Interactive Mathematics Miscellany and Puzzles is a frequently updated site. Here are the most recent addtions. For other changes please visit the site. There is a positive integer in every cell of a rectangular array. In each move, you may double every number in a row or subtract 1 from each number in a column. Prove that you can reach a table os zeros by a sequence of the permitted moves. The applet below is to assist you in getting insight into the problem http://www.cut-the-knot.org/Curriculum/Algorithms/TableToZeroGame.shtml Given two integers, N and M such that M divides N then M = gcd(N, M) and M = lcm(N, M). So, in this case, N times M = gcd(N, M) times lcm(N, M). Perhaps, surprisingly, this is true for any two integers. The applet below sheds more light on why this is so http://www.cut-the-knot.org/Curriculum/Arithmetic/GCDByBTree.shtml Euclid's algorithm is a famous procedure for finding the gcd, i.e., greatest common divisor (factor) of two integers. The idea is pretty simple. If N = M times s, with N, M, s, positive integers, then any divisor of M is also a divisor of N, making M their greatest common divisor http://www.cut-the-knot.org/Curriculum/Arithmetic/EuclidAlgorithm.shtml Factoring, i.e., listing all the prime factors, of an integer is a useful skill that often helps to solve math problems. For example, one of the ways to find the GCD or LCM of two integers is by listing all their prime factors. The GCD is then the product of all the common factors; the LCM is the product of all the remaining ones. The applet below is a tool for finding such prime factorizations with the device known as the Factor Tree http://www.cut-the-knot.org/Curriculum/Arithmetic/BTreeTesting.shtml Compare Fractions: Interactive Practice. The applet presents two fractions and leaves it to you to indicate the correct relationship (less than, equal to, greater than) by placing a suitable symbol between the two fractions http://www.cut-the-knot.org/Curriculum/Arithmetic/FractionComparisonExercise.shtml Nested Subsets: Every infinite set contains uncountably many nested subsets http://www.cut-the-knot.org/do_you_know/NestedSubsets.shtml In the proof we are going to use comple xnumbers. The proof comes from Bollobas where the author makes an observation that after the slogan 'let's use vectors and comple xnumbers' no more thinking is needed. While this is true that one of algebra's purposes and uses is to mechanize solving problems, this is a third proof of Napoleon's theorem that makes use of comple xnumbers. So that, perhaps, some deliberation as to which road to choose might follow a conscious decision to base a proof on comple xnumbers http://www.cut-the-knot.org/Curriculum/Geometry/NapoleonBollobas.shtml The number x_n is defined as the last digit in the decimal representation of the integer whole part of sqrt(n)^n (n = 1, 2, ...). Determine whether the sequence x_1, x_2, ..., x_n, ... is periodic http://www.cut-the-knot.org/proofs/ShiftingDigitsAndViewPoint.shtml Tangent as a Radical Axis: Let PQRS be a cyclic quadrilateral such that the segments PQ and RS are not parallel. Consider the set of circles through P and Q, and the set of circles through R and S. Determine the set I of points of tangency of circles in these two sets. http://www.cut-the-knot.org/Curriculum/Geometry/RadicalTangent.shtml APMO 1995 Problem 3 http://www.cut-the-knot.org/wiki-math/index.php?n=MathematicalOlympiads.APMO1995Problem3