Algebra

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

1. 17 rooks on an 8×8 chessboard
2. 3-Colors Tower of Hanoi [Java]
3. 3 glass puzzle [JavaScript]
   • 3 glasses puzzle, second, more general tool [JavaScript]
   • 3 Glasses Problem in Barycentric Coordinates
   • Three Jugs Problem [Java]
   • Two Pails Puzzle [Java]
4. 3-Term Arithmetic Progression
5. 4x4 coloring problem
6. A 49^th Degree Challenge
7. A Clique of Acquaintances
8. About numbers
9. Absolute Value
10. Achieve the Equilibrium [Flash]
11. Addition
    • Addition and Subtraction of Functions [Java]
12. Affirmative Action Problem [Java]
13. Algebraic Structure of Complex Numbers
14. All Integers Are Equal to 1
15. All Powers of x are Constant
16. Almost every integer has the digit 3 in it
17. Almost Integers - Gaussian Integers
18. Ambassadors at a Round Table [Java]
19. An Inequality for Grade 8
20. An Inequality With an Infinite Series
21. Arithmetic-Geometric Means Inequality
22. A simple integral
23. Another simple integral
24. Averages, Arithmetic and Harmonic Means [Java]
25. Averages in a sequence
26. Banzhaf Power Index Calculator [Java]
27. Barcode Encoding
28. Barcode Magic [Java]
29. Base Converter [JavaScript]
    • Binary System
    • Conversion of Fractions in Various Bases
    • Base (Binary, Decimal, etc.) Converter [Java]
30. Bayes' Theorem
31. Beatty Sequences
32. Bilateral Diagrams [Java]
33. Binomial Distribution [Java]
34. Binomial Theorem\
35. Bisector of an imaginary angle may be real
36. Byzantine Basketball
37. Calculation of the Digits of pi by the Spigot Algorithm [Java]
    • Calculation of the Digits of pi (Faster Version) [Java]
38. Cantor-Bernstein-Schroeder theorem
39. Cardinal Numbers
40. Cartesian Coordinate System [Java]
41. Cavalieri's Principle [Java]
42. Chinese Remainder Theorem
    • Chinese Remainder Theorem (Pigeonhole principle)
43. Clifford Algebras
44. Clubs in a Vector Space
45. Collatz Conjecture [Java]
46. Coloring a Graph [Java]
47. Combinatorial Proofs
48. Committee Chairs
49. Comparing complex numbers
50. Completing a Square and Quadratic Formula
51. Complex Numbers: A Dynamic Tool [Java]
52. Complex number to a complex power may be real
53. Constructible Numbers
54. Continued fractions
    Continued Fractions on the Stern-Brocot Tree
55. Countability of Rational Numbers
56. Counting Chips On a Circle [Java]
57. Counting Squares in a Square [Java]
58. Counting Triangles [Java]
59. Counting Triangles II [Java]
60. Covering A Chessboard With Domino
    • Dominoes on a Chessboard [Java]
61. Cubic Formula [JavaScript]
62. Cubics and Bézout's Theorem
63. Derivatives without Limits
64. Descartes' Rule of Signs
65. Distance between the Orthocenter and Circumcenter
66. Distance Between Strings [Java]
67. Distance From a Point to a Straight Line [Java]
68. Dividing Loot Fairly
69. Divisibility and Modulo Arithmetic at IMO 2007
70. Dollars in Pockets
71. Dot Patterns and Sierpinski Gasket [Java]
    • Arithmetic in Disguise [Java]
72. Dots and fractions [Java]
73. Ducci Sequences [Java]
74. EAN-13 Encoding [Java]
75. Eigenvectors by Inspection
76. Equations of the Staight Line [Java]
77. Equivalence Relations
78. Euclid's Algorithm
79. Euclid's Algorithm: An Interactive Illustration [Java]
80. Euler’s formula for e^iθ
81. Euler's Function and Congruence Theorem
82. Evolution of Algebraic Symbolism
83. Extension of Euclid's Algorithm
84. Factorials and division
85. Fair Division: Method of Sealed Bids II [Java]
86. Falsity implies anything
87. Farey Series
88. Fermat's Little Theorem
89. Fibonacci's Quickies
90. First Geometric Interpretation of Complex Numbers [Java]
91. Five Methods of Apportionment [Java]
92. Five Lattice Points [Java]
93. Fleury's Algorithm and Euler's Paths and Cycles [Java]
94. Flipping Items Simultaneously [Java]
95. Flipping Items Simultaneously II [Java]
96. Flipping pancakes [Java]
97. Floor Function
98. Formal Systems - an Example
99. Formula for Primes
100. Four Hinged Squares: using complex numbers
101. Four Pegs That Form a Square
     • Four Pegs That Form a Square [Java]
102. Four Voting Methods [Java]
103. Function
104. Fundamental Theorem of Algebra
     • Fundamental Theorem of Algebra - Yet Another Proof
105. Gasoline Stations on a Circular Trek
106. GCD(M, N) × LCM(M, N) = M × N [Java]
107. Generalization That Solves the Problem
108. Generating Functions
     • Generating Functions [Java]
     • Coin Change [Java]
     • Fast Power indices [Java]
     • Weighted Voting and Power Indices [Java]
109. Geometric Construction of Roots of Quadratic Equation [Java]
110. Gödel's Incompleteness Theorem
111. Graph and Roots of a Third Degree Polynomial [Java]
112. Graph and Roots of Quadratic Polynomial [Java]
113. Graph of a Polynomial [Java]
114. Graph of a Polynomial Defined by Its Roots [Java]
115. Group multiplication of permutations [Java]
116. Finding Hamilton Paths and Circuits [Java]
117. Harmonic Serires And Its Parts
118. Horner's Method [Java]
119. How to Ask an Embarrassing Question
120. How to define a permutation? [Java]
121. However You Solve It ... A Wonderful Equation
122. Hyperintegers and Hyperreal Numbers
123. Implementation of Base Conversion
124. Importance of the Absolute Value
125. Inclusion-Exclusion Principle
126. Inequality 1/2·3/4·5/6· ... ·99/100 < 1/10
127. Inequality (1 + 1^-3)(1 + 2^-3)(1 + 2^-3)...(1 + n^-3) < 3
128. Inequality 1 + 2^-2 + 3^-2 + 4^-2 + ... + n^-2 < 2.
129. Inequality from Math Magazine 2009
130. Infinite Discent for x⁴ + y⁴ = z⁴
131. Infinitesimals. Non-standard Analysis
132. 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 From Legendre's Formula
133. Infinity As a Limit
134. Inflection Points of Fourth Degree Polynomials [Java]
135. Integer Iterations on a Circle [Java]
     • What Is Your Answer to That Question? [Java]
136. Integer Iterations on a Circle III [Java]
137. Integer roots of integers are irrational
138. Integers and Rectangles [Java]
139. Integers and Rectangles: a Proof by Induction [Java]
140. Integers and Rectangles: Two Simple Proofs [Java]
141. Interest Calculations [Java]
142. ISBN Encoding [Java]
143. Irrational number to an irrational power may be rational
144. Is 0.999... = 1?
     Is .999... = 1? A Non-standard View
145. It's All About Powers of 10
146. Joachimsthal's Notations
147. Jumping Isn't Everything [Java]
148. Kaprekar's Iterations and Numbers [Java]
149. Lagrange's identity, Proof
150. Lagrange interpolation
151. Langford and Skolem Sequences [Java]
152. Latin Squares [Java]
     • Infinite Latin Squares
     • A Problem on an Infinite Checkerboard
153. Legendre's Formula
154. Leibniz and Pascal Triangles [Java]
155. Lewis Carroll's Game of Logic [Java]
156. Light Bulbs in a Circle [Java]
157. Lights Out [Java]
158. Linear functions
159. Linear Programming
160. Listing All Combinations [Java]
161. Listing all permutations [Java]
162. Listing All Permutations: Johnson-Trotter Method [Java]
163. Logistic Model applet [Java]
164. Long Runs of Composite Numbers
165. Long Up Short Down
166. Looking your best
167. Loop or Halt [Java]
168. Lucas' Theorem
169. Marriage Theorem
170. Marriage Problem, simulation [Java]
171. Math Lotto
172. Mathematical Induction
173. Mathematical Proof
174. Mathematicians Doze Off
175. Matrix Groups
176. Maximum Volume of a Cut Off Box [Java]
177. Means [Java]
178. Mediant Fractions
179. Method of Markers II [Java]
180. Minesweeper Theorem
181. Mini Lights Out [Java]
182. Minimax Principle [Java]
183. Minimax Theorem [Java]
184. Minumum Spanning Trees and Kruskal's Algorithm [Java]
185. Modular Arithmetic [Java]
186. Monotone Convergence Theorem
187. Monotone Number Sequences [Java]
188. Multiplication
189. Napoleon's Relatives [Java]
190. Nested Radicals
191. Nested Subsets
192. Newton's Method [Java]
193. Nine Squares [Java]
194. Number 8 Is Interesting
195. Number sequences and Pigeonhole [Java]
196. Number System With Base 36
197. On the Games of Hex and Y
198. One Dimensional Ants
199. One Trigonometric Formula and Its Consequences
200. Order. Well-ordered sets.
201. Ordinal Numbers
202. 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]
     Three digit twister [Java]
     Who Has Beard? [Java]
203. p-adic Expansions
204. p-adic Numbers
205. Parity counting [Java]
206. Parrondo Paradox Via Simpson Paradox [Java]
207. Party Acquaintances [Java]
208. Peano axioms
209. Pennies in Boxes [Java]
210. Percent
211. Permutations
     • Permutations as a Product of Transpositions
212. Pigeonhole Principle
     • Pigeonhole Principle and Extensions
213. Place Value [Java]
     • Letter Count in a Sentence [Java]
214. Probability of Divisibility
215. Problem 1 from the IMO 2009
216. Problem A1, 69th Putnam Competition
217. Problem A2, 69th Putnam Competition
218. Problem of Divisibility by 5ⁿ
219. Product of Diagonals in Regular N-gon
220. Product Of Four Consecutive Integers
221. Product of Simple Fractions [Java]
222. Product of permutations [Java]
223. Proizvolov's Identity [Java]
224. Proofs Without Words [Java]
225. Property of aⁿ
226. Property of the Powers of 2 [Java]
227. Yet Another Puzzle with Hats
228. Puzzlist vs. Mathematician: A Practical Arithmetic Problem
229. Pythagorean Triples [Java]
     • Divisibilty in Pythagorean Triples
     • Pythagorean Triples and Perfect Numbers
     • Pythagorean Triples Calculator [Java]
230. Quasigroups and block designs from triangular grid [Java]
231. Rabbits Reproduce; Integers Don't
232. Ramsey's Theorem
233. Recombining Permutations [Java]
234. Rectangle on a Chessboard [Java]
235. Riemann Sphere and Möbius Transformation
236. Riemann Sums - Function Integration [Java]
237. Right Replacement [Java]
238. Rolle's and Mean Value Theorems [Java]
239. Round Robin Tournament
240. Russell's Paradox
241. Scheduling and Critical Path Algorithm [Java]
242. Scrub Tile Puzzle [Java]
243. Self-descriptive Strings [Java]
244. Self-reference
245. Seven Terms Periodic Sequence
246. Shapley-Shubik Power Index [Java]
247. Shapley-Shubik Power Index Calculator [Java]
248. Shifting Digits and a Point of View
249. Ship-Dock Theorem
     • Ship-Dock Theorem,a formalization
250. Sierpinski Gasket and Tower of Hanoi
251. Sierpinski's Gasket and Dihedral Symmetry [Java]
252. Sieve of Squares [Java]
253. Σ2^-n = Σn·2^-n
254. Signs and Sums in a Sequence [Java]
255. Signs in a Matrix [Java]
256. Simple Cellular Automaton [Java]
257. Simple Graph Practice [Java]
258. Simpson Paradox [Java]
259. Sine of a Sum Formula, Proof Without Words
260. Single formula to express any integer with digits 0-9 each used just once
261. Six Balls, Two Weighings
262. Slider Puzzles [Java]
263. Solitaire in Two Colors [Java]
264. Solve x^x3 = 3
265. Soriteses
266. Sorting Algorithms
267. Sperner's Theorem
268. Splitting piles
269. Square root of 2 is irrational
270. Steinhaus' Problem [Java]
271. Step into the Elliptic Realm
272. Stern-Brocot Tree
273. Straight Tromino on a Chessboard [Java]
274. Structure of Hyperreal Numbers
275. Subtraction Game [Java]
276. Sums of Power Series
277. Sums Of Successive Squares
278. Sum without Adding
279. Sums and Products [Java]
280. Sums of powers of digits [Java]
281. Superposition Principle
282. Surreal Numbers
283. Taking Games Seriously [Java]
284. The ass and the mule [Java]
285. The Local-Global Principle
286. The Size is in the Eyes of the Beholder
287. Tool for Constructing Orthogonal Latin Squares [Java]
288. Three Dimensional Cross
289. Tower of Hanoi, the Hard Way [Java]
290. Transpositions
291. Traveling Salesman Problem [Java]
292. Triangular numbers that are also square
293. Trilateral Diagrams [Java]
294. Trinary Tree(s) underlying Primitive Pythagorean Triples
295. Twenty five boys and twenty five girls around a table.
296. Two Circles and a Limit [Java]
297. Two Colors in Two Rows [Java]
     • Two Colors in Two Rows II [Java]
298. Two Men of Tibet [Java]
299. Uncountability of the Reals - via a Game
300. Universal Coloring [Java]
301. Universal Product Code [Java]
302. Unproperty of the Powers of 2
303. USA 1975 Problem 3
304. Venn Diagrams
     • Venn Diagrams I [Java]
     • Venn Diagrams II [Java]
305. Ways To Count
306. Weierstrass Product Inequality
307. What, how, and the Web [Java]
308. What Is Algorithm?
309. What Is Infinity?
310. What Is Limit?
311. What is so natural about “natural logarithms”?
312. What Is Space Completion?
313. WHEN THE COUNTING GETS TOUGH, THE TOUGH COUNT ON MATHEMATICS
     • Matrices Help Relationships
     • An Airline Problem
314. Where To Start? [Java]
315. Why 17?
316. Wilson's Theorem
     • Geometric Proof of Wilson's Theorem
317. Word Problems [Java]
318. Zeros and Nines [Java]

Copyright © 1996-2010 Alexander Bogomolny