Suppose a n × n chessboard is missing two opposite corners. Prove that no matter what n is, you will not be able to cover the remaining squares exactly with dominoes.
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Suppose a n × n chessboard is missing two opposite corners. Prove that no matter what n is,
you will not be able to cover the remaining squares exactly with dominoes.
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- Mendoza's Pizzeria offers many wonderful topping options on their world-famous pizza pies. A cheese pizza costs $20, plus it costs $1 for any veggie topping (onions, green peppers, spinach, etc), and $2 for any meat topping (pepperoni, bacon, sausage, etc.). There must be atleast 2 toppings on each pizza pie, but there cannot be mor than 10 toppings on any one pie. Marty enters the scene with $30 in his pocket.Suppose we have a store that sells 4 types of footwear: sneakers, pumps, high heels, and sandals. How many ways are there to buy 6 pairs of footwear, when we buy AT LEAST one of each kind? We assume that the store has at least 6 of each kind of footwear. Show your work and explain and justify your answer.Sessa, an ancient wise man, visited the king of his nation and showed him a new game. The board was 8 squares by 8 squares. Two players would be given 16 pieces each, lined up on either side of the board. The pieces moved in set ways, and the purpose of the game was to move your pieces to trap the King piece of the other player. As you have probably figured out, Sessa invented chess. His king was utterly delighted at the game - war without bloodshed! - and asked Sessa what he would like in return. In the ancient bombastic style, the King said the request could include up to half of his kingdom. (That was not meant literally.) Sessa asked for the game board to be brought before him. "Highness," he said, "I ask for a grain of rice on the first square. Tomorrow, two grains of rice on the second square. The day after, 4 grains of rice on the third square. On the fourth day, 8 grains of rice. And so on, until the board is full in two months." The king and his ministers laughed…
- MilanbhaiFour men and a monkey spend the day gathering coconuts on a tropical island. After they have all gone to sleep at night, one of the men awakens and, not trusting the others, decides to take his share. He divides the coconuts into four equal piles, except for one remaining coconut, which he gives to the monkey. He then hides his share, puts the other piles together, and goes back to sleep. Each of the other men awakens during the night and does likewise, and every time there is one coconut left over for the monkey. In the morning all the men awake, divide what's left of the coconuts into four, and again there is one left over that is given to the monkey. Find the minimum number of coconuts that could have been in the original pile.An experiment is designed to see whether 3rd graders can write faster with a pen or pencil. Four third graders participate, 2 boys and 2 girls. For each child, a marble is drawn without replacement from a bucket containing 2 red marbles and 2 blue marbles. If a red marble is selected, the child gets a pen and if blue, the child gets a pencil. The children are assigned in this order: boy 1, girl 1, boy 2, girl 2. LetX be the number of boys assigned to a pen. Find E (X). Find SD(X). а. b.
- A tennis tournament has 85 participants. Players who lose a game are immediately eliminated; players who win a game keep playing. Still, the organizers have a lot of choices to make. They could give a first round bye to some players so that after the first round there are 26 = 64 players and no more bye are needed. Or they could give even a second round bye to the best players, or possibly even a third round bye to the very best ones. What is the best strategy for the organizers if they want to choose the winner of the tournament using as few games as possible?A shoe company wants to determine if the new tread on its top line of running shoes lasts longer than the original tread. The company recruits 50 runners for a study. Each runner will perform their typical workout wearing one shoe with the original tread on one foot and another shoe with the new tread on the other foot. The foot that wears the new type of tread will be decided by flipping a coin. After one month, the runner will wear the new type of tread on the opposite foot. At the end of the second month, the difference in tread wear (New - Original) will be calculated. What type of sampling is described in this study? What is the appropriate inference procedure? one-sample t-test for μ one-sample z-test for p one-sample t-test for diff two-sample t-test for new originalSessa, an ancient wise man, visited the king of his nation and showed him a new game. The board was 8 squares by 8 squares. Two players would be given 16 pieces each, lined up on either side of the board. The pieces moved in set ways, and the purpose of the game was to move your pieces to trap the King piece of the other player. As you have probably figured out, Sessa invented chess. His king was utterly delighted at the game - war without bloodshed! - and asked Sessa what he would like in return. In the ancient bombastic style, the King said the request could include up to half of his kingdom. (That was not meant literally.) Sessa asked for the game board to be brought before him. "Highness," he said, "I ask for a grain of rice on the first square. Tomorrow, two grains of rice on the second square. The day after, 4 grains of rice on the third square. On the fourth day, 8 grains of rice. And so on, until the board is full in two months." The king and his ministers laughed…
- A monkey has filled in a 3 × 3 grid with the numbers 1, 2, . . . , 9. A cat writes down thethree numbers obtained by multiplying the numbers in each horizontal row. A dog writesdown the three numbers obtained by multiplying the numbers in each vertical column.Can the monkey fill in the grid in such a way that the cat and dog obtain the same listsof three numbers? What if the monkey writes the numbers 1, 2, . . . , 25 in a 5×5 grid? Or1, 2, . . . , 121 in a 11 × 11 grid? Can you find any conditions on n that guarantee that itis possible or any conditions that guarantee that it is impossible for the monkey to writethe numbers 1, 2, . . . , n^2 in an n × n grid so that the cat and the dog obtain the same lists of numbers? Please provide a mathematically rigorous solution with proofs that could be understood by a person at a high school level.A shoe company wants to determine if the new tread on its top line of running shoes lasts longer than the original tread. The company recruits 50 runners for a study. Each runner will perform their typical workout wearing one shoe with the original tread on one foot and another shoe with the new tread on the other foot. The foot that wears the new type of tread will be decided by flipping a coin. After one month, the runner will wear the new type of tread on the opposite foot. At the end of the second month, the difference in tread wear (New – Original) will be calculated. What type of sampling is described in this study? What is the appropriate inference procedure? one-sample t-test for one-sample z-test for p one-sample t-test for two-sample t-test forOne hundred extremely intelligent male prisoners are imprisoned in solitary cells and on death row. Each cell is soundproofed and completely windowless. There is a separate room with one hundred small boxes numbered and labeled from 1 to 100. Inside each of these boxes is a slip of paper with one of the prisoners' names on it. Each prisoner's name only appears once and is in only one of the one hundred boxes. The warden decides he is going to play a game with all of the prisoners. If they win, they will all be let free, but if they lose the game, they will all be immediately executed. The hundred prisoners are allowed to enter this separate room with 100 boxes in any predetermined order they wish, but each can only enter the room once and the game ends as soon as the hundredth person enters the room. (At any time, only one prisoner is allowed to enter and remain in this room.) Once a prisoner enters the room, he is allowed to open and look inside as many as Xboxes, where X is a…