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

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