Algebra articles, problems, and puzzles.

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

1. 17 rooks on an 8×8 chessboard
2. 1=2 via Continued Fractions
3. 3-Colors Tower of Hanoi [Java]
4. 3 glass puzzle [JavaScript]
   • 3 glasses puzzle, second, more general tool [JavaScript]
   • 3 Glasses Problem in Barycentric Coordinates
   • 3 Jugs Problem - A Water Doubling Variant
   • Three Jugs Problem [Java]
   • Two Pails Puzzle [Java]
5. 3-Term Arithmetic Progression
6. 4x4 coloring problem
7. A 49^th Degree Challenge
8. A Clique of Acquaintances
9. A Diophantine Equation - Problem from 2000 Belarus MO
10. A Diophantine Equation with Square Roots - Problem from the 2008 British MO
11. A Low Bound for 1/2 * 3/4 * 5/6 * ... * (2n-1)/2n
12. A Mathematical Rabbit out of an Algebraic Hat
13. A Middle School Problem from a Moscow Olympiad
14. A Proof by Game for a Sum of a Convergent Series
15. A Property of the fusc() Function
16. A Representation of Rational Numbers
17. A simple integral
18. A Telescoping Series
19. About numbers
20. Absolute Value
21. Absolute Value and 1998 integers
22. Achieve the Equilibrium [Flash]
23. Addition
    • Addition and Subtraction of Functions [Java]
24. Addition and Subtraction Formulas for Sine and Cosine III
25. Affirmative Action Problem [Java]
26. Algebraic Structure of Complex Numbers
27. All Integers Are Equal to 1
28. All Integers Are Even
29. All Powers of 2 Are Equal to 1
30. All Powers of x are Constant
31. Almost every integer has the digit 3 in it
32. Almost Integers - Gaussian Integers
33. Ambassadors at a Round Table [Java]
34. An Equation in Radicals
35. An Equation in Reciprocals
36. An Identity between the Roots of a Cubic Polynomial - Problem from the 2007 Irish MO
37. An Identity Involving the Integer Part - Problem from the 1981 Canadian MO
38. An Inequality for Grade 8
39. An inequality from 1971 Canadian Mathematical Olympiad
40. 1962 IMO, Problem 2: an inequality with a twist
41. An Inequality with an Infinite Series
42. Another simple integral
43. Any Integer with Three 2s
44. Approximation Of Roots
45. Area of a Circle by Rabbi Abraham bar Hiyya Hanasi [Java]
46. Areas on the Graphs of Power Functions
47. Arithmetic-Geometric Means Inequality
    • AM-GM Inequality
    • An Extension of the AM-GM Inequality
    • An Extension of the AM-GM Inequality: A second look
48. Assigning Numbers to Points in the Plane
49. Averages, Arithmetic and Harmonic Means [Java]
50. Averages in a sequence
51. Ballot Lemma [Java]
52. Banzhaf Power Index Calculator [Java]
53. Barcode Encoding
54. Barcode Magic [Java]
55. Base Converter [JavaScript]
    • Binary System
    • Conversion of Fractions in Various Bases
    • Base (Binary, Decimal, etc.) Converter [Java]
56. Bayes' Theorem
57. Beatty Sequences
58. Bilateral Diagrams [Java]
59. Binet's Formula via Generating Functions
60. Binomial Distribution [Java]
61. Binomial Theorem
62. Bisector of an imaginary angle may be real
63. Byzantine Basketball
64. Calculation of the Digits of pi by the Spigot Algorithm [Java]
    • Calculation of the Digits of pi (Faster Version) [Java]
65. Calculus Proof of the Pythagorean Theorem
66. Cantor-Bernstein-Schroeder theorem
    • Cantor-Bernstein-Schroeder theorem, a Second Proof
67. Cantor's First Uncountability Proof
68. Cardinal Numbers
69. Cartesian Coordinate System [Java]
70. Cavalieri's Principle [Java]
71. Chinese Remainder Theorem
    • Chinese Remainder Theorem (Pigeonhole principle)
72. Clifford Algebras
73. Clubs in a Vector Space
74. Collatz Conjecture [Java]
75. Coloring a Graph [Java]
76. Combinatorial Proofs
77. Committee Chairs
78. Comparing complex numbers
79. Completing a Square and Quadratic Formula
80. Complex Numbers: A Dynamic Tool [Java]
81. Complex number to a complex power may be real
82. Constructible Numbers
83. Construction of Pascal's Triangle [Java]
84. Constructive Existence of the de Bruijn Cycles
85. Continued fractions
    • Continued Fractions on the Stern-Brocot Tree
86. cos(π/7)-cos(2π/7)+cos(3π/7) = 1/2
87. Countability of Rational Numbers
    • Counting Ordered Pairs
    • Countability of Rational Numbers: PWW
    • Countability of Rational Numbers via Conitnued Fractions
88. Countability Principle
89. Counting Chips On a Circle [Java]
90. Counting Squares in a Square [Java]
91. Counting Triangles [Java]
92. Counting Triangles II [Java]
93. Covering a Chessboard with a Hole with L-Trominoes [Java]
94. Covering A Chessboard With Domino
    • Dominoes on a Chessboard [Java]
95. Cubic Formula [JavaScript]
96. Cubics and Bézout's Theorem
97. Curious Identities Involving Integer Products
98. Decimal Sums of Successive Integers
99. Derivatives without Limits
100. Descartes' Rule of Signs
101. Determinant and Divisibility
102. Diagonal Argument
103. Diophantine Equations
104. Distance between the Orthocenter and Circumcenter
105. Distance Between Strings [Java]
106. Distance From a Point to a Straight Line [Java]
107. Distinct Multisets with the Same Pairwise Sum
108. Dividing Loot Fairly
109. Divisibility and Modulo Arithmetic at IMO 2007
110. Divisibility by Powers of 2
111. Dollars in Pockets
112. Dot Patterns and Sierpinski Gasket [Java]
     • Arithmetic in Disguise [Java]
113. Dots and fractions [Java]
114. Ducci Sequences [Java]
115. EAN-13 Encoding [Java]
116. Eigenvectors by Inspection
117. Equations of the Staight Line [Java]
118. Equivalence Relations
119. Escape of the Clones
120. Euclid's Algorithm
121. Euclid's Algorithm: An Interactive Illustration [Java]
122. Euler Cycles in Digraph
123. Euler's formula for e^iθ
124. Euler's Function and Congruence Theorem
125. Evolution of Algebraic Symbolism
126. Existence of the de Bruijn Cycles via Graphs
127. Extension of Euclid's Algorithm
128. Factorial Products
129. Factorials and division
130. Fair Division: Method of Sealed Bids II [Java]
131. Falsity implies anything
132. Farey Series
133. Fermat's Like Equation
134. Fermat's Little Theorem
135. Fibonacci Tilings
136. Fibonacci's Quickies
137. Filling a Grid with Good Neighbors [Java]
138. Filling an Array with 0s and 1s - and Counting [Java]
139. Filling Pool with Fluids
140. Finite Sums of Terms 2^(n-i) i²
141. First Geometric Interpretation of Complex Numbers [Java]
142. Five Methods of Apportionment [Java]
143. Five Lattice Points [Java]
144. Fleury's Algorithm and Euler's Paths and Cycles [Java]
145. Flipping Items Simultaneously [Java]
146. Flipping Items Simultaneously II [Java]
147. Flipping Items Simultaneously III [Java]
148. Flipping pancakes [Java]
149. Floor Function
150. Formal Systems - an Example
151. Formula for Primes
152. Four Hinged Squares: using complex numbers
153. Four Numbers, Six Differences, GCD of the Products
154. Four Pegs That Form a Square
     • Four Pegs That Form a Square [Java]
155. Four Voting Methods [Java]
156. Function
157. Function in the Plane That Vanishes
158. Fundamental Theorem of Algebra
     • Fundamental Theorem of Algebra - Yet Another Proof
159. Futurama Theorem and Puzzle [Java]
160. Gasoline Stations on a Circular Trek
161. Gauss and Euler Integrals
162. Gauss' Lemma
163. Gauss' theorem on constructibility of regular polygons
164. GCD(M, N) × LCM(M, N) = M × N [Java]
165. Generalization That Solves the Problem
166. Generating Functions
     • Generating Functions [Java]
     • Coin Change [Java]
     • Fast Power indices [Java]
     • Weighted Voting and Power Indices [Java]
     • Generating Functions from Recurrences
167. Geometric Construction of Roots of Quadratic Equation [Java]
168. Gödel's Incompleteness Theorem
169. Golden Ratio is Irrational
170. Graph and Roots of a Third Degree Polynomial [Java]
171. Graph and Roots of Quadratic Polynomial [Java]
172. Graph of a Polynomial [Java]
173. Graph of a Polynomial Defined by Its Roots [Java]
174. Group multiplication of permutations [Java]
175. Finding Hamilton Paths and Circuits [Java]
176. Getting a Scout out of Desert [Java]
177. 1/2 = 1
178. Harmonic Serires And Its Parts
179. Horner's Method [Java]
180. How to Ask an Embarrassing Question
181. How to define a permutation? [Java]
182. However You Solve It ... A Wonderful Equation
183. Hyperintegers and Hyperreal Numbers
184. Identities with Triangular Numbers
185. Implementation of Base Conversion
186. Importance of the Absolute Value
187. In Calculus too 1 = 0
188. Inclusion-Exclusion Principle
189. Inclusion-Exclusion Principle: an Example
190. Inequality 1/2·3/4·5/6· ... ·99/100 < 1/10
191. Inequality (1 + 1^-3)(1 + 2^-3)(1 + 2^-3)...(1 + n^-3) < 3
192. Inequality 1 + 2^-2 + 3^-2 + 4^-2 + ... + n^-2 < 2.
193. Inequality from Math Magazine 2009
194. Inequality of a Series of Square Roots
195. Inequality with Logarithms
196. Infinite Discent for x⁴ + y⁴ = z⁴
197. Infinitesimals. Non-standard Analysis
198. Infinitude of Primes
     • Infinitude of Primes - A Topological Proof
     • Infinitude of Primes - A Topological Proof without Topology
     • Infinitude of Primes Via *-Sets
     • Infinitude of Primes Via Coprime Pairs
     • Infinitude of Primes Via Fermat Numbers
     • Infinitude of Primes Via Harmonic Series
     • Infinitude of Primes Via Lower Bounds
     • Infinitude Of Primes From Legendre's Formula
     • New Proof of Euclid's Theorem
199. Infinity As a Limit
200. Inflection Points of Fourth Degree Polynomials [Java]
201. Integer Iterations on a Circle [Java]
     • What Is Your Answer to That Question? [Java]
202. Integer Iterations on a Circle III [Java]
203. Integer roots of integers are irrational
204. Integers and Rectangles [Java]
205. Integers and Rectangles: a Proof by Induction [Java]
206. Integers and Rectangles: Two Simple Proofs [Java]
207. Integers With Digits 1 and 2 and Residues Modulo Powers of 2
208. Interest Calculations [Java]
209. Is the Triangle Inequality Necessary?
210. ISBN Encoding [Java]
211. Irrational number to an irrational power may be rational
212. Is 0.999... = 1?
     Is .999... = 1? A Non-standard View
213. It's All About Powers of 10
214. Joachimsthal's Notations
215. Jumping Isn't Everything [Java]
216. Kaprekar's Iterations and Numbers [Java]
217. Lagrange's identity, Proof
218. Lagrange interpolation
219. Langford and Skolem Sequences [Java]
220. Latin Squares [Java]
     • Infinite Latin Squares
     • A Problem on an Infinite Checkerboard
221. Legendre's Formula
222. Leibniz and Pascal Triangles [Java]
223. Less than, Equal to, Greater Than Symbols
224. Lewis Carroll's Game of Logic [Java]
225. Light Bulbs in a Circle [Java]
226. Lights Out [Java]
227. Linear functions
228. Linear Programming
229. Linear System with Parameter
230. Listing All Combinations [Java]
231. Listing all permutations [Java]
232. Listing All Permutations: Johnson-Trotter Method [Java]
233. Logistic Model applet [Java]
234. Long Runs of Composite Numbers
235. Long Up Short Down
236. Looking your best
237. Loop or Halt [Java]
238. Lucas' Theorem
239. Lucky Numbers [Java]
240. Marriage Theorem
241. Marriage Problem, simulation [Java]
242. Math Lotto
243. Mathematical Induction
244. Mathematical Proof
245. Mathematicians Doze Off
246. Matrix Groups
247. Maximum Volume of a Cut Off Box [Java]
248. Means [Java]
249. Mediant Fractions
250. Method of Markers II [Java]
251. Minesweeper Theorem
252. Mini Lights Out [Java]
253. Minimax Principle [Java]
254. Minimax Theorem [Java]
255. Minumum Spanning Trees and Kruskal's Algorithm [Java]
256. Minus One But What a Difference
257. Modular Arithmetic [Java]
258. Monotone Convergence Theorem
259. Monotone Number Sequences [Java]
260. Multiplication
261. Napoleon's Relatives [Java]
262. Negative on the left, positive on the right: how come?
263. Nested Radicals
264. Nested Subsets
265. Newton's Method [Java]
266. Nine Squares [Java]
267. No Flipping without End [Java]
268. Number 8 Is Interesting
269. Number Comparison - from the 1969 Canadian MO
270. Number sequences and Pigeonhole [Java]
271. Number System With Base 36
272. On the Games of Hex and Y
273. One Dimensional Ants
274. One Trigonometric Formula and Its Consequences
275. Order. Well-ordered sets.
276. Order and Chaos in Multiple Pendulums
277. Ordinal Numbers
278. Outline Mathematics
     • Billy is twice as old as Sally [Java]
     • Coin Counting Word Problem [Java]
     • Crab's Weight [Java]
     • "Math trick" with two dice [Java]
     • Primes as differences of squares [Java]
     • Problem #24 from the Rhind Papyrus [Java]
     • Robbery #1 [Java]
     • Robbery #2 [Java]
     • Robbery #3 [Java]
     • Robbery #4 [Java]
     • Smallest multiple of 9 with no odd digits [Java]
     • The Joy of Homogeneitym
     • The Lucky Find [Java]
     • The Mean Property of the Mean
     • Three Colors on a Chessboard
     • Three digit twister [Java]
     • Who Has Beard? [Java]
279. p-adic Expansions
280. p-adic Numbers
281. Parabolic Sieve of Prime Numbers
282. Parity counting [Java]
283. Parrondo Paradox Via Simpson Paradox [Java]
284. Partial But General Case Of Fermat's Last Theorem
285. Party Acquaintances [Java]
286. Patterns in Pascal's Triangle
287. Peano axioms
288. Pennies in Boxes [Java]
289. Percent
290. Permutations
     • Permutations as a Product of Transpositions
291. Pigeonhole Principle
     • Pigeonhole Principle and Extensions
292. Pinocchio as Epimenides
293. Place Value [Java]
     • Letter Count in a Sentence [Java]
294. Presence of the Antipodes on a Circle
295. Probability of Divisibility
296. Problem 1 from the IMO 2009
297. Problem A1, 69th Putnam Competition
298. Problem A2, 69th Putnam Competition
299. Problem of Divisibility by 5ⁿ
300. Product of Diagonals in Regular N-gon
301. Product Of Four Consecutive Integers
302. Product of Simple Fractions [Java]
303. Product of permutations [Java]
304. Proizvolov's Identity [Java]
305. Proofs Without Words [Java]
306. Property of aⁿ
307. Property of the Powers of 2 [Java]
308. Puzzlist vs. Mathematician: A Practical Arithmetic Problem
309. Pythagorean Triples [Java]
     • Divisibilty in Pythagorean Triples
     • Pythagorean Triples and Perfect Numbers
     • Pythagorean Triples Calculator [Java]
310. Quasigroups and block designs from triangular grid [Java]
311. Rabbits Reproduce; Integers Don't
312. Ramsey's Number R(3, 3, 3)
313. Ramsey Number R(4, 3)
314. Ramsey Number R(5, 3)
315. Ramsey Number R(4, 4)
316. Ramsey's Theorem
317. Random Walks [Java]
318. Rascal Triangle
319. Rational Solutions to x^y = y^x
320. Recombining Permutations [Java]
321. Rectangle on a Chessboard [Java]
322. Rectilinear Paradox
323. Reflection Lemma [Java]
324. Riemann Rearrangement Theorem
325. Riemann Sphere and Möbius Transformation
326. Riemann Sums - Function Integration [Java]
327. Right Replacement [Java]
328. Rolle's and Mean Value Theorems [Java]
329. Rolle's Theorem in Action
330. Rooks on a Chessboard
331. Round Robin Tournament
332. Russell's Paradox
333. Scheduling and Critical Path Algorithm [Java]
334. Scrub Tile Puzzle [Java]
335. Self-descriptive Strings [Java]
336. Self-reference
337. Seven Terms Periodic Sequence
338. Shapley-Shubik Power Index [Java]
339. Shapley-Shubik Power Index Calculator [Java]
340. Shifting Digits and a Point of View
341. Ship-Dock Theorem
     • Ship-Dock Theorem,a formalization
342. Shuttle Puzzle Practice II [Java]
343. Sicherman Dice
344. Sierpinski Gasket and Tower of Hanoi
345. Sierpinski's Gasket and Dihedral Symmetry [Java]
346. Sieve of Squares [Java]
347. Σ2^-n = Σn·2^-n
348. Signs and Sums in a Sequence [Java]
349. Signs in a Matrix [Java]
350. Simple Cellular Automaton [Java]
351. Simple Graph Practice [Java]
352. Simple Somos Sequence - a Russian Olympiad problem
353. Simple Somos Sequence Calculator [Java]
354. Simpson Paradox [Java]
355. Sine of a Sum Formula, Proof Without Words
356. Single formula to express any integer with digits 0-9 each used just once
357. Six Balls, Two Weighings
358. Size of a Class: Two Viewpoints
359. Slider Puzzles [Java]
360. Solitaire in Two Colors [Java]
361. Solve x^x3 = 3
362. Solve cos²x + cos²2x + cos²3x = 1
363. Solving an Equation with Inverse Functions
364. Soriteses
365. Sorting Algorithms
366. Sperner's Theorem
367. Splitting piles
368. Square root of 2 is irrational
369. Star of David Theorems in Pascal Triangle
370. Steinhaus' Problem [Java]
371. Step into the Elliptic Realm
372. Stern-Brocot Tree
373. Straight Tromino on a Chessboard [Java]
374. Structure of Hyperreal Numbers
375. Subsets and Intersections
376. Subtraction Game [Java]
377. Sums and Products, a Generalization
378. Sums, Products, and 1-1 Functions
379. Sums of Geometric Series - Proofs Without Words [Java]
380. Sums of Power Series
381. Sums Of Successive Squares
382. Sum without Adding
383. Sums and Products [Java]
384. Sums of powers of digits [Java]
385. Superposition Principle
386. Surreal Numbers
387. Swapping Rows and Columns [Java]
388. Sylvester's Problem
     • Sylvester's Problem, a Second Look
389. Taking Games Seriously [Java]
390. The ass and the mule [Java]
391. The Basic Rules of Counting
     • Example: A Poker Hand
392. The Local-Global Principle
393. The Regula Falsi - Iterated
394. The Size is in the Eyes of the Beholder
395. Three Dimensional Cross
396. Tool for Constructing Orthogonal Latin Squares [Java]
397. Tower of Hanoi, the Hard Way [Java]
398. Transpositions
399. Traveling Salesman Problem [Java]
400. Treatise on Arithmetical Triangle
401. Triangular numbers that are also square
402. Trilateral Diagrams [Java]
403. Trinary Tree(s) underlying Primitive Pythagorean Triples
404. Tromino Puzzle: Deficient Squares [Java]
405. Twenty five boys and twenty five girls around a table.
406. Two Circles and a Limit [Java]
407. Two Colors in Two Rows [Java]
     • Two Colors in Two Rows II [Java]
408. Two Men of Tibet [Java]
409. Two Problems from a Serbian 7 Grade Math Competition
410. Two-Parameter Solutions to Three Almost Fermat Equations
411. Uncountability of the Reals - via a Game
412. Universal Coloring [Java]
413. Universal Product Code [Java]
414. Unproperty of the Powers of 2
415. USA 1975 Problem 3
416. Vandermonde Determinant
417. Venn Diagrams
418. Viète's Formulas
     • Venn Diagrams I [Java]
     • Venn Diagrams II [Java]
419. Ways To Count
420. Weierstrass Product Inequality
421. Weierstrass Substitution
422. What, how, and the Web [Java]
423. What Is Algorithm?
424. What Is Determinant?
425. What Is Finite?
426. What Is Infinity?
427. What Is Limit?
428. What is so natural about "natural logarithms"?
429. What Is Space Completion?
430. WHEN THE COUNTING GETS TOUGH, THE TOUGH COUNT ON MATHEMATICS
     • Matrices Help Relationships
     • An Airline Problem
431. Where To Start? [Java]
432. White and Black Balls in Urn. 1 in, 2 out. What Color Remains?
433. Why 17?
434. Wilson's Theorem
     • Geometric Proof of Wilson's Theorem
435. Word Problems [Java]
436. Yet Another Puzzle with Hats
437. Zeros and Nines [Java]

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