Algebra articles, problems, and puzzles.

What Is Arithmetic?
What Is Algebra?
Arithmetic: Articles and Problems

Note that in the list below the pages that carry Java applets are designated with (*).

  1. 17 rooks on an 8×8 chessboard
  2. 3-Colors Tower of Hanoi (*)
  3. 3 glass puzzle [JavaScript]
  4. 3-Term Arithmetic Progression
  5. 4x4 coloring problem
  6. A 49th Degree Challenge
  7. A Clique of Acquaintances
  8. A Diophantine Equation - Problem from 2000 Belarus MO
  9. A Diophantine Equation with Square Roots - Problem from the 2008 British MO
  10. A Telescoping Series
  11. About numbers
  12. Absolute Value
  13. Achieve the Equilibrium [Flash]
  14. Addition
  15. Addition and Subtraction Formulas for Sine and Cosine III
  16. Affirmative Action Problem (*)
  17. Algebraic Structure of Complex Numbers
  18. All Integers Are Equal to 1
  19. All Powers of x are Constant
  20. Almost every integer has the digit 3 in it
  21. Almost Integers - Gaussian Integers
  22. Ambassadors at a Round Table (*)
  23. An Identity between the Roots of a Cubic Polynomial - Problem from the 2007 Irish MO
  24. An Identity Involving the Integer Part - Problem from the 1981 Canadian MO
  25. An Inequality for Grade 8
  26. An inequality from 1971 Canadian Mathematical Olympiad
  27. An Inequality with an Infinite Series
  28. Approximation Of Roots
  29. Arithmetic-Geometric Means Inequality
  30. A simple integral
  31. Another simple integral
  32. Averages, Arithmetic and Harmonic Means (*)
  33. Averages in a sequence
  34. The Ballot Lemma (*)
  35. Banzhaf Power Index Calculator (*)
  36. Barcode Encoding
  37. Barcode Magic (*)
  38. Base Converter [JavaScript]
  39. Bayes' Theorem
  40. Beatty Sequences
  41. Bilateral Diagrams (*)
  42. Binomial Distribution (*)
  43. Binomial Theorem
  44. Bisector of an imaginary angle may be real
  45. Byzantine Basketball
  46. Calculation of the Digits of pi by the Spigot Algorithm (*)
  47. Cantor-Bernstein-Schroeder theorem
  48. Cardinal Numbers
  49. Cartesian Coordinate System (*)
  50. Cavalieri's Principle (*)
  51. Chinese Remainder Theorem
  52. Clifford Algebras
  53. Clubs in a Vector Space
  54. Collatz Conjecture (*)
  55. Coloring a Graph (*)
  56. Combinatorial Proofs
  57. Committee Chairs
  58. Comparing complex numbers
  59. Completing a Square and Quadratic Formula
  60. Complex Numbers: A Dynamic Tool (*)
  61. Complex number to a complex power may be real
  62. Constructible Numbers
  63. Construction of Pascal's Triangle (*)
  64. Continued fractions
  65. Countability of Rational Numbers
  66. Counting Chips On a Circle (*)
  67. Counting Squares in a Square (*)
  68. Counting Triangles (*)
  69. Counting Triangles II (*)
  70. Covering A Chessboard With Domino
  71. Cubic Formula [JavaScript]
  72. Cubics and Bézout's Theorem
  73. Derivatives without Limits
  74. Descartes' Rule of Signs
  75. Diagonal Argument
  76. Distance between the Orthocenter and Circumcenter
  77. Distance Between Strings (*)
  78. Distance From a Point to a Straight Line (*)
  79. Dividing Loot Fairly
  80. Divisibility and Modulo Arithmetic at IMO 2007
  81. Dollars in Pockets
  82. Dot Patterns and Sierpinski Gasket (*)
  83. Dots and fractions (*)
  84. Ducci Sequences (*)
  85. EAN-13 Encoding (*)
  86. Eigenvectors by Inspection
  87. Equations of the Staight Line (*)
  88. Equivalence Relations
  89. Euclid's Algorithm
  90. Euclid's Algorithm: An Interactive Illustration (*)
  91. Euler's formula for eiθ
  92. Euler's Function and Congruence Theorem
  93. Evolution of Algebraic Symbolism
  94. Extension of Euclid's Algorithm
  95. Factorial Products
  96. Factorials and division
  97. Fair Division: Method of Sealed Bids II (*)
  98. Falsity implies anything
  99. Farey Series
  100. Fermat's Little Theorem
  101. Fibonacci's Quickies
  102. First Geometric Interpretation of Complex Numbers (*)
  103. Five Methods of Apportionment (*)
  104. Five Lattice Points (*)
  105. Fleury's Algorithm and Euler's Paths and Cycles (*)
  106. Flipping Items Simultaneously (*)
  107. Flipping Items Simultaneously II (*)
  108. Flipping pancakes (*)
  109. Floor Function
  110. Formal Systems - an Example
  111. Formula for Primes
  112. Four Hinged Squares: using complex numbers
  113. Four Pegs That Form a Square
  114. Four Voting Methods (*)
  115. Function
  116. Function in the Plane That Vanishes
  117. Fundamental Theorem of Algebra
  118. Gasoline Stations on a Circular Trek
  119. GCD(M, N) × LCM(M, N) = M × N (*)
  120. Generalization That Solves the Problem
  121. Generating Functions
  122. Geometric Construction of Roots of Quadratic Equation (*)
  123. Gödel's Incompleteness Theorem
  124. Graph and Roots of a Third Degree Polynomial (*)
  125. Graph and Roots of Quadratic Polynomial (*)
  126. Graph of a Polynomial (*)
  127. Graph of a Polynomial Defined by Its Roots (*)
  128. Group multiplication of permutations (*)
  129. Finding Hamilton Paths and Circuits (*)
  130. Getting a Scout out of Desert (*)
  131. Harmonic Serires And Its Parts
  132. Horner's Method (*)
  133. How to Ask an Embarrassing Question
  134. How to define a permutation? (*)
  135. However You Solve It ... A Wonderful Equation
  136. Hyperintegers and Hyperreal Numbers
  137. Implementation of Base Conversion
  138. Importance of the Absolute Value
  139. Inclusion-Exclusion Principle
  140. Inequality 1/2·3/4·5/6· ... ·99/100 < 1/10
  141. Inequality (1 + 1-3)(1 + 2-3)(1 + 2-3)...(1 + n-3) < 3
  142. Inequality 1 + 2-2 + 3-2 + 4-2 + ... + n-2 < 2.
  143. Inequality from Math Magazine 2009
  144. Inequality with Logarithms
  145. Infinite Discent for x4 + y4 = z4
  146. Infinitesimals. Non-standard Analysis
  147. Infinitude of Primes
  148. Infinity As a Limit
  149. Inflection Points of Fourth Degree Polynomials (*)
  150. Integer Iterations on a Circle (*)
  151. Integer Iterations on a Circle III (*)
  152. Integer roots of integers are irrational
  153. Integers and Rectangles (*)
  154. Integers and Rectangles: a Proof by Induction (*)
  155. Integers and Rectangles: Two Simple Proofs (*)
  156. Interest Calculations (*)
  157. ISBN Encoding (*)
  158. Irrational number to an irrational power may be rational
  159. Is 0.999... = 1?
  160. It's All About Powers of 10
  161. Joachimsthal's Notations
  162. Jumping Isn't Everything (*)
  163. Kaprekar's Iterations and Numbers (*)
  164. Lagrange's identity, Proof
  165. Lagrange interpolation
  166. Langford and Skolem Sequences (*)
  167. Latin Squares (*)
  168. Legendre's Formula
  169. Leibniz and Pascal Triangles (*)
  170. Less than, Equal to, Greater Than Symbols
  171. Lewis Carroll's Game of Logic (*)
  172. Light Bulbs in a Circle (*)
  173. Lights Out (*)
  174. Linear functions
  175. Linear Programming
  176. Linear System with Parameter
  177. Listing All Combinations (*)
  178. Listing all permutations (*)
  179. Listing All Permutations: Johnson-Trotter Method (*)
  180. Logistic Model applet (*)
  181. Long Runs of Composite Numbers
  182. Long Up Short Down
  183. Looking your best
  184. Loop or Halt (*)
  185. Lucas' Theorem
  186. Lucky Numbers (*)
  187. Marriage Theorem
  188. Marriage Problem, simulation (*)
  189. Math Lotto
  190. Mathematical Induction
  191. Mathematical Proof
  192. Mathematicians Doze Off
  193. Matrix Groups
  194. Maximum Volume of a Cut Off Box (*)
  195. Means (*)
  196. Mediant Fractions
  197. Method of Markers II (*)
  198. Minesweeper Theorem
  199. Mini Lights Out (*)
  200. Minimax Principle (*)
  201. Minimax Theorem (*)
  202. Minumum Spanning Trees and Kruskal's Algorithm (*)
  203. Modular Arithmetic (*)
  204. Monotone Convergence Theorem
  205. Monotone Number Sequences (*)
  206. Multiplication
  207. Napoleon's Relatives (*)
  208. Nested Radicals
  209. Nested Subsets
  210. Newton's Method (*)
  211. Nine Squares (*)
  212. Number 8 Is Interesting
  213. Number Comparison - from the 1969 Canadian MO
  214. Number sequences and Pigeonhole (*)
  215. Number System With Base 36
  216. On the Games of Hex and Y
  217. One Dimensional Ants
  218. One Trigonometric Formula and Its Consequences
  219. Order. Well-ordered sets.
  220. Ordinal Numbers
  221. Outline Mathematics
  222. p-adic Expansions
  223. p-adic Numbers
  224. Parity counting (*)
  225. Parrondo Paradox Via Simpson Paradox (*)
  226. Partial But General Case Of Fermat's Last Theorem
  227. Party Acquaintances (*)
  228. Patterns in Pascal's Triangle
  229. Peano axioms
  230. Pennies in Boxes (*)
  231. Percent
  232. Permutations
  233. Pigeonhole Principle
  234. Place Value (*)
  235. Presence of the Antipodes on a Circle
  236. Probability of Divisibility
  237. Problem 1 from the IMO 2009
  238. Problem A1, 69th Putnam Competition
  239. Problem A2, 69th Putnam Competition
  240. Problem of Divisibility by 5n
  241. Product of Diagonals in Regular N-gon
  242. Product Of Four Consecutive Integers
  243. Product of Simple Fractions (*)
  244. Product of permutations (*)
  245. Proizvolov's Identity (*)
  246. Proofs Without Words (*)
  247. Property of an
  248. Property of the Powers of 2 (*)
  249. Puzzlist vs. Mathematician: A Practical Arithmetic Problem
  250. Pythagorean Triples (*)
  251. Quasigroups and block designs from triangular grid (*)
  252. Rabbits Reproduce; Integers Don't
  253. Ramsey's Theorem
  254. Random Walks (*)
  255. Recombining Permutations (*)
  256. Rectangle on a Chessboard (*)
  257. Reflection Lemma (*)
  258. Riemann Sphere and Möbius Transformation
  259. Riemann Sums - Function Integration (*)
  260. Right Replacement (*)
  261. Rolle's and Mean Value Theorems (*)
  262. Round Robin Tournament
  263. Russell's Paradox
  264. Scheduling and Critical Path Algorithm (*)
  265. Scrub Tile Puzzle (*)
  266. Self-descriptive Strings (*)
  267. Self-reference
  268. Seven Terms Periodic Sequence
  269. Shapley-Shubik Power Index (*)
  270. Shapley-Shubik Power Index Calculator (*)
  271. Shifting Digits and a Point of View
  272. Ship-Dock Theorem
  273. Sierpinski Gasket and Tower of Hanoi
  274. Sierpinski's Gasket and Dihedral Symmetry (*)
  275. Sieve of Squares (*)
  276. Σ2-n = Σn·2-n
  277. Signs and Sums in a Sequence (*)
  278. Signs in a Matrix (*)
  279. Simple Cellular Automaton (*)
  280. Simple Graph Practice (*)
  281. Simple Somos Sequence - a Russian Olympiad problem
  282. Simple Somos Sequence Calculator (*)
  283. Simpson Paradox (*)
  284. Sine of a Sum Formula, Proof Without Words
  285. Single formula to express any integer with digits 0-9 each used just once
  286. Six Balls, Two Weighings
  287. Size of a Class: Two Viewpoints
  288. Slider Puzzles (*)
  289. Solitaire in Two Colors (*)
  290. Solve xx3 = 3
  291. Soriteses
  292. Sorting Algorithms
  293. Sperner's Theorem
  294. Splitting piles
  295. Square root of 2 is irrational
  296. Steinhaus' Problem (*)
  297. Step into the Elliptic Realm
  298. Stern-Brocot Tree
  299. Straight Tromino on a Chessboard (*)
  300. Structure of Hyperreal Numbers
  301. Subtraction Game (*)
  302. Sums of Power Series
  303. Sums Of Successive Squares
  304. Sum without Adding
  305. Sums and Products (*)
  306. Sums of powers of digits (*)
  307. Superposition Principle
  308. Surreal Numbers
  309. Swapping Rows and Columns (*)
  310. Taking Games Seriously (*)
  311. The ass and the mule (*)
  312. The Basic Rules of Counting
  313. The Local-Global Principle
  314. The Size is in the Eyes of the Beholder
  315. Tool for Constructing Orthogonal Latin Squares (*)
  316. Three Dimensional Cross
  317. Tower of Hanoi, the Hard Way (*)
  318. Transpositions
  319. Traveling Salesman Problem (*)
  320. Treatise on Arithmetical Triangle
  321. Triangular numbers that are also square
  322. Trilateral Diagrams (*)
  323. Trinary Tree(s) underlying Primitive Pythagorean Triples
  324. Twenty five boys and twenty five girls around a table.
  325. Two Circles and a Limit (*)
  326. Two Colors in Two Rows (*)
  327. Two Men of Tibet (*)
  328. Uncountability of the Reals - via a Game
  329. Universal Coloring (*)
  330. Universal Product Code (*)
  331. Unproperty of the Powers of 2
  332. USA 1975 Problem 3
  333. Venn Diagrams
  334. Ways To Count
  335. Weierstrass Product Inequality
  336. What, how, and the Web (*)
  337. What Is Algorithm?
  338. What Is Infinity?
  339. What Is Limit?
  340. What is so natural about "natural logarithms"?
  341. What Is Space Completion?
  342. WHEN THE COUNTING GETS TOUGH, THE TOUGH COUNT ON MATHEMATICS
  343. Where To Start? (*)
  344. Why 17?
  345. Wilson's Theorem
  346. Word Problems (*)
  347. Yet Another Puzzle with Hats
  348. Zeros and Nines (*)

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Copyright © 1996-2010 Alexander Bogomolny

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