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The presentation should cover:
- 1. your methods used in solving a problem
- 2. the necessary mathematical knowledge needed to solve your problem?
- 3. the solution to the problem.
The presentation may be in the form of a paper or a slideshow (as in powerpoint or appleworks or ClarisWorks).
1. The Pythagorean theorem states that the sum of two numbers, each squared, equals a third number squared. If the smallest number is odd, can you find a pattern to predict the second and third numbers?
2. Create the 4 x 4 and the 5 x 5 magic squares. What is the pattern for all magic squares? There is so much information about magic squares on the web. You might want to start at: the Math Forum's site on magic squares
(http://mathforum.org/alejandre/magic.square.html)
3. How do you multiply magic squares and is the result another magic square?
4. What is a magic star, how is it generated, and what is its relation to magic squares.
5. Write down a 3 digit number. Write the number then in reverse order. Subtract the smaller of the two numbers from the larger to obtain a new number. Write the new number down. Reverse the digits again, but add these last two numbers. Do this process for another three digit number. Do you get a pattern, and does your pattern work for all three digit numbers?
6. You have 12 balls and a balance scale which only tells you whether one side is heavier. All but one of the balls are identical in weight. The oddball is either heavier or lighter than all the rest. You are allowed three weighings to discover which is the oddball and whether it is heavier or lighter. How do you do it?
7. In a certain high school, there were 1000 student lockers and 100 students. Each year on Homecoming Day, the students lined up in alphabetical order and performed the following strange ritual. the first student opened every locker. The second student closed every second locker. The third student changed every third locker (that is: opened every closed locker and closed the open ones). The fourth student changed every fourth locker, and so on. After all 1000 students had passed, which lockers were open?
8. The Ladies' Diary for 1739 -1740 had this following problem. There were three Dutch man named Henry, Ely, and Cornelius. Each was married, and the names of the women were Gertrude, Catharine, and Anna. Each person bought as many pigs as he or she paid dollars for each pig, and each man spent $63 more than his wife. Henry bought 23 more pigs than Catharine, and Ely bought 11 more than Gertrude. What was the name of each man's wife?
9. Amina entertained her female relatives at a Labor Day luncheon. Present were 1 great-grandmother, 2 grandmothers, 1 great-aunt, 4 mothers, 3 aunts, 3 first cousins, 4 sisters, 65 daughters, 4 nieces, 4 granddaughters, 1 grandniece, 2 first cousins once removed and one great-grandmother. How many people were at the luncheon?
10. The number six has four divisors: namely 1, 2, 3, and 6. List all the numbers less than twenty that have only four divisors. Then find a pattern or patterns in terms of prime numbers to predict or to describe all numbers that have only four divisors.
11. An interesting variation on the Crossing the River Problem is the following:
The five Family family members and their five dogs (each family member owned one of the dogs) were hiking when they encountered a river to cross. They rented a boat that could hold three living people or dogs. Unfortunately, the dogs were temperamental. Each was comfortable only with its owner and could not be near another person, not even momentarily, unless the owner was present. Dogs could be with other dogs, however. The crossing would have been impossible except that Lisa's dog has attended a first-rate obedience school and knew how to operate the boat. No other dogs were that well-educated.How was the crossing arranged and how many trips did it take?
from the new book Problem Solving Strategies, Chapter 11, Emeryville, CA: Keypress, 2002
12. Create the networks which correspond to the five Platonic solids. Are they all traversable? Do they have Euler circuits? Hamiltonian circuts?
13. One mathematical question is given any rectangle, can it be tiled by non-congruent squares. A rectangle can be tiled by using the following squares with sides:
18, 15, 14, 8, 1, 9, and three others.What are the lengths of the sides of the other three squares?
14. The Fibonacci Sequence has a simple rule, start with 1, 1 as the first two terms and then the next term is obtained by adding the previous two terms. This sequence was originally studied in the 13th century, and it continues to be of interest today.
1. Show that the square of any term differs by one from the product of its two adjacent terms.2. Show that the product of two adjacent terms differs by one from the product of the two terms preceding and following these terms.
3. Create your own sequence following the same rule but let the first two numbers be different from each other and different from one.
15. Determine how many zeros end the number 100!
16. One of the most famous unsolved problems is: are there infinitely many Fibonacci numbers that are prime. Research information about this problem and what Fibonacci numbers are known to be prime.
17. A service facility requires 2 minutes to service a customer. A new customer arrives every 5 minutes. when the service facility begins operation, there are 6 customers waiting and the first new customer arrives 1 minutes later. Which customer would be the first to arrive and find no queue? What is the answer for the general case (i.e. when variables are substituted for the numbers ?)
18. A social psychologist was interested in the custom of handshaking. He noticed that some people are more inclined than others to shake hands when they are introduced. One evening when he and his wife had joined four other couples at a party, he took advantage of the occasion to collect data. He asked each of the other nine people at the party how many people they had shaken hands with during the introductions. He received a different answer, from zero through eight, from each of the nine people. You can assume that husbands and wives do not shake hands with each other during the introductions, and, of course, people do not shake hands with themselves. Given this information how often did the psychologist's wife shake hands?
19. Three five-handed extra terrestrial monsters were holding three crystal globes. Because of the quantum-mechanical peculiarities of their neighborhood, both monsters and globes come in exactly three sizes with no others permitted: small, medium, and large. The medium-sized monster was holding the small globe; the small monster was holding the large globe; and the large monster was holding the medium-size globe. Since this situation offended their keenly developed sense of symmetry, they proceeded to transfer globes from one monster to another so that each monster would have a globe proportionate to its own size.
Monster etiquette complicated the solution of the problems since it requires that:1. only one globe be transferred at a time
2. if a monster is holding two globes, only the larger of the two may be transferred
3. a globe may not be transferred to a monster who is holding a larger globe
By what sequence of transfers could the monsters have solve this problem?
20. Tile an isosceles, right triangle with 6 similar, non-congruent triangles. Is 6 the smallest number of tiles needed?
21. There are ten stacks of gold bars with 10 bars and each stack. In 9 of the stacks, each bar weighs one pound, while in one stack each bar is exactly one ounce short of a pound. How could use an ordinary bathroom scale to determine which stack contains the light bars, if you were allowed only one weighing?
22. One day walking past an antique shop, you spot what you are certain, well, 80 percent certain, is an antique chair worth about $100. On the other hand, you recognize that there is a 20 percent chance that it is worthless. You know that the owner of the shop is reasonably competent in distinguishing antiques from junk. If it is antique, there is only one chance in 10 and that he will fail to recognize it. On the other hand, if it is junk, there is a very high probability, 0.98, that he will recognize it as junk. You enter the store and say, " how much for that chair?" the owner says, " $5.00 " Clearly, he thinks it is junk. Using this information, determine:
1) A new estimate of the probability that the chair is an antique2) Whether you should buy it or not.
23. A hot air balloon is coming to town. There are seventeen ropes holding the balloon to the ground. Take turns removing the ropes. Each person may take one, too, or three ropes. The person who has to take the last rope gets left behind. Can you find a way to get on the balloon every time? What happens if there are eleven ropes? Fifteen ropes?
24. The positive integers are arranged in groups as follows: (1), (2, 3), (4, 5, 6), (7, 8, 9, 10), and so forth with K integers in the Kth grouping. Find the sum of the integers in the Kth grouping.
25. There are five houses, five nationalities, five beverages, five animals, and five health foods. Given the following clues: who drinks Cutty Sark and who owns the zebra?
a) The Englishman lives in the red house.b) The Spaniard owns a dog.
c) Coffee is drunk in the green house.
d) The Ukrainian drinks tea.
e) The green house is immediately to the right of the ivory house.
f) The granola eater owns snails.
g) Tofu is eaten in the yellow house.
h) Milk is drunk in the middle house.
i) The Norwegian lives in the first house on the left.
j) The person who eat sprouts lives in the house next to the person with the fox.
k) Tofu is eaten in the house next to the house where the horse is kept.
l) The person who eats raw fish drinks orange juice.
m) The Japanese eats nuts.
n) The Norwegian lives next to the blue house.
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