Sieve of Erastothenese

Ok, you guys. Sorry I haven't posted in a LONG TIME. I have just been really busy. I am going to get you guys started on math history. You know about primes and composites right? Well, Erastothenese, a famous mathematician, figured out something that would change our lives FOREVER. He figured out how to do a number sieve. This sieve enables you to find prime numbers easily. Start out with a line of numbers starting with 2 and ending in any number, like this:
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
Then, erase or delete all of the numbers that are divisible by 2, besides 2. It should look like this:
2 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Now, go to the number 3. Do not delete 3, but find all of the numbers divisibe by 3 in you line. It should look like this:
2 3 5 7 11 13 17 19 23 25 27 29
See how I am doing this? Now go to the number five and do the same thing. You don't find the multiples of the crossed off numbers. When you are finished finding all of the multiples of the numbers, it should look like this:
2 3 4 5 7 11 13 17 19 23 29
The prime numbers would be 2,3,5,7,11,13,17,19,23,and 29, because they aren't divisible by any of the numbers less than them. And if any of those numbers look like composites to you, then there must be something wrong with your brain. JK!! Thats it. What? Why isn't this more complicated? Well, you would have to go back in time to ask Erastothenese that question, because I have absolutely no idea. What? What's that you say? You don't have a time machine?! That's crazy! Everyone has a time machine nowadays. JK!! Bye.

Logic Problems

Three cannibals and three anthropologists have to cross a river.
The boat they have is only big enough for two people. The cannibals will do as requested, even if they are on the other side of the river, with one exception. If at any point in time there are more cannibals on one side of the river than anthropologists, the cannibals will eat them.
What plan can the anthropologists use for crossing the river so they don't get eaten?
Note: One anthropologist can not control two cannibals on land, nor can one anthropologist on land control two cannibals on the boat if they are all on the same side of the river. This means an anthropologist will not survive being rowed across the river by a cannibal if there is one cannibal on the other side.



Three men in a cafe order a meal the total cost of which is $15. They each contribute $5. The waiter takes the money to the chef who recognizes the three as friends and asks the waiter to return $5 to the men.
The waiter is not only poor at mathematics but dishonest and instead of going to the trouble of splitting the $5 between the three he simply gives them $1 each and pockets the remaining $2 for himself.
Now, each of the men effectively paid $4, the total paid is therefore $12. Add the $2 in the waiters pocket and this comes to $14.....where has the other $1 gone from the original $15?



A mother is 21 years older than her child. In exactly 6 years from now, the mother will be exactly 5 times as old as the child.
Where's the father?



You are trapped in a room with two doors. One leads to certain death and the other leads to freedom. You don't know which is which.
There are two robots guarding the doors. They will let you choose one door but upon doing so you must go through it.
You can, however, ask one robot one question. The problem is one robot always tells the truth ,the other always lies and you don't know which is which.
What is the question you ask?



Cathy has six pairs of black socks and six pairs of white socks in her drawer.
In complete darkness, and without looking, how many socks must she take from the drawer in order to be sure to get a pair that match?


Now those are just the easy ones! Do you see how to work logic? Logic isn't like other math, you can't explain logic. Here are really hard ones:

You have twelve coins. You know that one is fake. The only thing that distinguishes the fake from the real ones is that its weight is imperceptibly different. You have a perfectly balanced scale that only tells which side weighs more than the other side.
What is the smallest number of times you must use the scale in order to always find the fake coin?
Use only the twelve coins themselves and no others, no other weights, no cutting coins, no pencil marks on the scale. etc.
These are modern coins, so the fake coin is not necessarily lighter.
Presume the worst case scenario, and don't hope that you will pick the right coin on the first attempt.



If you can figure out this problem and comment your answer, I will recognize you in a post afterwards.

I ask Alex to pick any 5 cards out of a deck with no Jokers.
He can inspect then shuffle the deck before picking any five cards. He picks out 5 cards then hands them to me (Peter can't see any of this). I look at the cards and I pick 1 card out and give it back to Alex. I then arrange the other four cards in a special way, and give those 4 cards all face down, and in a neat pile, to Peter.
Peter looks at the 4 cards i gave him, and says out loud which card Alex is holding (suit and number). How?
The solution uses pure logic, not sleight of hand. All Peter needs to know is the order of the cards and what is on their face, nothing more.

Answers

Cross Multiplying (the Bowtie Method)

Cross multiplying is when you have a problem like this:

The following information will be needed to solve the following problem:
Ms. Hamilton's eighth grade class wants to participate in the annual three-person-team basketball tournament.

After Sally takes 20 shots, she has made 55% of her shots. After she takes 5 more shots, she raises her percentage to 56%. How many of the last 5 shots has she made?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

What would 55% be out of 100? How can you put that into a fraction? Like this. 55/100. Now that was the easy part. If 55/100 is the amount of shots she made, how many shots did she make? Think about it. First, 50%=x. What is x? What is 1/2 of 20? 10. Now, we still have 5% more. If there is 100% all together, how many times can you put 20 into 100? 5 times. In that case, 5% is how much 1 shot is worth. You just used logic! So that would be 1 more shot, so Sally made 11 shots. So the fraction now is 11/20. So 55/100=11/20. Now we need to find out how many out of 5 shots Sally made. If you put 5 and 20 together, what do you get? 25. your problem is: 56/100=x/25. We need to know what x is. So what do you need to do? Multiply 56 by 25. The answer to that is what? 1400. so now we know that 1400=100x. Why do we need to know that? Because we need to know what x is, and we have to know what we need to multiply 100 by to get 1400, because whatever we multiply 100 my to get 1400 = x. What x100=1400? 14! You have your answer! This is how you would write it: 1400=100x 14=x. Now, how many more shots did she make than before? We know that she made 14/25 shots, but how many more did she make? 3! 3 is your answer. So, 56/100=14/25.

Do you get it now? Here is another example of cross-multiplying:


The problem that goes with this is: Handy Aaron helped a neighbor 1 1/4 hours on Monday, 50 minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and for a half-hour on Friday. He was payed $3.00 per hour. How much moola did he earn that week?
(A) $8 (B) $9 (C) $10 (D) $12 (E) $15

The correct answer is $15. Can you make up another problem this could be used for? If you can, post a comment with your idea and you will get recognition in the next post!

cheese

this reprisents me-i like cheese! this has nothing to do with math. i mean, i dont want you to find the area of the cheese or something. or do i...

Tessellations

Tessellations are shapes put together with no gaps and holes between them. For example:

That is a very complex tessellation. A more simple one would be:

Escher is one of the most famous people who does tessellations. He did this one:

You can also do tessellations with squares, triangles, hexagons and rectangles. There are many types of tessellations, but I will show you the main 2.

The easiest one is the slide method. Begin with either a rectangle or square. You can also use any other tessellating shape that has parallel sides for ALL SIDES. You will need a pencil or pen, scissors, tape, a sheet of blank computer paper, and an exact square or rectangle. Note: The shape must be exact and made of paper! Take your pencil and draw a random squiggly line on one side from corner to corner that is exact! Then, cut out what you drew and keep it where you cut it out from. Then, slide it over to the other side so that it does not change direction. tape it on, then do again for the other side(s). Now, you will need a sheet of paper. You trace your tessellating figure onto the paper, and do it until there is no more space left on the paper. You just made a tessellation!

Another kind of tessellation is called a 'Rotation About a Midpoint'. This time you MUST HAVE a square or rectangle. You fold the rectangle or square in half EXACTLY. Then, you draw a random squiggly line from the fold to the corner of that same side. then, cut out your squiggly line as before, and then rotate it on the midpoint(the fold). Tape it there, then do same procedure again until all of the sides are covered up. You may also combine the slide method with it if you are having trouble. Then to make the tessellation, you do the same thing as the slide. But this time, you will have to rotate your tessellating figure to finish the tessellation.