Chapters: Ch 1 · Ch 2 · Ch 3 · Ch 4 · Ch 5
Abacus / Soroban
Multiplication on Soroban — Jimmy Wu (YouTube) An instructional video on performing multiplication using the Japanese soroban (abacus), demonstrating the systematic digit-by-digit multiplication method that mental arithmetic practitioners use. The soroban encodes numbers across rods and performs addition through bead displacement, making the multiplication algorithm an elegant physical realization of positional arithmetic. https://youtu.be/ZiZ0VXf9Ar4
Division on Soroban — Jimmy Wu (YouTube) Companion video covering division on the soroban — a more complex operation requiring iterative quotient estimation and correction steps. Division on the abacus builds on the multiplication algorithm and provides deep intuition for long division, making it valuable both for mental arithmetic training and for understanding computational arithmetic at its foundations. https://youtu.be/1XwL-FGQxy0
Soroban Multiplication — Worked Example
45 x 123 = 5535
Step 1: 123 is a 3 digit number, 45 is a 2 digit number, Product is treated as a 3+2=5 digit number. Each digit from the left are multiplied one at a time to give a 2 digit number i.e. 1x4 = 04 which is written down in the 5th position from the right.
Step 2: 2 x 4 = 08 written from the 4th digit.
Step 3: 3x4 = 12 written from the 3rd digit from the right.
Step 4: 1x5 = 05 written into the 4th digit from right.
When any number causes overflow we split relative to 5 — add 10 to higher power, subtract 5 from lower power.
Step 5: 2x5 = 10. Adding 10 starting from the 3rd digit from the right.
Step 6: 3x5 = 15 added starting from 2nd digit.
123
x45
——
04 <- 1x4
08 <- 2x4
12 <- 3x4
05 <- 1x5
10 <- 2x5
15 <- 3x5
Read as 05535
Tomaibaby Kisangel Wooden Bead Abacus 23 Digit Rods A 23-rod wooden soroban purchased for abacus practice — the physical tool corresponding to the YouTube tutorials above. The 23-rod configuration allows working with large products without running out of rods. https://a.co/d/dHO78s7
Knight’s Tour & Graph Theory
The Knight’s Tour (boraberan.wordpress.com) A blog post exploring the knight’s tour problem — finding a sequence of knight moves that visits every square of a chessboard exactly once — and its solution via Warnsdorff’s heuristic. The post provides an accessible introduction to Hamiltonian path problems and the algorithmic strategies (backtracking, heuristics) used to solve them efficiently on standard and non-standard board sizes. https://boraberan.wordpress.com/2012/09/28/the-knights-tour/
Warnsdorff’s Algorithm for Knight’s Tour Problem — GeeksforGeeks Warnsdorff’s rule is a greedy heuristic that solves the knight’s tour by always moving to the square with the fewest onward moves — a beautifully simple rule that works remarkably well in practice. This GeeksforGeeks article covers both the theory and implementation, with analysis of why the heuristic succeeds and where it can fail on certain board configurations. https://www.geeksforgeeks.org/warnsdorffs-algorithm-knights-tour-problem/
Knight’s Tour — Wikipedia The comprehensive Wikipedia article on the knight’s tour, covering its history (dating to 9th-century Sanskrit texts), existence proofs, closed vs. open tours, Schwenk’s theorem (characterizing which board dimensions admit closed tours), and connections to Hamiltonian cycles in graph theory. An excellent reference for the full mathematical landscape of the problem. https://en.wikipedia.org/wiki/Knight%27s_tour
Warnsdorff’s Rule — Original Paper (PDF) The original 1996 paper by Douglas Squirrel and Paul Cull formally analyzing and proving properties of Warnsdorff’s heuristic for the knight’s tour, providing the theoretical underpinning for why this greedy approach achieves near-perfect success rates on standard chessboards. https://raw.githubusercontent.com/douglassquirrel/warnsdorff/refs/heads/master/5_Squirrel96.pdf
Graph Theory Origins in Sanskrit Poetry and Arabic — YouTube A highly entertaining pair of YouTube videos tracing the origins of graph theory to Sanskrit prosody (Pingala’s enumeration of poetic meters, equivalent to Fibonacci sequences and binary trees) and Arabic combinatorics — centuries before Euler’s Königsberg bridges. An essential corrective to the Eurocentric history of mathematics. https://youtu.be/DjZB9HvddQk
Physics
Field Theory Expansions of String Theory Amplitudes (Physical Review Letters) A research paper in Physical Review Letters studying how string theory scattering amplitudes reduce to field theory results in appropriate limits — a technical result connecting string theory’s infinite tower of resonances to quantum field theory predictions. The work advances the program of using string-inspired methods to compute particle physics amplitudes more efficiently. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.132.221601
How the Higgs Field (Actually) Gives Mass to Elementary Particles | Quanta Magazine Physicist Matt Strassler explains the Higgs mechanism through an analogy to music — specifically how fields can have “resonant frequencies” that give rise to particle masses. The article corrects common misconceptions about the Higgs as a “molasses” that slows particles, replacing it with a more accurate picture of spontaneous symmetry breaking and mass generation through field excitations. https://www.quantamagazine.org/how-the-higgs-field-actually-gives-mass-to-elementary-particles-20240903/
If the Universe Is a Hologram, This Long-Forgotten Math Could Decode It | Quanta Magazine A 1930s-era mathematical framework — twistor theory, developed by Roger Penrose — is being revisited by physicists working on the holographic principle (AdS/CFT). The article explains how quantum threads might weave together into a holographic spacetime fabric, with forgotten mid-century mathematics providing the key to decoding the correspondence between bulk gravity and boundary field theories. https://www.quantamagazine.org/if-the-universe-is-a-hologram-this-long-forgotten-math-could-decode-it-20240925/
Time Might Be a Mirage Created by Quantum Physics, Study Suggests (LiveScience) A theoretical study proposing that the subjective experience of time flowing forward could be an emergent illusion arising from quantum entanglement and decoherence — not a fundamental feature of reality. The work builds on the Wheeler-DeWitt equation (which contains no explicit time variable) and the “Page-Wootters mechanism” for relational time in quantum systems. https://www.livescience.com/physics-mathematics/quantum-physics/time-might-be-a-mirage-created-by-quantum-physics-study-suggests
‘Metaphysical Experiments’ Test Hidden Assumptions About Reality | Quanta Magazine Quanta explores a research program that tests physics and philosophy “as a single whole” — experiments probing contextuality, free will, and the nature of observation in quantum mechanics that simultaneously bear on longstanding philosophical questions about determinism and realism. The piece argues these experiments represent the only path to surefire knowledge about ultimate reality. https://www.quantamagazine.org/metaphysical-experiments-test-hidden-assumptions-about-reality-20240730/
Why Do Mirrors Flip Things Horizontally But Not Vertically? (ScienceAlert) A classic puzzle in physical optics: mirrors appear to flip left-right but not up-down, yet physically they flip front-back. The ScienceAlert piece unpacks why our brain’s body-centric reference frame creates the illusion of horizontal reversal, making this an elegant example of how perception and physics interact — and a good reminder that everyday phenomena can have non-obvious explanations. https://www.sciencealert.com/why-do-mirrors-flip-things-horizontally-but-not-vertically-here-s-the-physics
Johannes Brandstetter: Susskind on Riemannian Spaces and Tensor Calculus A tweet recommending Leonard Susskind’s coverage of Riemannian geometry and tensor calculus as an introduction to the mathematics of general relativity — particularly how the metric tensor encodes curvature and how this curvature is identified with gravity. Susskind’s pedagogical style makes differential geometry accessible to physicists and engineers without sacrificing rigor. https://x.com/jo_brandstetter/status/1802576302739775607
Massimo (@Rainmaker1973): Poincaré and the Three-Body Problem A tweet noting that Henri Poincaré proved the non-existence of a uniform first integral for the three-body problem — meaning no simple closed-form solution exists — yet stable solutions do exist empirically. This tension between theoretical chaos and observed stability (as in the Lagrange points and Trojan asteroids) remains one of the richest open areas in celestial mechanics. https://x.com/Rainmaker1973/status/1830309619966554272
Γ(z) (@gammaofzeta): No Model of Reality Is Without Metaphysical Baggage A philosophical tweet arguing that every descriptive model of reality carries implicit conceptual and cultural assumptions — no “view from nowhere” exists in science. This echoes themes from philosophy of science (Kuhn, Feyerabend) and is particularly relevant when evaluating mathematical models of physical phenomena or AI systems. https://x.com/gammaofzeta/status/1830429387885248789
Scientists Find a Fast Way to Describe Quantum Systems (Quanta) Coverage of a new technique for efficiently characterizing quantum states — a problem central to quantum computing, where full state tomography requires exponential measurements. The fast description method uses randomized measurement protocols to extract useful partial information about quantum systems, enabling practical benchmarking of quantum hardware.