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Last post: Sep 15, 2026
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Linkage
Sep 15, 2026 · original
Brick territories experiment ( \(\mathbb{M}\) ). What shapes do you get when (n) simultaneous breakout games compete against each other for pixels? Lillian Oppenheimer , 1898–1992 ( \(\mathbb{M}\) ), early and significant promoter of origami in the US, founder of the organization that became OrigamiUSA, mother of notable mathematicians William Kruskal, Martin David Kruskal, and Joseph Kruskal, and grandmother of computer scientist Clyde Kruskal. Now a Good Article on Wikipedia (not one that I wrote or edited; I merely reviewed it). Python sets and dictionaries can have quadratic-time performance ( \(\mathbb{M}\) ), for adversarially-chosen data. Terry Tao on what can be learned from studying problems rather than on what their solutions happen to be . Ig Nobel Prize goes to a study showing that rich people are more likely than others to steal candy from children ( \(\mathbb{M}\) ). The re
Periodic coloring of infinite planar graphs
Sep 14, 2026 · original
The de Bruijn–Erdős theorem states that the number of colors needed to color an infinite graph is the same as the maximum number needed for its finite subgraphs. So for any reasonable definition of an infinite planar graph, the 4-color theorem for finite planar graphs implies that every infinite planar graph is also 4-colorable. One way to construct infinite planar graphs is to make them periodic: start with any periodic tiling of the plane, decorate a single prototile by vertices and edges that may wrap from one tile to the next, and form an infinite graph from the copies of these decorations on all the tiles of the tiling. One possibility is to simply use one vertex in each tile, with edges that connect that vertex to its copies in each adjacent tile, in which case coloring the graph is the same as coloring the tiles of the tiling. For instance, here are two periodic colorings of the f
Linkage with two research problems
Aug 31, 2026 · original
In case you’ve been worrying that the recent publicity blitz of LLM solutions to open mathematics problems is causing us to run short, there are two more mixed in among my usual links here. Because I haven’t solved them, they are not very precisely formulated, and I don’t know how difficult or interesting they are nor even whether someone else might have already considered them; that’s often the way at the start of research. I think maybe I need to start bringing my DSLR along again when I go to the beach ( \(\mathbb{M}\) ) rather than relying on my cell phone camera (Pixel 6 Pro, yes I know it’s getting old now). Here’s what it thinks a crashing wave looks like, without additional processing except for a bit of a crop. To me it more resembles the kind of molded privacy glass that one uses for a bathroom window. Amazon buys rare books to destructively scan them for AI training ( \(\mathb
Fractional rings of tangent spheres
Aug 17, 2026 · original
Soddy’s hexlet consists of a ring of six spheres, tangent to each other consecutively around the ring, and another ring of three consecutively-tangent spheres, so that all the spheres in the first ring are tangent to all the spheres in the second ring. If you keep one ring fixed, you can rotate the other ring continuously, possibly changing the sizes of some of the spheres as they rotate but keeping the pattern of tangencies unchanged. Here’s a nice animation I found on Wikipedia, where the ring of six spheres rotates continuously while the other ring of three spheres (the central blue one and the two green planes, considered as degenerate spheres tangent at infinity) stays fixed. The larger red sphere is not part of this configuration and I don’t know why the author of this animation included it. We can describe the graph of tangencies of these nine spheres by using the join operation o
Linkage
Aug 15, 2026 · original
A permutation generation algorithm in the work of 13th-century Kabbalist Abraham Abulafia ( \(\mathbb{M}\) , via ). The resulting permutation sequence is the one you get by reversing suffixes whose lengths form the sequence \((((2, 3)^2, 2, 4)^3, 2, 5)^4, \dots\) but that’s not the generation rule. Instead the rule is: to generate the permutations of \(1, 2, 3,\dots, n,\) form its \(n\) cyclically rotated permutations (starting with \(1, 2, 3,\dots, n,\)) and for each one in order, recursively generate the permutations of its length-\((n-1)\) suffix. The link claims that this is the first method for systematically listing permutations (long before the discovery of the Steinhaus–Johnson–Trotter algorithm for generating a different ordering, by 17th-century bell ringers). However, I’m not entirely convinced: it’s not the same as an enumeration algorithm, but factorials and lists of permuta
Non-coplanar unit distances
Aug 2, 2026 · original
Reports that LLMs have killed the Erdős unit distance problem turn out to be greatly exaggerated. There is still plenty not yet understood about the problem. The problem asks, for \(n\) points in the Euclidean plane, how many pairs can be at unit distance from each other? When Paul Erdős posed the problem in 1946 , he observed that the graph of unit distances cannot contain a subgraph of the form \(K_{2,3}\), a complete bipartite subgraph with two vertices on one side and three on the other. One way to see this is to draw unit circles through the two vertices on one side of a supposed \(K_{2,3}\) subgraph. These cross each other at most twice, and their two crossing points are the only points that can be vertices on the other side of the subgraph. Through reasoning later generalized as the Kővári–Sós–Turán theorem , Erdős observed that this forbidden subgraph implies an \(O(n^{3/2})\) up
Linkage
Jul 31, 2026 · original
ACM considers allowing LLMs to read its digital library ( \(\mathbb{M}\) ) and requests feedback on this choice . Daedalus’ labyrinth ( \(\mathbb{M}\) ), a puzzle game dual to loopy/slitherlink. Voronoi go ( \(\mathbb{M}\) ), a variant of go where you can play anywhere on the board (not just intersections). Human mathematicians are being outcounterexampled ( \(\mathbb{M}\) ). Kevin Buzzard of the Xena Lean formalization project on recent counterexamples to the unit distance problem and Jacobian conjecture. 3-boundary surfaces for 3-component links ( \(\mathbb{M}\) ) with crochet, 3d-printed, and vector graphics illustrations. Innovations in Graph Theory added to DOAJ ( \(\mathbb{M}\) ). zbMATH digitizes 50 years of its old reviews ( \(\mathbb{M}\) ). Three new Wikipedia articles about mathematics books ( \(\mathbb{M}\) ), posted by Samuel Adrian Antz: Characteristic Classes (1974) by Joh
Integer complexity and cographs
Jul 19, 2026 · original
The integer complexity of a number \(n\) is the minimum number of ones needed to express \(n\) as a parenthesized combination of sums and products of ones. For instance, 10 has complexity 7 as it can be expressed using seven ones, but not fewer: \[10 = (1+1+1)(1+1+1)+1.\] The largest number with complexity \(k\) can be obtained by breaking up the sequence of \(k\) ones into subsequences of two and three ones (with as many threes as possible) and multiplying. For instance, for ten ones, you can’t do this with three groups of three (because you get an ungrouped one left over) but you can with two, giving \[(1+1+1)(1+1+1)(1+1)(1+1)=36.\] While looking at the integer complexity article on Wikipedia today, it occurred to me that I had seen the same formula for the maximum complexity before. It is the upper bound on the number of maximal cliques in an \(n\)-vertex graph. This upper bound was p
Linkage
Jul 15, 2026 · original
Another mathematics journal leaving its commercial publisher ( \(\mathbb{M}\) ), but with a twist: usually this is accomplished by a mass resignation of the editorial board. But in this case, Communications on Pure and Applied Mathematics is owned by the Courant Institute and was published by Wiley, so taking it in-house is just a matter of not renewing the contract. The causes of friction were increased publisher interference with editorial decisions (the usual), but also editor dissatisfaction with the publisher’s editorial management software. An American privacy emergency ( \(\mathbb{M}\) ): Cynthia Dwork on how new US government regulations forbidding the Census Bureau from masking its released data under differential privacy will give us less usable data, reduced protection against privacy-violating disclosures, or both. Cynthia also provides information about what you can do to he
Packing Latin squares into sudoku puzzles
Jul 7, 2026 · original
I have another new preprint, the result of a research project with UC Irvine undergraduate Cindy Zhang: “ Sudoku grids that require many clues ” ( arXiv:2607.05728 , to appear at JCDCG 3 2026 ). The main result is, I think, surprising: When generalized to \(n^2\times n^2\) grids, almost all sudoku puzzles must be almost entirely covered by clues, leaving only a logarithmic fraction of cells blank. This implies an average case time for solving randomly chosen puzzles that is exponential in \(n^4/\log n\), significantly better than the exponential in \(n^4\) that one gets for formulating the problem as an exact cover problem without using this bound on blank cells or the exponential in \(n^4\log n\) that one gets for a brute force search. The formatting requirements for JCDCG 3 are in one way quite free-form and in another way very strict: each submission can have only two a4 pages, and th

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