Category: Mathematical Playground
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The Unit Distance Problem: Erdős’s Lattices, AI-Aided Breakthrough, and Community-Optimized Solutions
Date: June 2026 Author: Michael T. M. Emmerich This report is about the unit disc problem, a problem that was receiving recently renewed attention because AI tools [OpenAI2026] helped to disprove a long open conjecture on this problem. We will discuss the problem and the geometrical conjecture, and how the main idea of how it…
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When Diversity Becomes Logic
How selecting diverse subsets can encode logical circuits Michael Emmerich, May 12th 2026. Figure 1 shows how the satisfiability of the 3-SAT clause (‘1’ OR ‘2’ OR ‘3’) can be encoded as a maximum-diversity subset selection problem. Each disc centre represents a candidate choice, and the task is to select as many non-overlapping discs as…
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Diversity Subset Selection on Lines and Staircases
My report, titled Exact Dynamic Programming for Solow–Polasky Diversity Subset Selection on Lines and Staircases, is now available on arXiv: https://arxiv.org/abs/2604.26929. In multiobjective optimization and decision analysis, we often face a deceptively simple task: from a larger set of candidate solutions, choose a smaller subset that is still representative, well-spread, and diverse. This sounds easy…
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Between Uniformity and Collision: A New Phase-Transition
A New Phase Transition for Energy-Minimal Point Distributions on the Interval Date: 2026-03-28 Author: Michael T. M. Emmerich There are at least two very natural ways to spread points on the unit interval . One can try to maximize the sum of all pairwise distances, or one can try to maximize the smallest pairwise distance.…
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A very, very simple method to estimate Pi
Date: 2026-03-15 Author: Michael T.M. Emmerich Every year, on March 14, mathematicians celebrate Pi Day. For my Mathematical Playground blog, I wished to post a short essay on a very, very simple Monte Carlo method for estimating . The nice aspect of this method is that it requires only a few lines of code, does…
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Isolating Noise from Signal in the Prime Number Distribution
A Deep Dive into Riemann’s Hypothesis: Isolating Noise from Signal in the Prime Distribution Date: 2026-02-14 Author: Michael Emmerich The primes look orderly from far away and chaotic up close. At a large scale they follow clean laws: the density of primes near is about , and the prime-counting function is well approximated by .…
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P-adic wheels
In the usual (real) number line, two numbers are “close” if their leading digits agree: is closer to than to . The p-adic world flips that intuition: closeness is governed by agreement of the low-order digits (divisibility by powers of a prime). This post introduces a small Pygame visualization (shown as circles inside circles inside…
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How to Optimally Stack Your Books and Avoid a ‘Bad Hangover’
Michael T.M. Emmerich Date: 2025-12-29 Abstract It is Christmas time: books appear in cheerful piles as gifts, and in some households the morning after brings not only a “hangover” but also an ambitious attempt to stack them and exploit “overhang” beyond the edge of a table. How far can a stack of identical “books” (idealized…
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The Hardness of Proving Hardness: Selecting Minimum Riesz Energy Subsets on a Line
Figure: Logarithmically scaled Riesz interaction matrix for a random set of 30 points on the unit interval, with cell sizes proportional to the distances between points. Each rectangle corresponds to a pair of points: darker cells indicate stronger interactions (points that are closer on the line), while lighter cells indicate weaker interactions. Off-diagonal entries show…
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Elegant Addition-only Prime Generation in LISP
Brief history of LISP. LISP is a functional programming language that was created in the late 1950s by John McCarthy at MIT. It quickly became the lingua franca for symbolic AI because of its tiny, expressive core: homoiconic syntax (code-as-data), first-class functions/closures, macros that extend the language, and automatic memory management. McCarthy emphasized writing programs…
Michael Emmerich 