Firstly, a manifold is an object or a generalisation of a space that is locally like Euclidean space. Poincaré ConjectureĮvery closed, simply connected, 3-manifold is homeomorphic to the 3-sphere. I will be presenting this conjecture (now theorem) first and then the remaining unsolved problems in order of increasing complexity. Grigori Perelman presented the proof in 2003 and he was officially awarded the Millenium Prize in 2010 which he declined. A few things will be assumed (like knowledge of groups and complex plane) but everything that I think is ‘new’ will be explained.Īt the time I am writting this, only the Poincaré conjecture has been solved. After a lot of umming and ahing I decided to present it at an elementary university level. Every day, the patch doubles in size.I have been wanting to write this article for quite some time, but struggled to decide at what level I should present the material. In a lake, there is a patch of lily pads. That means that once you solve everything inside the parenthesis and simplify the exponents, you go from left to right no matter what. PEMDAS says to solve anything inside parentheses, then exponents, and then all multiplication and division from left to right in the order both operations appear (that's the key). Which means they simplify the problem as follows: 6÷2(1+2) = 6÷ 2(3) = 6÷6 = 1.īut just because a number is touching a parentheses doesn't mean it should be multiplied before division that's to the left of it. Some people think that anything touching a parentheses should be solved FIRST. But the thing about PEMDAS is, some people interpret it different ways and in there lies the controversy behind this problem. Some people are POSITIVE the answer is 1 and some people are absolutely sure the answer is 9.Įxplanation: The handy order of operations rule you learned in grade school, PEMDAS, says you should solve a problem by working through the Parentheses, then the Exponents, the Multiplication and Division, followed by Addition and Subtraction. The masses are split on the answer to this stumper. Still confused? Let the genius UC Berkeley math professor Lisa Goldberg explain it even better with a bunch of diagrams! Sure, you aren't guaranteed to win if you switch, but if you play the game over and over, you'll win 2/3rds of the time using this method! So, basically, by switching your door choice, you're betting on the 2 in 3 chance you picked the wrong door at first. There's still a 1 in 3 chance you picked the right door and a 2 in 3 chance you picked an empty door, which means that when the host opened one of the empty doors, he eliminated one of the WRONG choices and the chances that the prize is behind the last closed door is still 2 in 3 - double what the chances you picked the right door at first are. In actuality, your chances never changed. What people get wrong here is thinking that because there are only two doors left in play, you have a 50% chance your first choice was correct. The Explanation: When you first picked one of the three doors, you had a 1 in 3 chance of picking the door with the prize behind it, which means you had a 2 in 3 chance of picking an empty door. The Answer: You should always switch your choice! Most people think the choice doesn't matter because you have a 50/50 chance of getting the prize whether you switch or not since there are two doors left, but that's actually not true! So, is it to your best advantage to stick with your original choice or switch your choice? He then says to you, "Do you want to stick with your choice or switch?" You pick door #1, and the host, who knows what's behind the doors, opens another door, say #3, and it has nothing behind it. Imagine you're on a game show, and you're given the choice of three doors: Behind one door is a million dollars, and behind the other two, nothing. If you take a moment to actually do the math, the only way for the bat to be a dollar more than the ball AND the total cost to equal $1.10 is for the baseball bat to cost $1.05 and the ball to cost 5 cents. But the mistake there is that when you actually do the math, the difference between $1 and 10 cents is 90 cents, not $1. Was your answer 10 cents? That would be wrong!Įxplanation: When you read the math problem, you probably saw that the bat and the ball cost a dollar and ten cents in total and when you processed the new information that the bat is a dollar more than the ball, your brain jumped to the conclusion that the ball was ten cents without actually doing the math.
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