IMP 2 POW 3: Divisor Counting.
I. Problem statement:
This POW is all about finding information and patterns about the way divisors of certain numbers are found and expressed. In this POW when we talk about divisors we usually are counting the number of divisors that a number has. The divisor is a number that a number can be divided by, of course every number is divisible by every other number but in these problems we are only talking about whole, positive numbers. Every number is divisible evenly by one or itself so every number has at least 2 divisors. Numbers that have only 2 divisors are called prime numbers, only uneven numbers can be prime numbers, all even numbers except two have at least 3 divisors, the number itself, 1 and 2. the task was to find information about different questions about divisors, to find patterns and to make our own question. II. Questions:

What kinds of numbers have exactly 3 divisors?, 4 divisors? and so on. Do bigger numbers necessarily have more divisors?
Is there a way to figure out how many divisors one million has, how many one billion has and so on without actually counting all the different ones? (in other words is there a system for counting divisors) What is the smallest number with 20 divisors?

Choose your own interesting question: Do all the rules for divisors work the same with negative numbers? III. Information gathering
For most of the questions I found the divisors or made generalizations based on other information I gained via other numbers divisors or I used logic to figure out simple rules to apply to the numbers. I gathered many different simple rules and the number of divisors in some numbers, and my brother showed me the equation to find any numbers divisor. (It wasn’t at all useful because I have no idea what most of the stuff in the equation means and I can’t write it into Word.) When I found information that I knew was correct by either finding consistent or obvious patterns or by realizing it could...

...11/15/09
Class G
Lauren McCarthy
Pow3: Eight Bags of Gold
Problem Statement
A king divides his gold among 8 trusted people. One of the trusted people is selling his gold. The king wants to find the thief but only has a pan balance.
Being conservative, he wants to use the pan balance as few times as possible. What is the least number of trials he will have to do in order to guarantee
that he has found the lightest bag?
Process
To solve the problem, I...

...Touchdowns = 3 pts. I did the same as above.
3,6,9,12,15,18,21,24,27
I looked at the patterns and knew that if the team only scored field goals they would go up by 5 pts. each score. If the team only scored touchdowns they would go up by 3 pts. each score. I decided that I was going to keep looking for patterns in the numbers for different combanations ex. one field goal the rest touchdowns. To keep track of the patterns i was going to make a chart 1-100 of all...

...POW 8
Problem Statement-
For this POW, our task was to find the best formula for finding the area of any polygon that is formed on a geoboard. In order to do this, there are two formulas given to help you. One tells how to get the area of a polygon based on the number of pegs on the boundary. This works as an In-Out table, where In is the amount of pegs on the boundary, and Out is the area. The other formula tells how to get the area by having a polygon...

...Problem Statement
My task was to find 3 equations, that would give me an answer, if I had certain information. The first was to find one that if you knew that there were four pegs on the boundary, and none on the interior, you could get the area. The second was if you knew that there were 4 pegs on the boundary, and you knew how many were on the interior, you could get the area. And last, if you had the number on the interior, and the number on the boundary, you could get the...

...POW 17- Cutting the Pie
Problem Statement-
If you were given a pie what is the maximum number of pieces you can produce from 4, 5, and 10 cuts? Keep in mind, that the slices do not have to be the same size and the cuts do not necessarily have to go through the center of the pie, but the cuts do have to be straight and go all the way across the pie. Include any diagrams you used to find the solution such as an In-Out table, or any patterns you found.
Process-...

...Pow2
Problem Statement:
There’s a standard 8 x 8 checkerboard made up by 64 small squares. Each square is able to combine with others squares to make other squares of different sizes. Our job is to find out how many squares there’s in total. Once you get all the number of squares get all the number of squares and feel confident with your answer you next explain how to find the...

...each part of the problem. When you know that one thing means you go on to the next part. When you figure out what that means you have to see how the two statements are related. If they are related then you can deduce a conclusion that makes sense.
2. Here are my conclusions for the 6 problems on page 7.
1. a. No medicine is nice
b. Senna is a medicine
Here I deduced that Senna is not a nice medicine. I think this because the first statement says that “no...

...“A Sticky Gum Problem” POW 4
Problem statement:
The next scenario is very similar. In this one, Ms. Hernandez passed a different gumball machine the next day with three different colors Once again her twins each want a gumball of the same color, and each gumball is still one cent. What is the most amount of money that Ms. Hernandez would have to spend in order to get each of her daughters the same color gumball?
In the last scenario, Mr. Hodges and his triplets pass...