CSE Blog - quant, math, computer science puzzles

Source:  www.puzzletweeter.com Problem: Let A,B be 2x2 matrices with integer entries. Suppose the matrices A,A+B,A+2B,A+3B,A+4B are all invertible and their inverses are also integer matrices. Then show that A+5B is invertible and it's inverse is an integer matrix. Update (24 June 2014): Solution: Posted by Yashoteja Prabhu (Ex-RA at Microsoft Research, IITB CSE 2011 Alumnus) and Sanchit Gupta in comments!

Art of Puzzle Solving from Pratik Poddar

Source: Interview Street Problem:   Find the number of positive integral solutions for the equations  (1/x) + (1/y) = 1/N!  (read 1 by x plus 1 by y is equal to 1 by n factorial)  Print a single integer which is the no of positive integral solutions modulo 1000007. Update (24 June 2014): Solution: Posted by JDGM (James D G Miles) in comments!

Source: Mailed to me by Sudeep Kamath (PhD Student, UC at Berkeley, EE IITB Alumnus 2008) - He found it at  http://puzzletweeter.com/ Problem: A group of students are sitting in a circle with the teacher in the center. They all have an even number of candies (not necessarily equal). When the teacher blows a whistle, each student passes half his candies to the student on his left. Then the students who have an odd number of candies obtain an extra candy from the teacher. Show that after a finite number of whistles, all students have the same number of candies. Update (20 May 2013): Partial Solution posted by JDGM in comments. Completed by me.

Source: Mailed to me by Sudeep Kamath (PhD Student, UC at Berkeley, EE IITB Alumnus 2008) - He found it at  http://puzzletweeter.com/ Problem: Consider an n x n chessboard, where each square is arbitrarily chosen to be either black or white. Your goal is to make all squares in the chessboard white. At each step, you are allowed to "switch" a square, but each switch will toggle not only the particular square being switched, but also the 4 squares that are adjacent to it: Two vertically up and down and two horizontally up and down the square being switched. Note : At corners only 4/3 squares are toggled, while at the center all 5 squares are toggled. Show how you can make the entire chessboard white. Update (May 20 2013): Solution: Posted by JDGM in comments! Very academic solution to a very interesting problem! :) Thanks

Source: Quantnet Problem: Roll a penny around another fixed penny in the center with edges in close contact. After moving half circle around the center penny, you will find the penny in motion has rotated 360 degrees. Why? Update (29/06/2013): Solution posted by Sanket Patel and Suyash Jain (IITB Mech 2008 Aumnus, Ex-Credit Suisse Analyst, Ex-Deutsche Bank Analyst) in comments!

Source:  Mailed to me by Sudeep Kamath (PhD Student, UC at Berkeley, EE IITB Alumnus 2008) - He found it at  http://puzzletweeter.com/ Problem: An integer point in a plane is a point whose coordinates are integers. Suppose we arbitrarily choose 5 integer points in a plane. Show that we can always find 2 among these 5 integer points such that the line segment joining the 2 points contains at least 1 more integer point.