If you walk into any book store you are bound to find entire shelves completely comprised with books that have 100+ different Sudoku puzzles for the reader to solve. Throughout the past 10 years Sudoku has become an internationally known puzzle game reaching the same amount if not surpassing in popularity as the crossword puzzle. In this paper I will explore how to determine how many fundamentally different completed Sudoku puzzles (known as Sudoku Squares) exist. In order to do this I will first give a brief history of where Sudoku puzzles originated from, then I will show how to determine how many Sudoku squares exist which will lead me into determining how many of the Sudoku squares that exist are fundamentally different from each other.
Latin Squares
Sudoku puzzles originated from Latin Squares which have been studied by mathematicians for centuries. A Latin Square of order n is an square with n rows and n columns where each row and column contain each of the n symbols appears exactly once. The following below are some examples.
Those who are adept in Sudoku puzzles will notice that a Sudoku puzzle is a 9x9 Latin Square without the 3x3 block criteria.
Counting Sudoku Squares
Shidoku Squares
At Barnes & Noble there is an entire section full of Sudoku puzzles. They include books like Treacherous Sudoku, Pocket Sudoku, Killer Sudoku, and Enslaved by Sudoku. With the amount of different Sudoku books, one question that I had was how many different Sudoku squares can there be? Since there are so many restrictions on what makes a Sudoku square, we are going to first consider Shidoku squares as they are going to be easier to visualize. A Shidoku square is a 4x4 square that follows the same restrictions as Sud...
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...allowed them to use Burnside’s Lemma and they discovered (56(0)+48(2)+4(6)+6(8)+4(10)+1(12))/128=256/128=2, where 56 of the transformations didn’t fix any squares and 1 transformation (the identity transformation) fixed 12 squares. Thus, there are only two fundamentally different Shidoku squares.
Now, there is also a less complicated way of determining how many fundamentally different Shidoku squares exist….pg 80, not sure I want to take time to explain this….
Sudoku Squares
Earlier, we determined that there are approximately 6.67 sextillion different valid Sudoku squares, our task is to use the six possible transformations (stated earlier) and Burnside’s Lemma to determine how many fundamentally different Sudoku squares exist. There are six basic transformations that can be applied to a Sudoku square, but we can combine multiple transformations to achieve a
On the second day of class, the Professor Judit Kerekes developed a short chart of the Xmania system and briefly explained how students would experience a number problem. Professor Kerekes invented letters to name the quantities such as “A” for one box, “B” for two boxes. “C” is for three boxes, “D” is for four boxes and “E” is for five boxes. This chart confused me because I wasn’t too familiar with this system. One thing that generated a lot of excitement for me was when she used huge foam blocks shaped as dice. A student threw two blocks across the room and identified the symbol “0”, “A”, “B”, “C”, “D”, and “E.” To everyone’s amazement, we had fun practicing the Xmania system and learned as each table took turns trying to work out problems.
...ngly opposite, the Chinese Landscape Painting depicts a boulder-filled mountainside with a waterfall, a river, a Chinese house, and trees spread throughout. The quote underneath is from Lao-tzu, (the founder of the Daoism philosophy). The complexity of the sentences by Lao-tzu is much higher than the sentence of Socrates.
As an example, he showed two squares with a dot in the middle, the one in the left was green and the one in the right is red. Under the two squares there was a picture of a desert with a dot in the middle as well. We needed to look at the dot between the colored squares
I learned about many significant artwork and artist in this class. This class provided me with a better understanding of the history of the world Art, but also helped me understand the development of art style. However, among all of these precious pieces of artwork, there are two special ones that caught my attention: The Chinese Qin Terracotta Warriors and The Haniwa. Each of them represents the artist’s stylistic characteristics and cultural context. Although they represented different art of rulers, historical values, and scenes, there were visible similarities.
Gaspare Traversi created his oil on canvas painting, Quarrel over a Board Game, in 1752. As the title states, the painting’s focus is on two gentlemen quarreling over a game of what appears to be checkers. This painting is currently being exhibited at the Wadsworth Atheneum in Hartford, CT. The painting is not large. It is 4 by 5 ft. in a gold patterned frame. The painting is placed on a wall in a brightly lighted room at a viewer’s eye level.
The last activity that we did was taking ten Q tips and made three attached squares and her assignment was to make a 4th enclosed box without adding an additional items. Once I told her to start she immediately started moving the Q tips around trying to create another box. After trying for a few minutes she then say there is no way to add another box.
A Cellular Automaton consists of a regular grid of cells, each as a finite number of states such as On and Off.An initial state [time t=0] is selected by assigning a state for each cell. The rule for updating the state of cells is the same for each cell and does not change over time.
The square is a shape that not only represents the plays character’s view of Othello but outlines his true inner character. First of al, the geometrical properties of a square support its representing character. The square, having straight rigid edges, appears to be a very strong shape. It’s sides and angle,
Traditional Chinese art is deeply rooted in its philosophy, encompassing Daoist, Buddhist and Confucian schools of thought. The goal of many traditional Chinese landscape artists, as described by Professor To Cho Yee of Michigan-Ann Arbor, is to “reveal the highest harmony between man and nature” through a balance of likeness and unlikeness (Ho). This metaphysical philosophy borrowed art as a vehicle to search for the truth or the “dao”, which is the path to enlightenment. As early as the 5th century, scholar artists such as Su Shi (1037-1101) of the Song dynasty realized that to create likeness, one must understand the object beyond its superficial state and instead capture the spirit of nature; only then can a point of harmony with nature
60 1,45 0,56 0,90 0,84 1,00 0,05 0,59 0,77 0,40 80 1,45 0,62 2,00 0,65 0,65
Pascal’s Triangle falls into many areas of mathematics, such as number theory, combinatorics and algebra. Throughout this paper, I will mostly be discussing how combinatorics are related to Pascal’s Triangle.
Let us see now how this algorithm works. The algorithms randomly creates solutions. Each one of these solutions has a fitness value based on some criteria. Those solutions of a specific problem are also called Phenotype, while the encoding of each solution is called Genotype. We refer on Representation as the procedure of establish the mapping between genotypes and phenotypes. Representation is used as in two different ways. As mentioned before, representation establish the mapping between the genotype and the phenotype. This means that representation could encode ore decode the candidate solutions.
Using a square, both the length & the width are equal. I am using a
There are six diagonal lines. At one end there are circles on them giving the impression of three circular prongs. At the other end the same size lines have cross connecting lines consistent with two square prongs. These perceptions can violate our expectations for what is possible often to a delightful effect.
Crossword puzzles and Sudoku in their own respect present different difficulties. As a young girl I have fond memories of my grandfather sitting at the table every morning completing his crossword puzzle before doing anything else. I, on the other hand prefer neither of the puzzles. If I had to choose, the Sudoku was easier to complete. There is a definite psychological answer as to why I particularly feel this way, and why I believe that one is easier to complete than the other.