Recent changes to Interactive Mathematics Miscellany and PuzzlesInteractive Mathematics Miscellany and Puzzles is a frequently updated site. Here are the most recent addtions. For other changes please visit the site.A Coin Tossing Surprise IA fair coin is tossed repeatedly. What is the expected number of tosses before $HT$ shows up for the first time? What is the probability that $HT$ shows up before $TT?$
https://www.cut-the-knot.org/Probability/CoinTossingSurprise1.shtml
Fair DuelTwo duelants take turns shooting each other until one is hurt. The weaker shooter starts first. The duel is fair. Estimate the ability of the second shooter
https://www.cut-the-knot.org/Probability/FairDuel.shtml
Is a Sum a Prime Number?Let a,b,c,d be positve integers with ab=cd. Can a+b+c+d be a prime?
https://www.cut-the-knot.org/m/Arithmetic/IsSumAPrimeNumber.shtml
A Moscow Olympiad Question with Two InequalitiesLet a,b,c be real numbers satisfying (a+b+c)c ≤ 0. Prove that ac ≥ b^2/4
https://www.cut-the-knot.org/m/Algebra/OlympiadQuestionTwoInequalities.shtml
Lucky Times at a Moscow Math OlympiadLucky times are defined by a sequence of the hour, minute and second hands. Which is more probable - lucky or unlucky?
https://www.cut-the-knot.org/Probability/LuckyTimes.shtml
Averages of Terms in Increasing SequenceLet k and n be positive integers with k ≤ n and let S consist of all strictly increasing k-tuples with each x_j an integer and 1 ≤ x_1 and x_n ≤ n. For j = 1, 2, ..., k, find the average value of x_j all sequences in S
https://www.cut-the-knot.org/Probability/AveragesInIncreasingSequence.shtml
An Inequality in Triangle with Radicals, Semiperimeter, Incenter and Inradius(AI + BI + CI)/r ≥ sum [sqrt{s-a}] * sum [1/sqrt(s-a)]
https://www.cut-the-knot.org/triangle/LorianLeoIncenter.shtml
An Equation in FactorialsFind all integer solutions of n! * (n-1)! = m!
https://www.cut-the-knot.org/m/Arithmetic/EquationInFactorials.shtml
Folding and Cutting a SquareFold an 8x8 square along the grid lines in an otherwise random sequence of folds until you arrive at an 1x1 square. Cut the latter from the midpoint of one edge to the midpoint of the opposite edge. What is the expected number of pieces do you get as the result?
https://www.cut-the-knot.org/m/Geometry/FoldingAndCuttingSquare.shtml
Getting from A to B via COn a square map a student goes from A to B. Find the probability that he'll pass C
https://www.cut-the-knot.org/Probability/GettingFromAtoB.shtml