I am making a computer program that is going to generate Sudoku puzzles of various types. One of these types is "jigsaw", in which the board is split into rows, columns and random 9-square contiguous regions. My regions are generated completely randomly and I have become concerned that maybe ther...

I was playing a casual game of Sudoku today when a friend came by and asked "What's the hardest game of Sudoku possible?" My response: "A Sudoku puzzle with the minimal amount of starting numbers where the puzzle is still solvable." However, I am not happy with this because I want to know the a...

Sudoku is a puzzle, with the objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 sub-grids that compose the grid (also "sudoku-blocks") contains all of the digits from 1 to 9. Let's define block as a 3x3 sub-grid (not necessarily forming one of th...

Let $A \in O(n).$ Show that $A$ is similar to a matrix which consists of $2 \times 2$ blocks down the diagonal of the form $$ \begin{pmatrix} \cos{\theta} & \sin{\theta}\\-\sin{\theta} & \cos{\theta} \end{pmatrix},$$ along with some diagonal elements which are $+1$ and $-1.$ For example, maybe...

I've researched this question but explanations that I have found were either not thorough enough or simply seemed downright incorrect. I thought it looked quite simple when I first saw it but it stumped me, Hopefully you can help me with this. A and B are 2 shapes which are mathematically simi...

My question regards this problem: http://i.stack.imgur.com/eBDPX.png I have already tried finding the determinant of the $ \lambda I -[T]$ matrix, getting the result: $(\lambda -2c)(\lambda -d) - (2c-b)(-a-c) = \lambda^2 - d\lambda-2c\lambda+2cd+2ac+2c^2-ab-bc$ I am lost at this point, I don'...

I am working on the following problem, but I can't seem to figure it out. The length of the sides in the triangle $T_1$ are $a_1$, $b_1$ and $c_1$. The length of the sides in the triangle $T_2$ are $a_2$, $b_2$ and $c_2$. Moreover: $$\sqrt{a_1 a_2} + \sqrt{b_1 b_2} + \sqrt{c_1 c_2} = \sqrt{(a...

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