Approximating Mathematical Constants using Minecraft

Michael Weselcouch; Molly Lynch

REVIEW 2 major objections 5 minor 19 references

Approximating Mathematical Constants using Minecraft

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper argues that Minecraft's hoppers, droppers, observers, and slimes can act as experimental instruments for approximating the mathematical constants square root of 2, pi, e, and zeta(3) in a college classroom.

desk verdict A genuinely fun and mostly sound classroom paper: the zeta(3) experiment rests on an identity its own random numbers violate, but the rest holds up and the paper deserves a serious referee. read the letter →

arxiv 2411.18464 v1 pith:DU2YF2BT submitted 2024-11-27 math.HO

Molly Lynch , Michael Weselcouch This is my paper · ORCID

classification math.HO

keywords MinecraftmathematicalconstantsMonteCarlomethodsderangementsApéry'sconstantrandomnumbergenerationexperimentalmathematicsuniversityeducation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that Minecraft, a sandbox video game, can serve as a working laboratory for undergraduate mathematics. The authors build four experiments that turn in-game mechanics into measuring instruments: a hopper timer measures travel time along the sides of a right triangle to approximate $\sqrt{2}$; slime deaths inside a circular arena run Monte Carlo integration for $\pi$; a dropper-based machine generates random permutations and counts derangements to approximate $e$; and bamboo growth observed by block-update detectors produces random integers used to estimate $\zeta(3)$ through coprime triplets. The resulting approximations have errors between about 0.008% and 4.5%, and the paper's stated purpose is to show that this kind of play can inspire learning rather than to set accuracy records.

What carries the argument

The load-bearing objects are four Minecraft blocks and mobs paired with four mathematical identities. A hopper transfers items at a fixed rate and so acts as a timer; a dropper ejects one of up to nine stored items at random and so acts as a randomizer; an observer fires when the block it faces changes state, turning random bamboo growth into a random-number source; and slimes, unlike most mobs, keep moving with no directional bias and serve as random scatter points for Monte Carlo integration. The associated identities are: the $45^\circ$–$45^\circ$–$90^\circ$ triangle ratio $\sqrt{2}:1$, the Monte Carlo area ratio $\pi/4$ for dots in a circle, the derangement probability $D(n)/n!$ approaching $1/e$, and the Euler product identity $P_3 = \prod_p (1-1/p^3) = 1/\zeta(3)$ for the probability that three uniformly random positive integers are coprime. Each experiment pairs one game mechanic with one identity to convert observed counts into a constant.

What would settle it

Repeat the bamboo and observer experiment with thousands of triplets and compare the proportion of coprime triples to $1/\zeta(3) = 0.8319\ldots$; if the empirical proportion converges to a value outside the sampling error of $0.8319$, the Minecraft number source is not equivalent to uniform integer sampling and that approximation is not measuring $\zeta(3)$.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes a proof of concept that Minecraft's mechanics can be used to experimentally approximate mathematical constants in a way suitable for higher education. The central claim is that four familiar randomized or geometric processes can be instantiated by ordinary game mechanics: hoppers release items at a constant rate and serve as timers, droppers eject one of up to nine items uniformly and serve as randomizers, observers detect random block-state changes such as bamboo growth, and slimes move randomly enough to act as scatter points. With these instruments the authors report $\sqrt{2} \approx 57/41 = 1.3902$, $\pi \approx 4\cdot 508/619 = 3.283$, $e \approx 647/238 = 2.71849$, and $\zeta(3) \approx 70/58 = 1.2069$, with errors of 1.70%, 4.49%, 0.00766%, and 0.4%, respectively. The paper frames these numbers as invitations to explore geometry, calculus, combinatorics, and number theory in a game world, not as competitive computations.

Load-bearing premise

The whole approach assumes that the in-game randomness used in each experiment faithfully models the mathematical randomness the identity requires; for the $\zeta(3)$ estimate this is the explicit assumption that bamboo-tick times, which follow a negative binomial distribution, behave like uniformly random positive integers, a gap the authors note but do not close.

Editorial extensions

If this is right

If the proof of concept holds, Minecraft becomes a low-stakes experimental surface where students can generate their own data for $\sqrt{2}$, $\pi$, $e$, and $\zeta(3)$ rather than only computing formulas.
The same constructions generalize: timing a diagonal across a 1-by-2 rectangle approximates $\sqrt{5}$, Monte Carlo integration over a plotted curve approximates definite integrals, and the coprime-triplet method approximates $\zeta(m)$ for any $m$, including $\zeta(2)=\pi^2/6$.
The randomized Minecraft mechanics can substitute for standard random sources in classroom probability experiments, as long as the distribution of the mechanic is matched to the model.
Because the paper deliberately trades accuracy for accessibility, its methods are suitable for first exposure to Monte Carlo integration, derangements, and the Riemann zeta function in an undergraduate setting.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A consequence the paper leaves implicit is that the same hopper and observer setup can quantify bias in Minecraft's randomness by comparing observed coprime fractions from bamboo-generated numbers against the uniform-integer prediction, turning the $\zeta(3)$ experiment into a lesson on distributional assumptions.
The derangement machine could be scaled to larger $n$ with more droppers, and the error bound of $1/10!$ mentioned in the paper suggests that even modest sample sizes make $e$ the most reliable approximation; the same scaling argument could be made for other constants.
The observer-based random number generator is not unique to bamboo: other random-tick block updates, such as crops growing or ice melting, could cross-check whether the negative binomial distribution changes the $\zeta(3)$ estimate, making the paper's method a template for studying pseudo-randomness in games.
A portable extension would be to reproduce the paper's experiments in any grid-based sandbox game with similar block mechanics, which would test whether the educational proof of concept depends on Minecraft specifically or on the general principles of timers and randomizers.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a charming proof-of-concept that Minecraft can be used for college math. The sqrt(2), pi, and e experiments are methodologically fine and reported honestly, with error percentages and concrete suggestions for improvement. The zeta(3) experiment has a real flaw: the coprimality identity used, P(gcd=1)=1/zeta(3), holds for integers chosen uniformly in the natural-density sense, but the bamboo growth times are negative binomial, as the authors themselves note in Section 5.1. They don't reconcile that. For a negative binomial law, the probability that three iid draws are coprime is not 1/zeta(3); it depends on the distribution's prime-divisibility probabilities. So the 70/58 estimate doesn't actually approximate zeta(3) by the stated argument. The authors flag it in one sentence and move on, but they don't fix it or reinterpret it.

That said, I don't think this sinks the paper. The educational proof-of-concept stands on three solid experiments plus the many ideas for extensions. The zeta(3) section could be fixed by using a uniform random source (e.g., item-dropping randomizers or a redstone-based PRNG) or by honestly framing the bamboo experiment as a demonstration of the concept of random numbers rather than an actual approximation of zeta(3). Either way, it's a minor-to-moderate revision, not a desk reject. My one other quibble: the slime-based Monte Carlo assumes the slime deaths are uniformly distributed over the square, and the paper doesn't verify that. For a proof-of-concept it's acceptable, but the authors should acknowledge the assumption.

Who's it for: math educators, recreational math folks, anyone teaching probability or number theory. The citations are appropriate — standard references for the constants, Minecraft wiki for mechanics. No citation problems I can see.

Recommendation: send to peer review. It's not a research breakthrough, but it's exactly the kind of thing a teaching journal or a general math magazine should consider. My referee report would be: revise with the zeta(3) fix, then accept.

Referee Report

2 major / 5 minor

Summary. The paper describes four Minecraft-based experiments for approximating mathematical constants: sqrt(2) via travel-time ratios on the leg and hypotenuse of a 45-45-90 triangle, pi via a slime-death Monte Carlo inside a circle, e via the proportion of derangements among randomly generated permutations using droppers, and zeta(3) via the proportion of triples of bamboo-growth waiting times that are relatively prime. Each section gives historical background, construction details, the measured value and percentage error, and suggestions for extensions. The authors frame the article as a proof of concept for using Minecraft in undergraduate mathematics education.

Significance. If the zeta(3) experiment were valid, the paper would be a charming and genuinely useful educational resource: the sqrt(2), pi, and e constructions are concrete, inexpensive, and tied to real mathematics (isosceles triangles, Monte Carlo integration, derangements). The e experiment is especially well matched to the mathematics, since D(n)/n! is exactly the nth partial sum of the series for 1/e, and the observed 647/238 is remarkably close. The paper is honest about its goal of inspiring experimentation rather than obtaining high precision, and it provides reproducible construction details and extension ideas. However, the zeta(3) section has a load-bearing statistical gap that undermines one of the four demonstrations; the proof-of-concept claim survives only if that section is repaired or removed.

major comments (2)

[Section 5.1, Eq. (5)] The identity P3 = ζ(3)^(-1) is derived in the text for 'three positive integers chosen uniformly at random.' Immediately after explaining their setup, the authors write, 'It should be noted that the random numbers that were generated did not follow a uniform distribution, instead they followed the negative binomial distribution.' They then use 58/70 as an estimate of ζ(3)^(-1). This is not justified: the coprimality probability of iid negative binomial draws is not 1/ζ(3) in general and depends on the distribution's parameter. For example, for iid geometric draws with P(X=k)=q(1-q)^(k-1), P(gcd(X,Y,Z)=1) = sum_{d>=1} mu(d) (q(1-q)^(d-1)/(1-(1-q)^d))^3, which differs from 1/ζ(3) for generic q. The agreement of 70/58 with ζ(3) is therefore coincidental unless a new calculation shows otherwise. This is a load-bearing defect because the zeta(3) section is one of only four demonstrations of the proof-of-concept thesis; the section needs either a correct probabilistic analysis or a different Minecraft mechanism that produces uniform natural-density samples.
[Section 5.1] The negative-binomial claim is not operational as stated. To repair the experiment or compute the correct coprimality probability, the authors must define the random variable precisely: what is counted (game ticks, hopper items, seconds), what constitutes a success/failure, and what the estimated success probability is. Without this information, the reader cannot reproduce the experiment or check whether a corrected identity applies.

minor comments (5)

[Section 1.1] The statement that 25 items released by a hopper implies a time between 10 and 10.4 seconds is slightly inconsistent with the stated constant rate of 2.5 items per second; please clarify how the endpoint of the interval is defined (e.g., time from first to last transfer).
[Section 4] The historical date 'Jacob Bernoulli in 1638' is a typo; Jacob Bernoulli was born in 1655 and worked on compound interest around 1683. Please correct.
[Section 3.1] Please report the dimensions of the circle used and the Monte Carlo standard error of the proportion; the observed 4.49% error is about two standard deviations away from pi/4, so a reader cannot tell how much of the discrepancy is sampling noise versus the blocky circle boundary.
[Section 5.1] The sentence 'the numbers generated varied between a minimum of 3 and a maximum of 838' is informative but should be tied to the definition of the random variable (ticks, hopper items, seconds), and the support of the negative binomial distribution should be stated.
[References] The Minecraft wiki citations are to fandom pages; consider adding access dates or archived URLs for stability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning found: all four constants are approximated by direct measurements or counts, with mathematical identities cited from standard external results.

full rationale

The paper contains no load-bearing circular step. Each approximation is an independent experiment against externally known constants: sqrt(2) is measured as a ratio of hopper-timer counts along the hypotenuse and leg of a right triangle, pi is measured as 4 times the fraction of slime deaths inside a circle, e is measured as the inverse of the empirical derangement proportion, and zeta(3) is measured as the inverse of the empirical coprime proportion. None of these estimates is fitted to the target constant, and no parameter is chosen using the known value of the constant being approximated. The mathematical identities used, such as the derangement probability and the coprimality probability, are standard results cited from the literature or derived in the paper from the Euler product. The zeta(3) section explicitly acknowledges that the generated numbers follow a negative binomial distribution rather than a uniform distribution; this is a correctness or assumption-violation concern, not circularity, because the paper does not redefine zeta(3) in terms of its Minecraft measurement. The only author-created resource cited is a Desmos page used as an optional tool for plotting functions, which is not load-bearing for any derivation. Accordingly, the paper is self-contained against external benchmarks and receives a circularity score of 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; the methods use in-game mechanics as measurement devices. The key additional assumptions are game-mechanic behaviors (hoppers, droppers, slimes, random ticks) plus the standard number-theoretic identity for coprime probabilities. The main weakness is that the latter identity is mismatched with the bamboo-generated distribution in the zeta(3) experiment.

assumptions (5)

standard math For three positive integers chosen uniformly at random, the probability that they are relatively prime is 1/zeta(3).
Invoked in Section 5 to justify approximating zeta(3) by the proportion of coprime triplets. This identity is standard but requires a uniform distribution over positive integers.
domain assumption Hoppers release items at a constant rate of 2.5 items per second, so item counts are proportional to elapsed time.
Used in Sections 1.1 and 2.1 to equate the ratio of hopper item counts to the ratio of travel times along the leg and hypotenuse of a right triangle.
domain assumption A dropper ejects one of its stored items uniformly at random.
Used in Sections 1.2 and 4.1 to generate uniformly random permutations of [9]: each ejected block sequence is assumed equally likely.
domain assumption Slime death locations are uniformly random within the test square.
Used in Section 3.1 for Monte Carlo estimation of pi. The paper selects slimes to avoid the southeast movement bias of other mobs, but does not empirically verify uniformity of death locations.
domain assumption Random block ticks cause bamboo growth at random intervals, and observer-based waiting times generate useful random numbers.
Used in Section 5.1 to generate triplets of random numbers. The paper notes the resulting distribution is negative binomial, not uniform, which is inconsistent with the required uniform-coprime probability.

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Cite this review

Pith. "Pith review of Approximating Mathematical Constants using Minecraft." pith.science (2026). https://pith.science/paper/DU2YF2BT

@misc{pith2026241118464,
  author       = {Pith},
  title        = {Pith review of: Approximating Mathematical Constants using Minecraft},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DU2YF2BT}},
  note         = {Machine review of arXiv:2411.18464}
}

abstract

In this article we will use Minecraft to experimentally approximate the values of four different mathematical constants. The mathematical constants that we will approximate are $\sqrt{2}, \pi$, Euler's number $e$, and Ap\'{e}ry's constant $\zeta(3)$. We will begin each section with a brief history of the number being approximated and describe where it appears in mathematics. We then explain how we used Minecraft mechanics to approximate the constant. At the end of each section, we provide some ideas for how to apply our techniques to the approximation of other mathematical constants in Minecraft or elsewhere. This article is a proof of concept that Minecraft can be used in higher education. We should note that the goal of this article is not to have the most accurate approximations possible, the goal is to inspire people to have fun while learning about various mathematical topics. We hope you learn something new in this article and feel inspired to try some of these techniques on your own.

Figures

Figures reproduced from arXiv: 2411.18464 by the authors.

**Figure 2.** The leg and hypotenuse of a large 45◦ − 45◦ − 90◦ right triangle. of the hypotenuse to the length of the leg of the triangle is √ 2 : 1. Therefore, to approximate the value of √ 2, we can simply measure the time it takes for a player to travel at a constant rate along one of the legs and the time it takes for that player to travel along the hypotenuse. Since the length of the hypotenuse is √ 2 times as long as the l… view at source ↗

**Figure 3.** The circumference of the circle is bounded below by the perimeter of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: After 43270 dots, 33943 in the circle, the approximation of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: An approximation of a circle with radius 11 in Minecraft [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Slimes being killed by zoglins. The location of each death was recorded [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: The graph of f(x) = x 2 sin(x) + √3 x and how to make it in Minecraft. continuous compound interest. In doing so, he tried to evaluate the limit seen in Equation 1. Using the binomial theorem, he was able to show that 2 < e < 3, but at this time e still did not have a …

**Figure 8.** Figure 8: A dropper can contain up to nine different blocks. [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: The observer detects the exact moment that the bamboo grows. [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Agarwal, R.P, Agarwal, H. (2021). Origin of Irrational Numbers and Their Approximations, Computation 9(3)

work page 2021

[2] [2]

Ap´ ery, R. (1979). Irrationalit´ e deζ2 et ζ3, Journ´ ees Arithm´ etiques de Lu- miny, Ast´ erisque61. 14

work page 1979

[3] [3]

Beckmann, P. (1977). A history of π. St. Martin’s Press

work page 1977

[4] [4]

Coolidge, J. L. (1950) The Number e, The American Mathematical Monthly 57(9)

work page 1950

[5] [5]

Damini D.B., Dhar, A. (2022). How Archimedes showed that pi is approxi- mately 22 by 7, Azim Premji University At Right Angles

work page 2022

[6] [6]

https://academo.org/ demos/estimating-pi-monte-carlo/

Estimating Pi Using the Monte Carlo Method. https://academo.org/ demos/estimating-pi-monte-carlo/

work page

[7] [7]

Euler, L. (1748). Introductoio in Analysin Infinitorum. Lausannae, M. M. Bousquet

work page

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impulseSV. YouTube. (2014). Minecraft - Tutorial: Quick and easy item sorter (with overflow protection). https://www.youtube.com/watch?v= NHjWYCDYJ4s

work page 2014