CSE Blog - quant, math, computer science puzzles

Source: Quantnet Forum Problem: There is a calculator in which all digits(0-9) and the basic arithmetic operators(+,-,*,/) are disabled. However other scientific functions are operational like exp, log, sin, cos, arctan, etc. The calculator currently displays a 0. Convert this first to 2 and then to 3. Update (8/7/2012): Solution posted in comments by Siddhartha, Sumit, Salman Khan and Kapil Dubey. Thanks. Interesting generalisation proposed by Siddhartha.

Source: IBM Ponder This July 2012 Problem very similar to   CSE Blog: Painting Coloured Balls  ( Link:  http://pratikpoddarcse.blogspot.in/2010/10/painting-coloured-balls.html  ) Problem: Alice and Bob are playing two games: they start simultaneously and the first one to win his game is the winner. Alice is given an urn with N balls, colored in N different colors and in every second she randomly picks two balls, colors the first ball in the color of the second ball and returns both to the urn. Her game is over once all the balls in the urn have the same color. Bob is given an urn with M balls, all colorless, and in every second he picks a random ball, color it, and puts it back to the urn. Bob's game is over once all his balls are colored. Our question is: what are the possible values of M for which (on average) Bob is expected to lose for N=80 and win for N=81?  We ask for possible Ms for which the expected time of Bob's game is smaller than Alice's...

Source: Nick's Mathematical Puzzles Problem: Let n be a positive integer, and let S n  = {n 2  + 1, n 2  + 2, ... , (n + 1) 2 }.  Find, in terms of n, the cardinality of the set of pairwise products of distinct elements of S n . For example, S 2  = {5, 6, 7, 8, 9}, 5 × 6 = 6 × 5 = 30, 5 × 7 = 7 × 5 = 35, 5 × 8 = 8 × 5 = 40, 5 × 9 = 9 × 5 = 45, 6 × 7 = 7 × 6 = 42, 6 × 8 = 8 × 6 = 48, 6 × 9 = 9 × 6 = 54, 7 × 8 = 8 × 7 = 56, 7 × 9 = 9 × 7 = 63, 8 × 9 = 9 × 8 = 72, and the required cardinality is 10. Update (Sept 12, 2012): Solution posted by Parakram Majumdar (CSE IITB Alumnus, Morgan Stanley Strats and Modeling Analyst) - Detailed explanation of the solution by unknown in comments!