Please help with a few questions!

[loca]

I've been going crazy with these few problems:

1. I have to find how many squares are in a figure. There are 9 squares. How do i go about solving the problem?

2. If you raise 3 to the 324th power, what is the units digits of the result?

3. Find the name of a state whose letters all have vertical and horizontal symmetry.

4. 50bananas= 20 coconuts
30 coconuts=12 fish
100fish=1hammock
how many banans=1hammock?

5. volumes 1 and 2 of the complete works of wally smart are standing in numerical order from left to right. volume 1 has 450 pages and volume 2 has 475 pages. excluding the covers, how many pages are between page 1 of volume 1 and page 475 of volume 2?

Thanks in advance for you help!!!!

 

[stapel]

1. I have to find how many squares are in a figure. There are 9 squares. How do i go about solving the problem?

What is the actual problem? What are you supposed to do with the squares?

2. If you raise 3 to the 324th power, what is the units digits of the result?

What is the units digit of 3⁰? Of 3¹? Of 3²? Of 3³? Keep going until you find a pattern. Then figure out where "324" fits in that power pattern.

3. Find the name of a state whose letters all have vertical and horizontal symmetry.

Written in all capital letters? You're probably supposed to make that assumption.... Just look at a listing of the fifty states, and see which one fits.

4. 50 bananas = 20 coconuts, 30 coconuts = 12 fish, and 100 fish = 1 hammock. how many bananas = 1 hammock?

Work through the list: How many fish equal one hammock? How many coconuts is that? How many bananas is that?

5. volumes 1 and 2 of the complete works of wally smart are standing in numerical order from left to right. volume 1 has 450 pages and volume 2 has 475 pages. excluding the covers, how many pages are between page 1 of volume 1 and page 475 of volume 2?

Take two books off the shelf. Note where the beginnings and endings are. Then put them back on the shelf, noting where their beginnings and endings go, relative to each other. (There's a trick to this question, so go to a bookshelf to figure it out.)

Eliz.

 

[Denis]

1. I have to find how many squares are in a figure. There are 9 squares. How do i go about solving the problem?

How many if the figure had 4 squares?
4 plus the bigger square by itself, which means 5, right?

Now LOOK at your figure with 9 squares:
9 small squares, right?
1 big square, right?
Now how many 2by2 squares can you see? I see 4; do you?

 

[stapel]

So this isn't nine separate squares, but one big square divided into a three-by-three grid? That possibility should have occurred to me.

Eliz.

 

[TchrWill]

1. I have to find how many squares are in a figure. There are 9 squares. How do i go about solving the problem?
Not quite sure whether you mean a square measuring 3 by 3 squares or a square measuring 9 squares on each side. Either way, the following will enable you to determine the total number of squares within a given square disected figure, such as a chessboard.

If you determine the number of squares on smaller boards starting with one square you will readily discover a pattern that leads to a simple formula for a board of any number of squares.
A one square board obviously has only one square.
A 2x2 square board has 5 squares, the 4 basic ones and the one large 2x2 one.
A 3x3 square board has 14 squares, the smaller 9 plus 4 2x2's plus 1 3x3 one.
A 4x4 square board has 30 squares, the smaller 16 plus 9 3x3's, plus 4 2x2's plus 1 4x4 one.
Are you beginning to see the pattern?
1x1 - 1^2 = 1
2x2 - 2^2 + 1^2 = 5.
3x3 - 3^2 + 2^2 + 1^2 = 14.
4x4 - 4^2 + 3^2 + 2^2 + 1^2 = 30.
5x5 - 5^2 + 4^2 + 3^2 + 2^2 + 1^1 = 55.

What would your guess be for the number of squares on an 8x8 board? Can you derive a general expression for the answer? If not, see below.

Let N = the total number of squares in a square of nxn squares. Then

N = n^2 + (n-1)^2 + (n-2)^2................(n-n+1)^2 = n(n + 1)(2n + 1)/6

So for the typical chess board problem with 8x8 squares, the total number of definable squares is

N = 8^2 + 7^2 + 6^2 + 5^2 = 4^2 + 3^2 + 2^2 + 1^2 = 8(9)(16+1)/6 = 204.


2. If you raise 3 to the 324th power, what is the units digits of the result?

calculating the first 8 cubes of 3 you will find that the units digit repeats in the pattern of 3-9-7-1-3-9-7-1-etc.

Therefore, 324/4 = 81 meaning that the units digt of 3^324 is 1.