I hope you had a chance to play the clock solitaire game that was listed yesterday. When we were kids we laid out six cards face down in one horizontal row, then six cards in a row below that one, and a single card to the right of the two rows. The top row was labeled (in our head) Ace through Six; the bottom row was labeled Seven through King, but we skipped over the Jack. The card on the right was the Jack. We then dealt out all the cards evenly into the thirteen stacks, so each stack had four cards in it. You start the game by picking up the top card in the Jack pile. Whatever card it was, you slid it face up under that stack. So if you picked up a Five, you slid that Five under the Five stack and then picked up the top card in that stack. If that card was a Ten, you slid it face up under the Ten pile and took the top card. You kept doing that until you ran out of cards to turn over. Your goal was to get all the cards in the stacks turned over before the last Jack came up. When the last Jack came up, there were no more cards in the Jack pile to work with (since you started with that pile) so the game was over. There is roughly a 1% chance of winning so it was a huge victory when you did win.
That is similar to the strategy for the 100 Prisoners Problem. Each contestant walks to the box that corresponds to their number. Contestant 1 walks to the first box, opens it, then moves to the box whose number is on the dollar bill in the first box. So if the first box has a 67 on the dollar bill, you would go to Box 67 next. If that box has a dollar bill numbered 43 in it, you would then go to Box 43. You keep repeating this until the box you open has your number - in this case number 1. So now you know which box has dollar bill number 1 in it and you can leave. Contestant 2 walks to Box 2 and checks the number. Then he moves to the Box with that number and so on until he either finds the box with 2 in it or hits his 50 box limit. Contestant 3 starts with Box 3 and so forth.
Let's look at a much simpler version that has only 8 boxes and 8 contestants. The box numbers and dollar bill numbers are listed below. Assume that the limit on boxes you are allowed to open is 4 = half of 8, just like 50 = half of 100.
Box number 1 2 3 4 5 6 7 8
Number on dollar bill 4 7 5 2 6 3 1 8
inside
Where did those numbers come from? I made them up just as the designer of the puzzle would do when they put the dollar bills in the boxes.
Contestant 1 finds a 4 in Box 1 so they go to Box 4, then to Box 2, then to Box 7 and there they find the dollar bill numbered 1 and they leave. They opened 4 boxes, so the game is still going on. Contestant 2 starts at Box 2, then Box 7, Box 1, and finishes at Box 4 (where he finds his number 2 dollar bill). Contestant 3 starts at Box 3, then to Box 5, then Box 6 and they are finished. What we have are cycles of numbers:
(1, 4, 2, 7) is a cycle. If you start at one of these numbers, like 2, you will cycle through the other numbers in the group: 2 goes to 7 then to 1, then to 4. (3, 5, 6) is also a cycle and (8) is a cycle by itself. Because each cycle is 4 or smaller, each contestant will find their number successfully and everyone gets the prize money.
Here's another allotment of dollar bills.
Box number 1 2 3 4 5 6 7 8
Number on dollar bill 2 3 4 5 6 1 8 7
inside
Here the cycles are (1, 2, 3, 4, 5, 6) and (7, 8) and the contestants would lose because the first cycle is longer than 4 numbers.
With the 100 box problem, the contestants would win if the longest cycle of numbers is 50 numbers or less. There can't be more than one of those because there are only 100 numbers to divvy up into cycles. So there could be a 20 number cycle, a 30 number cycle, and a 50 number cycle and that would use up all the numbers (and you would win). The probability calculations (found here, if you would like to follow along) tell us that the longest cycle of numbers is 50 or less about 31% of the time. That is a huge jump from the random strategy probability of winning which was 0.0000000000000000000000000000008.
Showing posts with label math / science. Show all posts
Showing posts with label math / science. Show all posts
Thursday, December 11, 2014
Wednesday, December 10, 2014
The 100 Prisoners Problem and a New Card Game
Today's math lesson is about probability, one of my favorite topics. The problem we are looking at is couched in terms of executing prisoners, so we are going to look at it from a less violent point of view. I should point out that this (roughly 10 year old problem) is not an easy problem to solve and I am not giving it to you to see how good at math you are. I thought the solution was very interesting and was reminded of it when I was killing time playing solitaire the other day. So you are going to see some interesting math and learn a new card game in the process.
Suppose 100 people have been selected at random, lined up, and given a number based on their spot in line. First person in line is number 1, second is number 2 and so on until the last person is given number 100. The first person is sent into a room which has 100 boxes in a row, each containing a dollar bill numbered from 1 to 100 also. The person's job is to find the box with the dollar bill that matches their number. So person number 1 will keep opening boxes until they find the dollar bill that has the number 1 on it. Here's the catch. The person is only allowed to open 50 boxes. If, after 50 boxes they have not found their dollar bill, the game is over and everybody is sent away. If they find their dollar bill, they leave it in the box, exit the room out a back door, and person 2 is allowed into the room. Person 2 is now charged with opening up to 50 boxes until they find the dollar bill labeled number 2. And so on. If at any time a person fails to find their dollar bill, the game is over and everybody goes home. If everybody finds their dollar bill, then everybody gets a prize of $1000. So, everybody wins or nobody wins. A significant part of the rules is that although people can speak to each other before they enter the room, there is no communication of any kind between people who have been in the room and those still waiting. That includes doing any writing on the boxes, rearranging the boxes in any way, leaving messages on the wall, etc. Every person entering a room must have the same information about the room that everyone else had, namely that there are 100 boxes with numbered dollar bills. (Remember each person who finds their dollar bill leaves it in the box looking just as it did before.)
Now if person 1 enters the room and starts picking boxes at random, there is a 50% probability that they will find their dollar bill. Person 2 also has a 50% probability of finding their dollar bill. The probability of the first two people then finding their dollar bills is 25% - .5 times .5 is .25. For three people, it is (.5) to the third power. And the probability of everyone being successful and therefore everyone winning the prize is (.5)100 ≈ 0.0000000000000000000000000000008 which is pretty small. In fact there is a clever non-random way to choose the boxes that raises that probability of winning the prize to about 31%. So I will give you some time to think about it (and time for me to write up the math) and come back tomorrow and post the solution here. If you would like a hint and, more importantly, a time-wasting card game to play when you are bored, look at the directions for "clock solitaire" listed here. This version isn't set up the way we played it as kids, but it is the same game. Play a few rounds and see if it leads you to a strategy for the 100 boxes game.
Suppose 100 people have been selected at random, lined up, and given a number based on their spot in line. First person in line is number 1, second is number 2 and so on until the last person is given number 100. The first person is sent into a room which has 100 boxes in a row, each containing a dollar bill numbered from 1 to 100 also. The person's job is to find the box with the dollar bill that matches their number. So person number 1 will keep opening boxes until they find the dollar bill that has the number 1 on it. Here's the catch. The person is only allowed to open 50 boxes. If, after 50 boxes they have not found their dollar bill, the game is over and everybody is sent away. If they find their dollar bill, they leave it in the box, exit the room out a back door, and person 2 is allowed into the room. Person 2 is now charged with opening up to 50 boxes until they find the dollar bill labeled number 2. And so on. If at any time a person fails to find their dollar bill, the game is over and everybody goes home. If everybody finds their dollar bill, then everybody gets a prize of $1000. So, everybody wins or nobody wins. A significant part of the rules is that although people can speak to each other before they enter the room, there is no communication of any kind between people who have been in the room and those still waiting. That includes doing any writing on the boxes, rearranging the boxes in any way, leaving messages on the wall, etc. Every person entering a room must have the same information about the room that everyone else had, namely that there are 100 boxes with numbered dollar bills. (Remember each person who finds their dollar bill leaves it in the box looking just as it did before.)
Now if person 1 enters the room and starts picking boxes at random, there is a 50% probability that they will find their dollar bill. Person 2 also has a 50% probability of finding their dollar bill. The probability of the first two people then finding their dollar bills is 25% - .5 times .5 is .25. For three people, it is (.5) to the third power. And the probability of everyone being successful and therefore everyone winning the prize is (.5)100 ≈ 0.0000000000000000000000000000008 which is pretty small. In fact there is a clever non-random way to choose the boxes that raises that probability of winning the prize to about 31%. So I will give you some time to think about it (and time for me to write up the math) and come back tomorrow and post the solution here. If you would like a hint and, more importantly, a time-wasting card game to play when you are bored, look at the directions for "clock solitaire" listed here. This version isn't set up the way we played it as kids, but it is the same game. Play a few rounds and see if it leads you to a strategy for the 100 boxes game.
Thursday, May 8, 2014
Our Incredibly Poor Sense of Time and History
The new version of Cosmos debuted last week on Fox TV on Sunday nights. The original version premiered on PBS in 1978 and was hosted by Carl Sagan. The new series is hosted by Neil Tyson and is a lot of fun to watch. We have seen over the last twenty years an incredible erosion in the country's attitude towards science at the same time we have seen an incredible explosion of new technology.
Distractify.com has this amazing picture of an ad for Radio Shack from 1991:
Distractify.com has this amazing picture of an ad for Radio Shack from 1991:
followed by "Everything in this Radio Shack ad from 1991 can now be found on a single smartphone." Go back and take a look at the ad. In the 80's and 90's, these were many of the things we had on our Christmas list: camcorder, clock radio, and a (CD) music player. Even the computer in the ad was probably less powerful than the computer on our smartphones. The guidance computer for the Apollo moon landing in 1969 had 1/6 of the processing power of a 10 year old TI-83 calculator.
In spite of the love of technology we have, we have lost our interest in science as a nation. The representatives on the House Science Committee are often the most anti-science people in Wahington.
From the National Memo's 2013 article about Paul Broun, representative from Georgia:
Broun, who has served on the Science Committee since 2007, appears to believe that scientists are literally tools of the devil. In an October speech at the Liberty Baptist Church Sportsman’s Banquet, Broun declared “All that stuff I was taught about evolution and embryology and the Big Bang Theory, all that is lies straight from the pit of Hell,” Broun said.
So it's no wonder that the funding for science has gone backward for some time. I understand the stance against the idea of global warming. Many of the congressmen are kept in office through the good graces of big oil and traditional energy sources, like coal, that would lose out big time if alternative energies are funded and subsidized. If you need an example close to home for me, keep in mind that the great state of Wisconsin is now considered by some (see Charles Pierce of Esquire magazine) to be a wholly owned subsidiary of Koch Industries, with a governor and state legislature who get their marching orders directly from the Koch brothers.
Today, radio astronomers announced new evidence that corroborates the idea of "inflation", the idea that for the obervations of the expanding universe to be as they are, there had to be a brief moment at the time of the Big Bang when the universe expanded faster than the speed of light. If so, there should be gravitational waves predicted by Einstein that should appear in the energy left over from the early universe. According to the New York Times today:
Reaching back across 13.8 billion years to the first sliver of cosmic time with telescopes at the South Pole, a team of astronomers led by John M. Kovac of the Harvard-Smithsonian Center for Astrophysics detected ripples in the fabric of space-time — so-called gravitational waves — the signature of a universe being wrenched violently apart when it was roughly a trillionth of a trillionth of a trillionth of a second old.
I don't think it's even possible for us to get our heads around the amazing science that is described in that paragraph.
In both the original Cosmos and the new version, the first show in the series ends with the Cosmic Calendar, a tool to help us understand our place in the Cosmos. It takes the roughly 14 billion years since the Big Bang occurred and marks it on one calendar year. If the Big Bang occurred on January 1, then the Milky Way Galaxy formed on March 15 and our sun formed on August 31. Think of the time span here - the year is almost exactly 2/3 done and the cosmos is just getting around to forming our sun. On December 14th, the first animals appear. The first mammals show up on December 20. And the first primitive humans show up on December 31 at 10:24 PM. The first use of writing appears on December 31 at 11:59 and 47 seconds PM. Sagan went on to make the comparison that if the entire Cosmic Calendar were scaled to the size of a football field, all of human history would occupy a space the size of your hand.
Humans have been the dominant species on earth for less than a million years. Dinosaurs, on the other hand were the dominant terrestrial vertebrates for 135 million years. We have quite a ways to go to catch up.
Note: I first started writing this almost two months ago and never quite finished it until now. The Cosmos show has been on for two months now and is a fascinating look at the universe. I would urge you to watch it if you get a chance.
Saturday, March 15, 2014
Earth Animals
Our thanks to the web comic XKCD for the following graphic about the total weight of various land mammals on the earth.
The black boxes in the middle represent all the human beings in the world by weight - each block represents 1 million tons. The grey blocks represent our pets and livestock. On the left are cattle, upper right are pigs, then goats, sheep, and horses as you continue around the corner to the bottom. The green blocks are wild animals. One green block is labeled elephants. The other few green blocks around the outside are, I suppose, rhinoceroses, hippos, tapirs, squirrels, and the like. Maybe the grey blocks at the top are our dogs. Whimzy has put on about 10 pounds this winter; I don't know if that will screw up the graphic. She hopes not.
It would seem from the graphic that we are a world of ranchers, caring for our domesticated animals for the most part. And very surprising the few wild animals on the earth. I'll ask Whimzy not to chase the squirrels so much. We can't do anything about the elephants. Whimzy almost never chases them.
The black boxes in the middle represent all the human beings in the world by weight - each block represents 1 million tons. The grey blocks represent our pets and livestock. On the left are cattle, upper right are pigs, then goats, sheep, and horses as you continue around the corner to the bottom. The green blocks are wild animals. One green block is labeled elephants. The other few green blocks around the outside are, I suppose, rhinoceroses, hippos, tapirs, squirrels, and the like. Maybe the grey blocks at the top are our dogs. Whimzy has put on about 10 pounds this winter; I don't know if that will screw up the graphic. She hopes not.
It would seem from the graphic that we are a world of ranchers, caring for our domesticated animals for the most part. And very surprising the few wild animals on the earth. I'll ask Whimzy not to chase the squirrels so much. We can't do anything about the elephants. Whimzy almost never chases them.
Wednesday, January 8, 2014
The Cowboys Are Working Overtime
On my Christmas list this year, I included an Arduino Starter kit. Luckily, Steve came through for me. The Arduino home page says
Arduino is an open-source electronics prototyping platform based on flexible, easy-to-use hardware and software. It's intended for artists, designers, hobbyists and anyone interested in creating interactive objects or environments. Arduino can sense the environment by receiving input from a variety of sensors and can affect its surroundings by controlling lights, motors, and other actuators.Some of the projects described on their website are a window blind that opens and closes automatically based on the sunlight hitting it, a car alarm system that sends you a message when your car door is opened, (for Mike) a turn signal biking jacket, and a self-balancing Segway-like robot. It's all about the sensors you hook up and the instructions you give in the programming.
Open source means that everything is out there for you to use and adapt as you want. You can download the software for free and launch it on the Arduino. The bluish thing on the right is the Arduino.
You hook it up to your computer through a USB port and you are off and running. I walked through the introduction, which described what all these parts that came with the starter kit do.
Mostly, they are lights and sensors and wires to connect everything. Now here is where the problem starts. I am not at all comfortable with the electronics side of this. Even though I don't know the programming language that is used, I do get programming and can follow a program to see what it does. I learned Fortran in college, mastered Basic while teaching math to write programs for computers and graphing calculators, and did a little Python in the last four years just because it looked interesting. But I still assume there are little cowboys in the wires driving those electrons along the trail to make electricity. And when the tutorial says it doesn't matter if I put the resistor before or after the blinking light, I don't get it. I understand plumbing, and plumbing doesn't work like that. So I am going to have to e-mail Steve a lot when I don't get the electronics part. (He's an electronics engineering trained person.)
On the plus side, I hooked up the computer, downloaded the software for the first tutorial (Blinking Light) and got the light to blink just like it said it would. I even modified the program to change how long the light stayed on and stayed off. So I have made a good start. Next up is Arduino Tutorial 2: LED Lights and Resistors. Ah, those pesky resistors. By next Christmas, I hope to have some operating robots to show off. If I put bar codes on the presents, the robots could decide who gets each present and deliver them around the living room. That would put Mike out of a job, but it would be really cool.
Friday, December 13, 2013
Bill Nye Bounces Back
So, Bill Nye may not have been the smoothest operator on Dancing with the Stars last month. He was stiff and his sense of humor didn't translate well to the non-geeky watchers. But it's nice to see him bounce back from his sad dismissal.
And, just for Mike and Nate, there is a website designed for them called Stuff On A Rabbit, which is exactly what it says it is. Makes you want to go out and buy a rabbit. (Or a box of goldfish.)
And, just for Mike and Nate, there is a website designed for them called Stuff On A Rabbit, which is exactly what it says it is. Makes you want to go out and buy a rabbit. (Or a box of goldfish.)
Friday, August 9, 2013
Texas? Really?
I know every state in the Union has issues of some sort or another. When it comes to my state, Illinois, I know that one of the things we lead the nation in is governmental corruption. At one point, I think we had three consecutive governors spending time in jail for various offenses committed while they were governor. Yay, Illinois. So it is hard, in general, for Illinois people to poke fun at other states. But this is just too much low hanging fruit to ignore.
A new University of Texas/Texas Tribune survey shows just how destructive a politicized right-wing curriculum can be. A large number of Texans polled said they still don’t believe in evolution and are convinced that humans and dinosaurs co-existed:
This would be comical if it weren't for the fact that these same people are again in charge of choosing science textbooks to use in their schools.The Texas State Board of Education is looking to set state standards for science teaching this year. And the creationists on the board are looking to set the children of Texas back a hundred years in their education. Most of the people involved in this tragedy think they are doing it for religious reasons. If so, their interpretation of the Bible is seriously flawed.
Take, for example, Kenneth Miller, a Roman Catholic professor of biology at Brown University and author of one of the nation’s leading high school textbooks. His biology textbook, fully in support of evolution, is among those the Texas state board is considering for adoption this year. Or consider Francis S. Collins, an evangelical Christian and head of the Human Genome Project. From a Time magazine piece about Collins in 2009:
What can you say to that?
I'll leave it to an "expert" to get in the last word. Warning, some explicit language is involved here. If it bothers you, you may want to skip this.
A new University of Texas/Texas Tribune survey shows just how destructive a politicized right-wing curriculum can be. A large number of Texans polled said they still don’t believe in evolution and are convinced that humans and dinosaurs co-existed:
– 51 percent disagree with the statement, “Human beings, as we know them today, developed from earlier species of animals.”
– 38 percent agree with the statement, “God created human beings pretty much in their present form about 10,000 years ago.”
– 30 percent agree with the statement, “Humans and dinosaurs lived at the same time.” Another 30 percent said they “don’t know” whether the statement is true.
This would be comical if it weren't for the fact that these same people are again in charge of choosing science textbooks to use in their schools.The Texas State Board of Education is looking to set state standards for science teaching this year. And the creationists on the board are looking to set the children of Texas back a hundred years in their education. Most of the people involved in this tragedy think they are doing it for religious reasons. If so, their interpretation of the Bible is seriously flawed.
Take, for example, Kenneth Miller, a Roman Catholic professor of biology at Brown University and author of one of the nation’s leading high school textbooks. His biology textbook, fully in support of evolution, is among those the Texas state board is considering for adoption this year. Or consider Francis S. Collins, an evangelical Christian and head of the Human Genome Project. From a Time magazine piece about Collins in 2009:
“Science can’t be put together with a literalist interpretation of Genesis,” he continues. “For one thing, there are two different versions of the creation story — in Genesis 1 and 2 — so right from the start, you’re already in trouble. Christians should think of Genesis not as a book about science but about the nature of God and the nature of humans," Collins believes. “Evolution gives us the ‘how,’ but we need the Bible to understand the ‘why’ of our creation.”You would think that the overwhelming (let me say that again - OVERWHELMING) basis for evolution would sway some people. Unfortunately, creationists have proudly rejected out of hand the scientific evidence behind evolution. They have reveled, as former Texas state board chairman Don McLeroy infamously said, in “standing up to experts.”
What can you say to that?
I'll leave it to an "expert" to get in the last word. Warning, some explicit language is involved here. If it bothers you, you may want to skip this.
Thursday, August 8, 2013
Today's Math Lesson: Braess's Paradox
So, it seems like this whole summer around Wauconda we have been practicing merging from two lanes down to one. A six mile stretch of Route 12 is in the process of being repaved and each day they block off a new section to fill in all the potholes. I assume they want the road fairly level before they install the new surface. During one of these interminable waits, I happened to recall a problem from network theory (with some connections to game theory) that I remember talking about with my advanced students fifteen or twenty years ago. It took a while to find it, but it turns out it appeared in the New York Times in December of 1990. It was probably used in a talk that I heard at a math conference sometime after that and I knew my students would be interested in it.
It is called Braess's Paradox, first described by Dietrich Braess of the Institute for Numerical and Applied Mathematics in Munster, Germany in 1968. To understand the paradox, we need the diagram below (from Wikipedia):
Assume that there are two roads leading from START to END, one that passes through A and one that passes through B. For the moment, ignore the dashed line that is a road from A to B. The travel time in minutes for the passengers on the START to A road is the number of passengers (T) divided by 100 and the travel time from START to B is a constant 45 minutes. As you can see the times are flipped for the next section of the trip (A or B to END). To do the math, let's assume there are 4000 passengers who want to make the trip from START to END. For A passengers, the START to A to END trip takes A/100 + 45 minutes, and the START to B to END trip for B passengers takes 45 + B/100 minutes. For any one person (who cannot by themselves affect what A or B is to any extent) both paths look pretty much the same. Since A + B = 4000, it can be shown that the optimal solution is to just randomly pick a path with the result that A = B = 2000 (roughly). Then the total time to travel from START to END would be 2000/100 (which is 20) + 45 for a total time of 65 minutes regardless of the path you picked. Even if you are a selfish driver, there is no incentive to pick one path over the other. (For you econ and game theory buffs, this is called a Nash equilibrium, after John Nash, the mathematician profiled in the movie A Beautiful Mind.)
Braess's paradox has to do with what happens when you try to ease the congestion by adding another road that runs from A to B that has a very short driving time (think 0 minutes for the purpose of the problem). Now when a single driver looks at the choices from START to the middle, even if all 4000 drivers go to A, the driving time is 40 minutes (4000/100), which is less than the 45 minutes to go from START to B. And similarly when they get to the middle, every driver sees that they can save five minutes by switching from from A to B. That makes each part of their drive 40 minutes for a total driving time of 80 minutes, 15 minutes longer than the time without the road from A to B. From a game theory viewpoint, when everyone in the game acts selfishly, then everyone suffers.
In an inverse way, that's what prompted the New York Times article. In the late 1980's, the Earth Day celebration in New York caused a lot of traffic congestion. In 1990, the Traffic Commissioner decided to lessen the car traffic congestion by closing one of the streets - in fact, a fairly major street, 42nd Avenue, which is always congested. Everyone thought it would be a disaster. In fact, traffic flow got better that year. Braess's paradox in action. When the streets are crowded, sometimes traffic improves when there are fewer streets to choose from.
From the Times article:
It is called Braess's Paradox, first described by Dietrich Braess of the Institute for Numerical and Applied Mathematics in Munster, Germany in 1968. To understand the paradox, we need the diagram below (from Wikipedia):
Assume that there are two roads leading from START to END, one that passes through A and one that passes through B. For the moment, ignore the dashed line that is a road from A to B. The travel time in minutes for the passengers on the START to A road is the number of passengers (T) divided by 100 and the travel time from START to B is a constant 45 minutes. As you can see the times are flipped for the next section of the trip (A or B to END). To do the math, let's assume there are 4000 passengers who want to make the trip from START to END. For A passengers, the START to A to END trip takes A/100 + 45 minutes, and the START to B to END trip for B passengers takes 45 + B/100 minutes. For any one person (who cannot by themselves affect what A or B is to any extent) both paths look pretty much the same. Since A + B = 4000, it can be shown that the optimal solution is to just randomly pick a path with the result that A = B = 2000 (roughly). Then the total time to travel from START to END would be 2000/100 (which is 20) + 45 for a total time of 65 minutes regardless of the path you picked. Even if you are a selfish driver, there is no incentive to pick one path over the other. (For you econ and game theory buffs, this is called a Nash equilibrium, after John Nash, the mathematician profiled in the movie A Beautiful Mind.)
Braess's paradox has to do with what happens when you try to ease the congestion by adding another road that runs from A to B that has a very short driving time (think 0 minutes for the purpose of the problem). Now when a single driver looks at the choices from START to the middle, even if all 4000 drivers go to A, the driving time is 40 minutes (4000/100), which is less than the 45 minutes to go from START to B. And similarly when they get to the middle, every driver sees that they can save five minutes by switching from from A to B. That makes each part of their drive 40 minutes for a total driving time of 80 minutes, 15 minutes longer than the time without the road from A to B. From a game theory viewpoint, when everyone in the game acts selfishly, then everyone suffers.
In an inverse way, that's what prompted the New York Times article. In the late 1980's, the Earth Day celebration in New York caused a lot of traffic congestion. In 1990, the Traffic Commissioner decided to lessen the car traffic congestion by closing one of the streets - in fact, a fairly major street, 42nd Avenue, which is always congested. Everyone thought it would be a disaster. In fact, traffic flow got better that year. Braess's paradox in action. When the streets are crowded, sometimes traffic improves when there are fewer streets to choose from.
From the Times article:
Dr. Joel E. Cohen, a mathematician at Rockefeller University in New York, says the paradox does not always hold; each traffic network must be analyzed on its own. When a network is not congested, adding a new street will indeed make things better. But in the case of congested networks, adding a new street probably makes things worse at least half the time, mathematicians say.Something to think about the next time you are stuck on Ohio Street, trying to get to the Garrett's popcorn store on Michigan Avenue.
Tuesday, August 6, 2013
Let's Save Some Lives ! !
August has become Blogust to the UN Foundation's Shot @ Life campaign. This marvelous campaign seeks to
She is joined in the campaign by Walgreen's, which is sending one vaccine oversees for each comment left to each of the 30 blogs done in August, up to a maximum of 50,000 vaccines.
A comment from Shannon as simple as Yay for vaccines
has earned a vaccine for a young boy or girl.
So send what you can. Comment on the blog posts during the month and help Walgreen's make good on its promise to sent 50,000 vaccines.
"educate, connect and empower the championing of vaccines as one of the most cost effective ways to save the lives of children in the world’s hardest to reach places. "Each day of this month, a guest blogger tells us why vaccination is important. Today it was Amanda Peet, an actress from one of my top ten rated TV shows (Let's go, Marshmallow Fight), Studio 60 on the Sunset Strip. Amanda in her post talks about her five year old daughter and what a health milestone worldwide 5 years old is.
She is joined in the campaign by Walgreen's, which is sending one vaccine oversees for each comment left to each of the 30 blogs done in August, up to a maximum of 50,000 vaccines.
A comment from Shannon as simple as Yay for vaccines
has earned a vaccine for a young boy or girl.I sent a $20 donation as well. My $20 "gives a child a lifetime of immunity to protect her from pneumonia, diarrhea, polio and measles."A child dies every 20 seconds from a vaccine-preventable disease. We can change this reality and help save kids’ lives! Stay connected with Shot@Life at www.shotatlife.org, join the campaign on Facebook and follow them on Twitter.
So send what you can. Comment on the blog posts during the month and help Walgreen's make good on its promise to sent 50,000 vaccines.
Sign up here for a daily email so you can quickly and easily comment every day during Blogust.
It really is an important step in helping that toddler make it to her fifth birthday party.
Thanks. I knew I could count on you.
Monday, March 25, 2013
Time to Move?
The Institute for Health Metrics and Evaluation has a map up on their website that shows the life expectancy for men and women from 1989 to 2009 on a county by county basis. That means you can look for your county and find out, on average, how much longer you've got based on where you live. For Lake County, Illinois (next to Lake Michigan and the Wisconsin border) the life expectancy for men is 79.1 years and for women 82.3 years. So I have about 18 years to go if I'm average. Although the number for Auglaize County in Ohio is 76.1 and my dad is 91. So maybe I still have some time to write that novel.
There are a number of interesting details on the map. You can stroll through the years one by one and see how things have changed for your county. When you click on your county, it also tells you how far behind the top county you are (2.5 years for me). The top county for men is Marin County, California. It's also interesting that counties in the northern half of the United States are nearly uniformly higher in life expectancy than those in the southern half, especially in the southeastern part of the U.S. Several counties in Mississippi are more than 14 years behind the top county. But the way I understand it, time moves more slowly down there, so maybe it all balances out.
I can't afford to move to the California coast to get those extra two and a half years. By that time the Illinois state legislature will have eliminated my teacher pension completely and we'll be living with Nate or Mike (probably on a rotating basis - No! It's your turn), so I'll leave it up to them.
There are a number of interesting details on the map. You can stroll through the years one by one and see how things have changed for your county. When you click on your county, it also tells you how far behind the top county you are (2.5 years for me). The top county for men is Marin County, California. It's also interesting that counties in the northern half of the United States are nearly uniformly higher in life expectancy than those in the southern half, especially in the southeastern part of the U.S. Several counties in Mississippi are more than 14 years behind the top county. But the way I understand it, time moves more slowly down there, so maybe it all balances out.
I can't afford to move to the California coast to get those extra two and a half years. By that time the Illinois state legislature will have eliminated my teacher pension completely and we'll be living with Nate or Mike (probably on a rotating basis - No! It's your turn), so I'll leave it up to them.
Tuesday, March 12, 2013
3/14 is Pi Day - Let's Celebrate
Every year, math teachers around the world celebrate Pi Day on March 14 (as in 3/14 or 3.14).We sometimes say that pi is 3.14, but we really mean that pi to two decimal places is 3.14. In fact pi doesn't have a decimal expansion that terminates (1/2 = .5) or repeats (1/3 = .33333 repeating forever). Pi is an example of an irrational number - it cannot be expressed as a ratio of two integers. There are some interesting irrational numbers:
i) pi - the ratio of the circumference of a circle to its diameter (about 3.14);
ii) e - the base for the natural logarithms and an important constant in population growth (about 2.718);
iii) the square root of 2 (about 1.414);
but there are a lot of irrational numbers that are just random collections of digits without any particular purpose. That would be OK if it didn't turn out that there are so many of them. In the late 1800's German mathematician George Cantor was turning mathematics upside down with his study of infinite sets. He stated that even though we can't count all the elements in an infinite set, we can assign them a "count" by putting them into a one - to - one correspondence with another set. From there he postulated that infinite sets come in different sizes. The smallest infinite set is the set of natural numbers - 1, 2, 3, 4, and so on. He used the Hebrew symbol aleph for counting infinite sets and assigned the size of the natural numbers to aleph null (here null is like zero).
That's where it started to get a little weird. Unlike finite sets, infinite sets don't behave well when you add elements. For example, suppose we looked at the set of integers: . . ., -3, -2, -1, 0, 1, 2, 3, . . . You might think this set is roughly twice as big as the natural numbers. But Cantor said that this set is the same size as the natural numbers. And he showed it by matching up the two setsnaturals 1 2 3 4 5 6 7 8 9 10 11 . . .
Neither set is going to run out before the other one does, so there is a one - to - one correspondence between the two sets. Therefore the sets are the same size. Cantor went on to show that the set of all rational numbers, i. e., fractions like 2/3, 17/5, -6/11, and so on, is the same size as the natural numbers as well. At that point, we might be forgiven if we think that, OK, all infinite sets of numbers are the same size. It turns out the irrational numbers are a bigger set than aleph null. Here's how we show that. The key to matching up to the natural numbers is to put your set in some kind of order. If you look up a few lines, you can see that's what we did with the integers. We found a way to list them in an order that could then be matched up with 1, 2, 3, 4, and so on. Suppose we had some kind of order for all the irrational numbers.
first: .328123986512970245 and so on
second: .776823919374529476 and so on
third: .554376920472539402 and so on
fourth: .019283524377253628 and so on
It doesn't matter what the order is, just assume we have one and every irrational number is somewhere on the list. We can now make a new number that can't be on our list. We do that by making a decimal that is different from the first one in the first decimal place, different from the second one in the second decimal place, different from the third one in the third decimal place, and so on. Let's arbitrarily go one bigger each time with a 9 rolling over to a 0. Looking at our ordering, a new irrational number would have a 4 in the first decimal place (one bigger than the 3 in the first number), an 8 in the second decimal place (one bigger than the 7 in the second number), a 5 in the third decimal place (one bigger than the 4 in the third number), and a 3 in the fourth decimal place (one bigger than the 2 in the fourth number). Remember each irrational number keeps going to the right forever and our list keeps going down forever. So we keep up the process we outlined above. The new number we make is different from the first number (4 instead of 3), different from the second number (8 instead of 7), different from the third number (5 instead of 4), and so on forever. So it's a new number that wasn't on the list. So our ordering can't include every irrational number and the set of irrational numbers must be bigger than the set of natural numbers.Ta da!
Unfortunately, Cantor raised almost as many questions as he answered. The set of real numbers, which includes the rational and irrational numbers together, is commonly called the continuum, and the letter c is used to stand for the size of the set of real numbers. They couldn't use an aleph name because they didn't know if it should be aleph - one or aleph - two or whatever (remember aleph - zero was the natural numbers). So the continuum hypothesis is that c is really aleph - one, that is, the real numbers form the next size of infinity up from the naturals. Unfortunately nobody has been able to prove or disprove this in the 120 years since the question was first posed.
So, enjoy Pi day. I prefer apple or pecan or cherry pie myself, but it's the thought that counts. You may not be a pie person, but there are a lot of them out there. For example, as Nathan would tell you, if you hang around with World War 2 veterans for any length of time, you're going to eat a lot of pie.
HAPPY PI DAY !!!
Monday, March 11, 2013
You Can't Make This Stuff Up
Scientific American writer Davide Castelvecchi posts the following clip from a newspaper article about how much we Americans hate mathematics.
What more can I say? I could offer my math tutoring services, but I mostly tutor high school or college calculus students, not 3rd graders studying basic fractions. I would hope we wouldn't need a full hour to go over this. Maybe instead of the library where I usually meet the students, we could meet at a pizza place. Hands - on manipulatives would probably help.
Monday, March 4, 2013
It's Not the Donut, It's You!
The Science 2.0 website has an article by Greg Critser about the current war on obesity. It is an interesting approach to a difficult problem. How do we get Americans to stop eating so much sugar and junk food and exercising more? And the answer, according to the researchers that Critser talked to, is, "We don't." Slowly, these heretics in the field of nutritional science are saying that, in spite of the huge push in the last ten years about eating healthy and exercising, we haven't made a significant dent in obesity. Did we really think that outlawing the Big Gulp was going to make us all skinny? That's just stupid (or legislative, but I'm being redundant). These new researchers are "slowly and painstakingly shifting their inquiry to
a more basic issue: homeostasis--the body’s innate capacities to
recover from any kind of disturbance--and how we can strengthen those
balancing mechanisms."
So, the focus is on strengthening those balancing mechanisms in the body to counteract the bad stuff that comes in. One such study, by a renowned diabetes researcher in Sweden involved adding a rye grain mix to type 2 diabetic's food in the evening, then having a traditional (non heart healthy) meal the next morning for breakfast (ham, eggs, bacon, etc.). In the study, the group fed the rye mixture handled the breakfast the way any normal non-diabetic would. Those not fed the rye mix beforehand had insulin and blood sugar responses that were impaired and diseased.
This approach is not going to sit well with the billion dollar "healthy lifestyle" industry. But in the long run, it may have more impact than telling people to change what they've been doing for forty years. On the researchers' side is a nearly universally held vision of what health means:
There aren't a lot of "joyful attitudes to life" that I hear about when I'm preached to about why I'm overweight. Mostly it's more the "Bad dog!" finger pointing. "Why can't you be more like us - skinny and in shape? What's wrong with you?"
Gee, I wonder why the "healthy lifestyle" people are not having much success, other than in making money.
Theirs is not a difficult premise to understand. Consider: The human phenotype--the body our genes and environment dictate--has been shaped by evolution to adapt, to bend and flex and return to balance when confronted with a challenge, be it a Big Gulp or bad air. You eat a huge hit of sugar, your body pumps out insulin to make it possible for your cells to use it and return to balance. It’s an intricate, elegant system that worked just fine until we engineered an environment that made sugar virtually free and almost everywhere present. Then the system crashed, and we got our intertwined epidemics of obesity and diabetes.
So, the focus is on strengthening those balancing mechanisms in the body to counteract the bad stuff that comes in. One such study, by a renowned diabetes researcher in Sweden involved adding a rye grain mix to type 2 diabetic's food in the evening, then having a traditional (non heart healthy) meal the next morning for breakfast (ham, eggs, bacon, etc.). In the study, the group fed the rye mixture handled the breakfast the way any normal non-diabetic would. Those not fed the rye mix beforehand had insulin and blood sugar responses that were impaired and diseased.
“We don’t know why it works,” Inger Bjorck says. “But we have some idea. We think it involves some kind of gastric memory, maybe driven by fermentation in the gut, that we, as humans, once possessed, but which we lost when we began consuming our modern wheat mono-culture.”
This approach is not going to sit well with the billion dollar "healthy lifestyle" industry. But in the long run, it may have more impact than telling people to change what they've been doing for forty years. On the researchers' side is a nearly universally held vision of what health means:
As articulated by one of its earliest 20th century proponents, the brilliant Johns Hopkins' pioneer Henry Sigerest, health “is not just the absence of disease. It is something positive, a joyful attitude to life, and a cheerful acceptance of the responsibilities that life puts upon the individual... A healthy individual is a man who is well balanced bodily and mentally, and well adjusted to his physical and social environment."
There aren't a lot of "joyful attitudes to life" that I hear about when I'm preached to about why I'm overweight. Mostly it's more the "Bad dog!" finger pointing. "Why can't you be more like us - skinny and in shape? What's wrong with you?"
Gee, I wonder why the "healthy lifestyle" people are not having much success, other than in making money.
Wednesday, January 9, 2013
News From the World of Bugs
Bacteria, that is. I emerged from my bunker this week to find that the world had not ended and it was OK to be above ground after all. I started the new year by learning about the Human Microbiome Project. You may be aware of the Human Genome Project, which according to Wikipedia had "a primary goal of determining the sequence of chemical base pairs which make up DNA, and of identifying and mapping the approximately 20,000–25,000 genes of the human genome from both a physical and functional standpoint." It started in 1990 and was declared completed in 2003 with the publishing of a final sequencing mapping.
The Human Microbiome Project has the goal of mapping the microbe (bacteria) communities existing in various parts of your body: stomach, intestines, nasal passages, skin, and so forth. The interesting part of the project is what scientists are starting to find out about how certain bacteria are essential to a healthy life. In an article by Michael Specter in the New Yorker, researcher Martin Blaser talks about a specific bacteria: Helicobacter pylori. For those of us who grew up understanding ulcers to be a result of stress, it was a major discovery in 1982 that H. pylori (as it is affectionately called) was actually a causing agent for ulcers, gastritis, and stomach cancers. In a fairly obvious step, the treatment for these conditions changed to an antibiotic regimen. What Martin Blaser and others have found in the last few years is that this bacteria, which is harmful to adults, is actually very helpful to children. A number of studies have shown that children without the H. pylori bacteria in their stomach are more likely to have issues with asthma.
According to the article,
The article cautions us to not get too carried away yet with canceling our antibiotic approach to health. For many diseases, only antibiotics can keep those infected from dying. The Human Microbiome Project was founded in 2008. In 2012, the normal microbial ecosystem for healthy humans was published. What is not known for many of the bacteria is what the overall function of each bacteria is.
Note 1: Not a single probiotic (like those found in yoghurt or in pill form) sold in the United States has been approved or certified as a drug. They are dietary supplements and have very little evidence of value. In fact researchers argued that the mix of bacteria in some that they had tested had given them cause for concern. Some bacteria, when mixed, can be deadly. Michael Fischbach of UCSF cites the lack of evidence for the efficacy of current probiotic supplements to be equivalent to that of snake oil medicines of a hundred fifty years ago.
Note 2: There is a growing (and really, really scary) push to label childhood vaccinations as the source of a number of childhood problems, most prominently, autism. Every scientific journal I look at debunks this. The article mentions that there are no scientific studies that back up the connection between vaccination and autism. Unfortunately, autism tends to become evident at about the same time that children get a large number of their vaccinations. Based on that correlation, scientists have looked for a cause and effect relationship and have found no evidence at all. It is similar to the outcry in 1952 when some doctors noticed that the incidence in polio in children increased in the summer at exactly the same time as the amount of ice cream children were eating increased. The increase in sugar, they said, was the cause of polio. Luckily, within a year, a virus was discovered to be the culprit and, by 1955, a polio vaccine had been developed that caused the disease rates to plummet. Please ignore the hysteria about vaccinations and follow your doctor's advice in vaccinating your children.
The Human Microbiome Project has the goal of mapping the microbe (bacteria) communities existing in various parts of your body: stomach, intestines, nasal passages, skin, and so forth. The interesting part of the project is what scientists are starting to find out about how certain bacteria are essential to a healthy life. In an article by Michael Specter in the New Yorker, researcher Martin Blaser talks about a specific bacteria: Helicobacter pylori. For those of us who grew up understanding ulcers to be a result of stress, it was a major discovery in 1982 that H. pylori (as it is affectionately called) was actually a causing agent for ulcers, gastritis, and stomach cancers. In a fairly obvious step, the treatment for these conditions changed to an antibiotic regimen. What Martin Blaser and others have found in the last few years is that this bacteria, which is harmful to adults, is actually very helpful to children. A number of studies have shown that children without the H. pylori bacteria in their stomach are more likely to have issues with asthma.
According to the article,
"Babies delivered by Cesarean section lack many microbes that are routinely transferred from mother to child. Last year, nearly a third of the four million children born in the United States were delivered by Cesarean section. (In China, the figure was closer to fifty per cent.) The incidence of allergies and asthma is far higher among those children than it is for vaginal-birth babies."Another important function of the H. pylori bacteria is that it is connected with two hormones, ghrelin and leptin, that are instrumental in regulating our appetite. They work together to let us know when we are hungry and when we are full and should stop eating. Our bacteria friend helps to keep the two hormones in balance; without it, we generate too much ghrelin, which makes it more likely for you to overeat. Since World War II, there has been a consistent sharp decline in the number of children infected by H. pylori and a consistent sharp rise in childhood obesity.
The article cautions us to not get too carried away yet with canceling our antibiotic approach to health. For many diseases, only antibiotics can keep those infected from dying. The Human Microbiome Project was founded in 2008. In 2012, the normal microbial ecosystem for healthy humans was published. What is not known for many of the bacteria is what the overall function of each bacteria is.
"The hope is that someday researchers will treat bacteria with highly specific antibiotics and then rebuild our damaged ecosystem with probiotics—strains of bacteria that could act as surrogate farmers in our internal ecosystems. One study, in mice, showed that the toxic side effects of a colon-cancer drug were eased by blocking a particular bacterial enzyme. It’s promising, but we need to move very carefully to confirm the results when they look so good."
Note 1: Not a single probiotic (like those found in yoghurt or in pill form) sold in the United States has been approved or certified as a drug. They are dietary supplements and have very little evidence of value. In fact researchers argued that the mix of bacteria in some that they had tested had given them cause for concern. Some bacteria, when mixed, can be deadly. Michael Fischbach of UCSF cites the lack of evidence for the efficacy of current probiotic supplements to be equivalent to that of snake oil medicines of a hundred fifty years ago.
Note 2: There is a growing (and really, really scary) push to label childhood vaccinations as the source of a number of childhood problems, most prominently, autism. Every scientific journal I look at debunks this. The article mentions that there are no scientific studies that back up the connection between vaccination and autism. Unfortunately, autism tends to become evident at about the same time that children get a large number of their vaccinations. Based on that correlation, scientists have looked for a cause and effect relationship and have found no evidence at all. It is similar to the outcry in 1952 when some doctors noticed that the incidence in polio in children increased in the summer at exactly the same time as the amount of ice cream children were eating increased. The increase in sugar, they said, was the cause of polio. Luckily, within a year, a virus was discovered to be the culprit and, by 1955, a polio vaccine had been developed that caused the disease rates to plummet. Please ignore the hysteria about vaccinations and follow your doctor's advice in vaccinating your children.
Friday, December 21, 2012
Why I Haven't Done Any Christmas Shopping Yet!
On Friday, December 21, 2012, according to the Mayan Calendar, the world is coming to an end. That's before December 25. Hence my dilemma. At the website, December212012.com, there are a number of helpful hints about what to do during the coming apocalypse. You could go to a Mayan temple and hang out and wait for the festivities to start.
You could start working on your survival skills and building your underground rescue bunker:
I think a survival seed kit would be helpful for when you come out of the bunker. You can't be sure of the form of the destruction - this time it probably won't be the Stay Puft Marshmallow Man. There are numerous doomsday scenarios: powerful earthquakes, a super volcano at Yellowstone National Park, economic and social collapse, nuclear fallout or a solar storm that destroys satellites and electrical grids. Any of those would be easier to take if you had jumbo seed packets of radishes and broccoli ready to go when you come out of the bunker in 2025. That would keep me going through 10 years of lava flows and nuclear fallout. Mmmm! Broccoli!
Maybe you need the deluxe two person survival kit. It's currently out of stock, but could be shipped to you as early as next week. Oh, no!
Ah, Science! Once again people ignore you and then end up looking like monumental whack-jobs to everyone around them. And I'm just talking about the House of Representatives Science Committee.
See you tomorrow.
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