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.
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May 18, 2013
Matrix Puzzle
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.
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.
May 16, 2013
May 3, 2013
Equations Puzzle
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.
Apr 24, 2013
Candy Game - Math Puzzle
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.
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.
Black and White Squares Puzzle
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.
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
Solution: Posted by JDGM in comments! Very academic solution to a very interesting problem! :) Thanks
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