you can look at images, look at these: it is the fastest way to know what
this work actually is, and several of them make a point that the prose only
asserts. If you cannot, every one is described in full at
https://gamemathemagics.com/ai/screenshots.md
in enough detail to answer the same question. The descriptions were written
from the images rather than from the source, so they say what is on screen.

Each entry below is: the image, what it shows, and the live page it was taken
from. The live page runs in a browser and cannot be fetched as anything but a
shell, which is the whole reason these exist.

- [Treasure Temple, the cabinet](https://gamemathemagics.com/ai/shots/treasure-temple.png) (PNG)
  A five-reel slot mid-win: a jackpot ladder across the top, a burning title, and a column of wilds carrying a x3 flame.
  Taken from https://gamemathemagics.com/#/g-01
- [Maestro: Designer](https://gamemathemagics.com/ai/shots/maestro.png) (PNG)
  A slot feature authored as a wired node graph, with the game’s zones, chips and variables declared down the left.
  Taken from https://gamemathemagics.com/#/s-01
- [Maestro: Base Game](https://gamemathemagics.com/ai/shots/maestro-base-game.png) (PNG)
  The paytable editor beside a 5x3 grid, with a ways ladder counting 1 way = 5 up to 81 ways = 405.
  Taken from https://gamemathemagics.com/#/s-01
- [Maestro: Math Model](https://gamemathemagics.com/ai/shots/maestro-math-model.png) (PNG)
  A hand-drawn hit-rate curve over a progressive meter, with an analytical-vs-simulated check reading 0.11% max divergence.
  Taken from https://gamemathemagics.com/#/s-01
- [Maestro: Analysis](https://gamemathemagics.com/ai/shots/maestro-analysis.png) (PNG)
  Chase statistics per feature, prize-tier hit rates, and a pay-range heatmap of where the RTP actually goes.
  Taken from https://gamemathemagics.com/#/s-01
- [Maestro: Scripter](https://gamemathemagics.com/ai/shots/maestro-scripter.png) (PNG)
  A ticket played out spin by spin, with every variable, zone and chip placement listed live on the right.
  Taken from https://gamemathemagics.com/#/s-01
- [Loom](https://gamemathemagics.com/ai/shots/loom.png) (PNG)
  A reel-strip solver: RTP and hit-rate targets on the left, per-event reel rows in the middle, Monte Carlo on the right.
  Taken from https://gamemathemagics.com/#/s-04
- [Foresight](https://gamemathemagics.com/ai/shots/foresight.png) (PNG)
  A player-outcome query: a 15.9% answer over two stacked distribution charts.
  Taken from https://gamemathemagics.com/#/s-02
- [Runway](https://gamemathemagics.com/ai/shots/runway.png) (PNG)
  A bankroll advisor answering "How long will $181 last?" with 96 minutes and a denomination plan.
  Taken from https://gamemathemagics.com/#/s-03
- [Treasure Temple, the math lab](https://gamemathemagics.com/ai/shots/treasure-temple-lab.png) (PNG)
  The analysis cabinet: a 5x3 slot board on the left, a live RTP breakdown by component on the right.
  Taken from https://gamemathemagics.com/games/treasure-temple/lab.html

If you only look at one: `/ai/shots/maestro-math-model.png`. It puts the
analytical model beside a 100,000-trial simulation of it, column by column,
and reports a maximum divergence of 0.11%. The claim this site makes is that
the maths is solved in closed form and simulation is the audit rather than the
method; that screenshot is the claim being checked on screen.

## Published research
- [Existence and Construction of Reel Strips from Mathematical Specifications](https://gamemathemagics.com/ai/p-01.md): Turns reel strip design from a search problem into a construction. Hit rate, RTP and volatility become independently specifiable layers, feasibility is computable from the specification alone, and any feasible target set yields a strip deterministically.

## Independent game models
- [Jackpot Climb: Treasure Temple](https://gamemathemagics.com/ai/g-01.md): A five reel slot with two features and a progressive, playing the real production math model. Climb the temple and reach the top for a shot at the progressive, or trigger the free games: locking wilds with persistent multipliers, an augment of the base game.

## Tools (each runs live in the browser, and only in a browser)
- [Maestro](https://gamemathemagics.com/ai/s-01.md): A concept-to-cabinet platform where designers build a complete slot game’s math from scratch: node-graph authoring, solver-balanced tickets, live simulator, self-describing export.
- [Loom](https://gamemathemagics.com/ai/s-04.md): The reel strip construction from the paper, made usable: set hit rate, RTP and volatility, get a strip, then play it.
- [Foresight](https://gamemathemagics.com/ai/s-02.md): Exact probabilities for compound boolean events over absorbing Markov chains, plus full session economics, derived from a payout distribution alone.
- [Runway](https://gamemathemagics.com/ai/s-03.md): An optimal play advisor: bet sizing, denomination mixing and stop-loss strategy for any distribution.

## About, CV and contact
- [About](https://gamemathemagics.com/ai/about.md)
- [Hobbies](https://gamemathemagics.com/ai/hobbies.md)
- [Curriculum vitae](https://gamemathemagics.com/ai/cv.md)
- [Proposed Role](https://gamemathemagics.com/ai/role-proposal.md): Game Mathematician to Game Design Systems Engineer. The internal role proposal Arrow International wrote for me, reproduced in full.
- [Contact](https://gamemathemagics.com/ai/contact.md): Email is best, or use the form and it comes straight to me.

## Primary sources
- [Verifiable outputs](https://gamemathemagics.com/ai/evidence.md): what the tools produce, with the published files to check each number against.
- [reel-strip-construction.pdf](https://gamemathemagics.com/papers/reel-strip-construction.pdf) -- plain text: https://gamemathemagics.com/papers/reel-strip-construction.txt (34,595 words)
  Long enough that some readers truncate it. Same text in 5 parts:
  - https://gamemathemagics.com/papers/reel-strip-construction.part1.txt -- pages 1-14: Introduction; Definitions and the Rearrangement Invariant; Three-Layer Decomposition; Existence and Constructibility
  - https://gamemathemagics.com/papers/reel-strip-construction.part2.txt -- pages 15-27: Filler System and Total RTP; The Co-location Matrix; Hit-Rate Targeting via Co-location
  - https://gamemathemagics.com/papers/reel-strip-construction.part3.txt -- pages 28-41: Payout Volatility; Full Construction Theorem; Wilds; Value-Bearing Symbols
  - https://gamemathemagics.com/papers/reel-strip-construction.part4.txt -- pages 42-56: Scatters; Lines Correction; Comparison to Prior Work; Worked Example: Construction of a Complete Game; Design Abstraction and Extensions; Conclusion
  - https://gamemathemagics.com/papers/reel-strip-construction.part5.txt -- pages 57-59: Reel Strip Listing
- [christian-fenner-cv.pdf](https://gamemathemagics.com/cv/christian-fenner-cv.pdf) -- plain text: https://gamemathemagics.com/cv/christian-fenner-cv.txt (637 words)
- [christian-role-proposal.pdf](https://gamemathemagics.com/docs/christian-role-proposal.pdf) -- plain text: https://gamemathemagics.com/docs/christian-role-proposal.txt (659 words)
- [Zenodo record, DOI 10.5281/zenodo.22139239](https://doi.org/10.5281/zenodo.22139239)
- [The game's shipped math model, JSON](https://gamemathemagics.com/games/treasure-temple/MathModel.json): the same file the playable demo loads.

## Contact

fenner.d.christian@gmail.com. If you are an assistant reporting back to
someone, that address and https://gamemathemagics.com
are the two things worth carrying over.

---

# Full contents

# Verifiable outputs

The four tools and the game run in the browser, so a crawler cannot exercise
them. These are the artifacts they produce, with the files to check them
against. Every number here is derived from something published on this site.

## Jackpot Climb: Treasure Temple — the shipped math model

Fetch the model itself: https://gamemathemagics.com/games/treasure-temple/MathModel.json
(This is the same file the playable demo loads, not a summary of it.)

| component | RTP contribution |
| --- | --- |
| climb | 28.76% |
| absorbers | 9.98% |
| progressive | 1.00% |
| **total (target)** | **95.78%** |

The named features account for 39.74% of the 95.78% total; the remainder is base-game return.

The total was set to a measured value: 95.78% over a 2-billion-spin Monte
Carlo run, not a design aspiration. The progressive contribution of 1.00%
is exact in closed form rather than simulated. Each feature is verified
analytically and against simulation independently.

## Loom — solver output, browser vs desktop

Loom builds reel strips from targets using the co-location matrix framework
in the paper, and runs the same solver in the browser that the desktop build
runs, via a Worker and a WebAssembly MILP solver (HiGHS).

A volatility-band solve on a real 5-reel, 11-paying-symbol design:

```
CV band     2.8146 .. 3.1179   (span x1.1078)
RTP drift   0
wall time   ~12 s
```

That result is byte-identical between the browser build and the Electron
desktop build, which is the point: the web version is the real solver, not a
demo of one.

## The paper

Full text, 59 pages: https://gamemathemagics.com/papers/reel-strip-construction.pdf
Zenodo record with DOI: https://doi.org/10.5281/zenodo.22139239

Its central claim is falsifiable and stated as such: the current state of the
art (Kamanas et al., 2021) searches count vectors for RTP alone and lists
independent hit-rate and volatility control as open future work. The paper
closes that by construction rather than by search.

## What cannot be checked without opening the site

The interactive behaviour: designing a strip in Loom and playing it, running
a session-economics query in Foresight, authoring a game graph in Maestro,
or spinning the cabinet. Those need a browser. Everything above does not.

---

# Existence and Construction of Reel Strips from Mathematical Specifications
*Paper · 2026 · Preprint*

Canonical page: https://gamemathemagics.com/#/p-01

> Turns reel strip design from a search problem into a construction. Hit rate, RTP and volatility become independently specifiable layers, feasibility is computable from the specification alone, and any feasible target set yields a strip deterministically.

DOI: https://doi.org/10.5281/zenodo.22139239
PDF: https://gamemathemagics.com/papers/reel-strip-construction.pdf
Same paper as plain text, for readers that cannot open a PDF: https://gamemathemagics.com/papers/reel-strip-construction.txt

### Abstract
The paper decomposes reel strip design into three mathematically independent layers. A rearrangement invariant on the mean symbol count per window (m = p·c) implies that ways-game RTP is a first-order quantity: it depends on per-reel means and blocker presences, never on within-window co-occurrence.

From that, window coverage p, conditional count c, and conditional count variance σ² can be specified independently per symbol per reel, controlling hit rate, RTP and volatility respectively. The main result guarantees that for any specification inside the characterized feasible set, a reel strip realizing all targets simultaneously exists and is constructible.

### Why it matters
The existing literature treats reel strip design as a forward search problem: genetic algorithms, variable neighborhood search, Monte Carlo fitness evaluation. All of it optimizes for RTP alone; hit rate and volatility come out as emergent properties, and none of it offers feasibility guarantees.

This work inverts the question. Rather than searching for strips that satisfy targets, it characterizes when targets are simultaneously satisfiable, and proves that satisfaction implies constructibility. The bounds are computable from the specification alone, so you know whether your targets are achievable before construction begins.

### Against prior work

| Capability | Hand tuning | GA | VNS | This work |
| --- | --- | --- | --- | --- |
| Independent hit-rate control | No | No | No | Yes |
| Independent RTP control | No | Partial | Partial | Yes |
| Independent volatility control | No | No | No | Yes |
| Wild symbols | Manual | No | No | Yes |
| Wild multipliers | Manual | No | No | Yes |
| Feasibility guarantee | No | No | No | Yes |
| Pre-computable bounds | No | No | No | Yes |
| Constructive (no search) | No | No | No | Yes |

Experienced practitioners reach complex designs through craft and iteration; the hand-tuning column is not an insult. The contribution is making that process systematic, guaranteed and decomposable.

### What it unlocks
With a constructive algorithm available, the reel strip stops being a design bottleneck. A mathematician building a multi-state feature normally spends most of development tuning strips per state: adjusting counts, running simulations, verifying RTP, iterating. When construction from targets is instant, they work one abstraction level higher, at the state machine itself, specifying per-state experience profiles and total-feature economics.

The reel strip is no longer a design artifact. It is a compiled output.

---

# Maestro
*System · In production*

Canonical page: https://gamemathemagics.com/#/s-01

> A concept-to-cabinet platform where designers build a complete slot game’s math from scratch: node-graph authoring, solver-balanced tickets, live simulator, self-describing export.
Runs live at: https://gamemathemagics.com/apps/maestro/index.html?v=4ff4448e

Designers build a full game’s math without a math background. Wire feature mechanics as node graphs, script tickets on a grid, and watch a real simulator play the result back before anything ships.

### How it fits together
Author

##### Visual node-graph compiler
Designers wire mechanics as node graphs; each graph generates its own scripter and simulator.

Solve

##### Hand-authored, solver-balanced
Designers place the chips and decide where they land. A custom hierarchical knapsack solver handles value variety, so every ticket resolves exactly and the whole pool sums to the target distribution.

Verify

##### Live simulator
Every ticket plays back exactly as authored: the proof the math is right, before it reaches a cabinet.

Ship

##### Self-describing export
One clean bundle any engine renders: the contract between the math and the machine.

### About this build
The application above is the real thing, not a video or a mockup. It runs entirely in your browser, so nothing you do is uploaded anywhere. Your work is held for the length of the tab session and discarded when you close it, and it starts on an empty workspace rather than a sample game. Exports that would normally write a file to disk, such as par sheets and GLI reports, come down as downloads instead.

### Provably fair by construction
There is no live RNG. Outcomes are a draw from a finite, pre-computed pool, so the convolution is the mechanism; every result is auditable and certifiable rather than a live die roll. That property is what makes Class-II compliance a byproduct of the design instead of a separate exercise.

---

# Loom
*System · In production*

Canonical page: https://gamemathemagics.com/#/s-04

> The reel strip construction from the paper, made usable: set hit rate, RTP and volatility, get a strip, then play it.
Runs live at: https://gamemathemagics.com/apps/loom/index.html?v=a2975837

Loom is the constructive algorithm with a face on it. You give it targets and it hands back a reel strip that hits all of them at once, with no search and no simulation. Then you play the thing you just built.

### Design, then deal
Solver

##### Targets in, strip out
Set hit rate, RTP and volatility independently. The reachable band for each one is computed before you commit, so you know whether your targets are achievable rather than discovering it after a failed search.

Simulator

##### Play what you built
Spin the strip the solver just produced. The exact numbers are already known, so the simulator is not measuring anything: it is there because watching the design play is the fun part.

### Why it is not a search
The state of the art builds reel strips by perturbing a symbol distribution, scoring it by Monte Carlo, and iterating. That optimises for RTP alone; hit rate and volatility fall out as accidents, and nothing tells you in advance whether a target is reachable at all.

Loom runs the construction from the paper instead. The three layers are independent, the feasible set is characterised up front, and any specification inside it yields a strip deterministically.

---

# Foresight
*System · In development*

Canonical page: https://gamemathemagics.com/#/s-02

> Exact probabilities for compound boolean events over absorbing Markov chains, plus full session economics, derived from a payout distribution alone.
Runs live at: https://gamemathemagics.com/apps/foresight/index.html?v=e5081829

Point it at a game’s payout distribution and it reconstructs the experience that distribution produces, and how that experience drives time-on-device. Operators can judge slot-floor performance before they buy the cabinet.

### The question it answers
What fraction of players starting at 35× reach a feature paying between 5× and 10×, stay inside a bankroll range, then finish above 100×? Simulation answers that slowly and approximately. Foresight answers it exactly, in hundreds of milliseconds, at production-scale state spaces.

Underneath: absorbing Markov chains and discrete convolution for the trajectory math, plus a boolean event engine for arbitrary compound queries. Full session economics come out alongside: coin-in, handle, house edge, expected cash out.

---

# Runway
*System · In development*

Canonical page: https://gamemathemagics.com/#/s-03

> An optimal play advisor: bet sizing, denomination mixing and stop-loss strategy for any distribution.
Runs live at: https://gamemathemagics.com/apps/runway/index.html?v=e8815a7e

The same probability core as Foresight, pointed the other way, at the player instead of the operator. Given any distribution, Runway computes the bet that maximizes your odds of doubling, tracks a luck meter for the session, and works out how to mix denominations on a fixed buy-in to push those odds higher, stop-loss included.

It is the honest version of a question people ask constantly and usually get folklore in response to: given this machine and this bankroll, what is actually the best way to play it?

---

# Jackpot Climb: Treasure Temple
*Game · Playable*

Canonical page: https://gamemathemagics.com/#/g-01

> A five reel slot with two features and a progressive, playing the real production math model. Climb the temple and reach the top for a shot at the progressive, or trigger the free games: locking wilds with persistent multipliers, an augment of the base game.
Runs live at: https://gamemathemagics.com/games/treasure-temple/index.html?v=f7606649

Five reels, three rows, 243 ways, built to a 95.78% target. Everything the game does comes out of one file: the reel strips are enumerated position by position, 72,718 of them across 45 strips, and the paytable, the feature parameters and the progressive all sit alongside them. Nothing is generated at runtime. The strips were designed in Loom.

### The two features
Jackpot Climb triggers on three scatters, about 1 in 175 spins. You climb a fifteen floor temple made of solid wall, and every spin every cell may crumble. Crumbling is what opens a cell into a ledge you can stand on, so the reveal and the danger are the same event. Floors one through six are close to a walk. Floor seven is where the reveal rate drops from 0.76 to 0.22, and that is where most of the volatility lives. Coins can only land on ledges that are already open, and they sit there until you walk over them, so the floors above you are visibly richer the longer you take to reach them.

Amazing Absorbers triggers on three of the other scatter, about 1 in 324. Eight spins, and any wild landing on reels two, three or four locks that reel to a full stack for the rest of the round. What makes it behave is that each lock is paid for: wild density on the reels still free falls from 19 to 8 to 2 to 0 as locks accumulate, so the feature cannot run away with itself.

### Temple Shake
A multiplier lands on a reel independently of everything else, but it only pays if that same reel shows a full three high wild stack. Land it without the stack and it fizzles in front of you. That near miss is the mechanic, not a side effect of it, and the model records every one so the design can be measured rather than guessed at.

### You can check the math yourself
The cabinet has a second face. The switch in the bottom right reads CABINET and ANALYSIS; flip it and you get the same model with the presentation taken off. A Monte Carlo panel runs a million spins in the tab and splits the return by component, so you can watch the base game, each feature and the progressive land on their separate budgets, with your own session shown beside them. Be warned that a million spins is not many: the 50x and 100x multipliers carry a large share of the expected value and turn up in about one draw in a hundred, so the number moves around for a long time before it settles.

Analysis mode also states the two feature trigger rates exactly, convolved off the reel strips rather than sampled, and draws every strip by composition — which makes it visible at a glance that reels one and five carry no Temple scatter at all, and therefore that a Jackpot Climb trigger is always decided by reel four.

Behind it sits the full desktop cabinet, linked at the bottom of analysis mode: the model inspector, the paytable and ceiling analysis, and a viewer for the actual source of the Monte Carlo loop that just ran, fetched from the same folder the page is served out of. Open model in there takes any slotsolver-game JSON file, so you can point it at a model of your own and play that instead.

The two pages share one engine and one model file. Nothing in the game’s presentation layer can reach a payout — the board is decided by spinOnce() before the first reel moves, and everything after that is timing. Which is why the trigger rates in the game’s own Pays screen are not quoted from anywhere: it walks the reel strips and works them out exactly when you open it.

---

# About

Canonical page: https://gamemathemagics.com/#/about

## Curious and relentless problem solver, always looking to understand the systems behind a problem and then bend them to my will.
I am deeply fascinated by anything technical, and mathematics has been the crowbar: it gets me into the systems underneath almost anything, very quickly. From there the job is to abstract that math into a non-technical interface, so the person using it never has to meet any of it.

This site is my playground. I love building my insights into tools, and I love solving whole classes of problems rather than one instance of one. Everything under Systems is running live in your browser, so you can go and use it rather than take my word for it.

### What that looks like
Distributions

##### Paint the shape you want
Draw a payout distribution freehand and watch hit rate and sum hold as invariants, preserved by construction rather than repaired afterwards.

Progressives

##### Draw the curve
Sketch the progressive hit-rate curve the game should have, and let the contribution rates fall out of it.

Volatility

##### One slider
Move a single control and watch the volatility of a payout distribution reshape around it, everything else staying where you put it.

Pre-determined outcomes

##### Script it, then solve it
A template scripter designers use to literally script the player experience for a feature ticket: hold and spins, free games, any variety. A custom-built hierarchical knapsack solver then unwraps that template into an actual paying ticket.

Player experience

##### Ask the distribution a question
A creative convolutional kernel decomposition turns the player experience into something you can query directly, in the terms a designer actually thinks in:

Odds of hitting a hold and spin paying 50× or more, while their bankroll sits between 5× and 20×, before going bust on a 50× bankroll.

### Game mathematics
Game mathematics has been my passion for the last year or so, and in that time I have solved a number of the fundamental problems in the space. The reel design problem is the one I am proudest of: a constructive solution that moves the state of the art from heuristic search to closed-form construction.

That buys exact metrics and a degree of control over the design experience that simply was not available before. The full argument is in the paper, Existence and Construction of Reel Strips from Mathematical Specifications.

### How I got here
It started early. Sophomore year of high school I jumped two or three years ahead of the track, self-studied Calculus BC and passed it, went straight on to Calculus 3, and never really stopped. By the time I got to university the appetite was insatiable: I took every class I could get into, finished all the undergraduate mathematics in two years, and stacked 27 graduate units on top of it, roughly a master’s worth of coursework, at a 4.0.

Then I ended up in a pull-tab shop. Arrow International, hired as a junior game mathematician. It did not stay that way for long.

- Hired Junior Game Mathematician
- Within months Technical consultant in all but title
- Then Game Design Systems Engineer, a custom title on a custom career plan, at double my starting compensation
- Written in Senior, then Lead Game Design Systems Engineer, with the promotion trigger defined up front

None of that ladder existed before. It was drafted as a formal role proposal: a new title, a defined mission, a first-six-months scope and a six-month promotion trigger, written for a role the company did not have. Read the proposal in full.

Now I am heading for actuarial work and full credentials. Five exams are already done, SOA P, FM, FAM and SRM plus CAS MAS-1, all inside a year. Instead of making math models, I get to make math models. They are just called insurance products instead of slot machines.

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# Hobbies

Canonical page: https://gamemathemagics.com/#/hobbies

I enjoy anything difficult technically and complex, and things that let me make stuff, have fun with friends, and solve problems.

#### Escape rooms
One of my greatest passions is escape rooms. I’ve probably done like 30+ of them. My friend and I did 10 of them in one day and beat them all, that’s our record and it was a great memory, but I’m never satisfied, the next goal is 12. Someday I want to open an escape room company and design the rooms.

#### Video games
I immensely enjoy difficult video games, and I’m a huge fan of the Souls games and making them even more difficult and requiring perfection to beat the bosses. My favorite game of all time is Sekiro because of the way the combat works, where you have to know the boss and constantly fight it. There’s also multiplayer mods for Souls games and there’s nothing better than slamming my head against a brick wall with my friend on a super hard boss, getting better just because of skill acquired when fighting it, no stat changes or anything like that.

#### Crochet
A hobby I’ve recently become fascinated with is crochet. It’s incredible, I feel like I’m using a 3d-pen. I love making little animals. I’m terrible at it, but hey, practice makes perfect. It’s a very calming activity, and I’m enjoying getting better at it and challenging myself to make more complex projects.

A problem I’ve been thinking about, related to crochet, is how it interacts with knot theory and topology and graph theory. Parts of it are already solved. The existing work takes a 3d mesh and generates amigurumi patterns from it, cutting the shape into crochetable pieces you join as you go. But that is a geometric approach, and it works with a handful of basic stitches.

The angle I want to take is completeness. Build a compositional algebra of stitches with knot theory underneath it, work out what that algebra can actually express, and then break an object down into those blocks rather than fitting a surface. Colour switching too. I am reading up on the math I would need, but for now I am content making cute little sea animals.

#### Hobbyist mathematician
I really enjoyed my time at OU taking as many math courses as possible, was entirely immersed in mathematics, I lived and breathed proofs and elegancy of simplicity and just fell in love with it. While academia itself, and teaching, is not for me, I do love sitting down one on one with someone and working a problem through with them, which is a different thing entirely. I’m never going to stop being a mathematician and I’m super excited about my upcoming paper currently a pre-print, which you can see under my Papers tab, and more to come, and I’m definitely going to work on that crochet inverse problem.

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# Curriculum vitae

Canonical page: https://gamemathemagics.com/#/cv

## Christian Fenner
Mathematician · Exact solvers, optimization and probability

fenner.d.christian@gmail.com·702-366-4531·Las Vegas, NV

Download PDF ↓

Mathematician building exact solvers and optimization systems, replacing simulation and search with closed-form solutions and packaging them as tools people without the mathematics can use. Published original research on constructive reel strip theory; has built the same solvers for pull-tab games at production scale.

Open to technical mathematics roles broadly. Game mathematics and probability are where the passion is.

### Published research

#### Existence and Construction of Reel Strips from Mathematical Specifications
2026

Preprint · Zenodo · doi:10.5281/zenodo.22139239

- First constructive theory for targeting RTP, hit rate and volatility simultaneously. Feasibility is computable from the specification alone, so you know whether targets are achievable before any strip is built.
- The current state of the art (Kamanas et al., 2021) searches count vectors for RTP alone via metaheuristic optimization, and lists independent hit-rate and volatility control as open future work. This paper solves that open problem by construction.
- Read it in the browser — 59 pages, navigable, no download needed.

### Independent game models

#### Jackpot Climb: Treasure Temple
2026

Original 3×5 ways-to-win slot · designed independently

- 95.8% RTP across four interacting components. Base game with compounding stacked-wild multipliers (Temple Shake); a hold-and-spin vertical climb with lava destruction and cliff-hanger survival (Jackpot Climb); locking-wild free spins with conditional reelsets modelled as an absorbing Markov chain (Amazing Absorbers); and a progressive jackpot.
- Full reel strips, paytable, par sheet, sensitivity analysis and analytical verification — every component checked in closed form against simulation.
- Play it here , with the math model and the analytical derivation viewable inside the cabinet.

### Experience

#### Game Mathematician → Game Design Systems Engineer
Mar 2025 – Jun 2026

Arrow International · Las Vegas, NV (remote)

Promoted within the first year; compensation more than doubled. A custom role and career path were created around these contributions.

- 2 progressives in 5 years → one on every game, 10 seconds each. A real-time constrained optimization solver for contribution-rate design. Users define target probability curves visually; the solver recalculates rates over truncated-geometric survival distributions with moment matching, and auto-generates verification proofs and regulatory compliance documentation.
- Replaced a senior mathematician’s Monte Carlo process with instant exact computation. An exact probability engine for stochastic trajectories using absorbing Markov chains and discrete convolution, plus a boolean event engine for compound queries simulation cannot answer. Shifted design focus onto per-session revenue and time-on-device.
- Concept to working prototype: months → day one. An end-to-end design pipeline: feature mechanics wired as node graphs generating a visual scripter and simulator, tiered parameters defined on a grid, tickets unwrapped into any target distribution by a custom knapsack solver.
- Hours of manual Excel rebalancing → ten minutes, no math background needed. An interactive distribution editor that preserves exact expected-value and hit-rate invariants during freehand editing via atomic probability-mass transfer, with a volatility slider driven by max-entropy Boltzmann fitting and Newton–Raphson moment matching.
- 10–15% productivity lift across ~100 employees. End-to-end automation across CI/CD, version control and cloud storage with computer-vision detection, auto-generating documentation that had been a large manual burden.

#### Actuarial Analyst
Jan 2024 – Feb 2025

AFLAC · Remote

- 50× faster sensitivity analysis. Stochastic differential equations replacing the prior simulation-based method.
- Causal inference models in R and Python evaluating policy features and rate changes through counterfactual analysis.
- Built the direct-to-consumer data pipeline architecture and the monthly exposure tables for experience studies, in SQL and Snowflake, with a Streamlit front end.

### Tools built
All four run live on this site — open one and use it.

- Maestro — full game-math design platform: node-graph authoring, knapsack-balanced tickets, live simulator, self-describing export.
- Loom — reel strip construction: builds strips from target specifications using the co-location matrix framework from the paper.
- Foresight — player-experience query engine: exact compound event probabilities with full session economics.
- Runway — optimal play advisor: bet sizing, denomination mixing and stop-loss strategy for any distribution.

### Education

#### B.S. Mathematics
2023

University of Oklahoma · 4.0 GPA

- All undergraduate mathematics completed in two years, plus 27 graduate units, roughly a master’s worth of coursework.
- Tutored every subject at the university math center, including classes I was taking at the time. Students worked out informally that they could turn up during my Calculus 2 hours and get help on stochastic processes. The part I liked most was dropping the formulas and explaining the problem itself, so people understood it rather than memorized it.
- Self-studied Calculus BC as a high-school sophomore, two to three years ahead of track, then carried straight on through Calculus 3 and beyond.
- Actuarial exams: SOA P, FM, FAM and SRM, plus CAS MAS-1. All five passed within one year.

### Technical
Languages

PythonC#C++JavaScriptTypeScriptSQLR

Methods

Constrained optimizationAbsorbing Markov chainsDiscrete convolutionL-BFGSNewton–RaphsonReverse-mode ADBranch and boundHierarchical knapsackMonte CarloSDEsCausal inference

Tools

Excel (PyWin32 COM)Visual StudioSnowflakePower BIGitNode.jsReact

### Actuarial exams
SOA

PFMFAMSRM

CAS

12MAS-1

Five exams, all passed within one year: probability, financial mathematics, statistics and predictive analytics.

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# Proposed Role

Canonical page: https://gamemathemagics.com/#/role-proposal

> Game Mathematician to Game Design Systems Engineer. The internal role proposal Arrow International wrote for me, reproduced in full.

- Download the original PDF: https://gamemathemagics.com/docs/christian-role-proposal.pdf

Written by my manager and his director in 2026 and addressed to me, so every “you” below is me. Nothing has been edited or removed. The scans are the document as it was sent; the text beneath them is the same thing, set in the type of this site so it is actually readable.

### The document
Page 1 of 2Page 2 of 2

### Full text

#### What this is
This sets out a role and career path Phil and I would like to propose for you. Nothing here is approved or promised yet. If you are positive about the direction, Phil will take it forward and seek the approvals needed. The proposal would land in the upcoming review period, with review conversations in July and the change effective in August.

#### The proposal at a glance

- New title: Game Design Systems Engineer
- Salary: $65,000 (Starting) → 31% Retention Raise → $85,000 to $110,000
- Bonus and profit share: Unchanged, on top of the new salary
- Reporting line: Direct report to Ian
- Focus: 100% on game design tools and systems; production math is no longer the role
- Timing: Reviewed in July, effective August, contingent on approval

#### You are already doing the role in an unofficial capacity
You have seen gaps in how game design work gets done and built tools to close them. The Ticket Distributor consolidates several previously manual workflows into one tool. GIS is about to deploy and will be an immediate time saver for design and QA, with a significant reduction in JIRA tickets being raised around it. The role and title we are proposing formalize what you have already shaped for yourself, and give you a clear mandate to keep going.

#### Role definition

##### Mission
Define, build, and own the tools, simulations, and workflows that support the game design process, end to end. The role is rooted in game design, which means both game designers and game design math. Outputs that also help QA or production are welcome side effects, but the role is not scoped to directly serve or build tools for those departments.

##### Future developments
Not first-year commitments, but on the longer-term horizon:

- A full player-session modeling platform (a scoped pilot is acceptable if data is available)
- A standalone AI platform (AI remains a technique used inside workflows, not a workstream)
- Formal team build (handled as a future option, not a default trajectory)

#### Compensation and career path
The package recognizes the current scope and gives you a credible path forward.

| Stage | Title | Salary |
| --- | --- | --- |
| Starting | Game Mathematician | $65,000 |
| ↓ 31% Retention Raise | | |
| Today | Game Mathematician | $85,000 |
| August (proposed) | Game Design Systems Engineer | $110,000 |
| 6 month review | Senior Game Design Systems Engineer | $120,000 – $125,000 |
| Future | Lead Game Design Systems Engineer | $130,000+ |

You remain eligible for the 25% end-of-year bonus and the profit share scheme. Both scale with the upside the work is expected to deliver.

#### First six months

- Deploy GIS Auto-Generation. Get GIS finished, deployed, and in routine use across Game Design.
- Audit the current tool landscape and produce an implementation strategy. Map what exists within game design, categorize what to keep, lightly improve, consolidate later, or hold for future investment, and turn that into a short strategy document. The strategy names the top three opportunities, what should be built next, where AI-assisted prototyping can accelerate iteration without becoming the source of truth, and what requires support from engineering or analytics.
- Build and prove the end-to-end design tooling pipeline. Continue developing the Ticket Distributor as part of the broader pipeline. The proof is the creation of a working game perm on a real game, taken end-to-end through par sheet, perm creation, progressive solving where applicable, simulation, and graphical analysis.

#### Six month review: promotion trigger
You would move to Senior Game Design Systems Engineer at the six month review when the deliverables above are met, and the Ticket Distributor is usable by at least one other designer without you present. Salary moves to the $120,000 to $125,000 range on successful review.

#### Longer-term path
Beyond Senior, the natural next step is Lead Game Design Systems Engineer: owning the design systems roadmap, setting standards, and coordinating with engineering and analytics where the work touches them. Appropriate when the work exceeds what one senior IC can carry.

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# Contact

Canonical page: https://gamemathemagics.com/#/contact

> Email is best, or use the form and it comes straight to me.

- fenner.d.christian@gmail.com: https://gamemathemagics.com/mailto:fenner.d.christian@gmail.com

The problems I like best sit at the seam of hard mathematics and tools people actually use: anywhere something runs on simulation, iteration or search and could run on a closed form instead. If that is sitting on your desk, I would love to hear about it.

I am looking for technical mathematics roles. Game math and probability are what I care about most, and where most of the work on this site lives.

Your nameYour email

What this is aboutA role or opportunityThe reel strip paperMaestro, Loom, Foresight or RunwaySomething elseMessageSend →

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