December 17, 2003
Interesting Mathematical Calculations

Chad Orzel writes about interesting mathematical calculations... "What's the goal?" he asks. I guess the goal is to come up with a set of problems, puzzles, and calculations that are (a) fun to do, (b) teach interesting facts and relationships, and (c) convince pre-teenagers and early teenagers that math is worth learning because it is a useful tool. Uncertain Principles: Brad DeLong asks the all-important question "Can there possibly be as many as one hundred interesting math problems?" Well, OK, strictly speaking, he's assuming that there are at least one hundred interesting math problems, and is compiling a list as an aid to getting his kids to see the utility of math. This makes it more of a proof-by-contradiction problem: since he's asking for help from his readers, he may be close to the end... Part of the problem with trying to compile such a list is defining just what counts as an interesting math problem. A bunch of the list to date are more puzzles than calculations, a few others are demonstration exercises, a few are Fermi Problems, and four are freshman physics problems (numbers 3, 14, 15, and 23). Very few of them bear any resemblance to math...

Posted by DeLong at 09:16 PM

December 15, 2003
Request for Help

My two children get what the payoff to reading well is immediately and completely. My two children get what the payoff to writing well is as well: they understand that it is fun now and it will be important later on if they want to have options to be able to write quickly, clearly, and coherently. But math. Math textbooks are remarkably dry. How can I persuade them that math can be fun, that they will be able to learn and calculate interesting things if math is their friend, and that their options later on will be much, much greater if only they apply themselves to math now? So far, I only have twenty-three problems that I regard as interesting and amusing enough to hand them in an attempt to propagandize for math. But I want more: I want to have one hundred... Brad DeLong's Collaborative Website: OneHundredInterestingMathCalculations: [1. World War II Bomber Pilot Survival Odds] [2. How Many Extraterrestrial Civilizations Are There?] [3. Gravity and "Weighing the Earth"] [4. Economic Growth Since 1500] [5. Exponential Growth and Human Populations] [6. How Much Blood Is There in the World?] [7. Julius Caesar's Last Breath] [8. The Birthday Fact] [9. The...

Posted by DeLong at 04:01 PM

December 14, 2003
One Hundred Interesting Mathematical Calculations, Puzzles, and Diversions: Number 23: The Muddy Parent Problem

One Hundred Interesting Mathematical Calculations, Puzzles, and Diversions: Number 23: The Muddy Parent Problem Suppose a 250-pound parent climbs a muddy 45-degree hill... 1: Calculate the force tending to make the parent slide down the muddy hill. Answer to 1: The first force to be taken into account is the force of gravity. We represent the force of gravity by an arrow, G, pointing downward toward the center of the earth. The earth is pulling the parent downward. By how much? By 250 pounds--that's what it means to say that this is a 250-pound parent. But the parent does not fall straight down through the hill. The molecules of dirt are pushing on the parent, counteracting the force of gravity. This force is represented by an arrow, H, pushing the parent away from the hill. We add up the forces G and the forces H by putting the tail of the H arrow at the head of the G arrow. The sum of these two arrows ("vector addition") is the arrow S--the total force that is tending to make the parent slide down the hill. How big is this force arrow S? Well, we know that the force arrow G...

Posted by DeLong at 08:21 PM

December 08, 2003
One Hundred Interesting Mathematical Problems, Puzzles, Diversions, and Amusements: Number 21: Ancestors

One Hundred Interesting Mathematical Problems, Puzzles, Diversions, and Amusements: Number 21: Ancestors "Dad?" "Yes?" "Were my ancestors... famous back in the Middle Ages?" "Your ancestors? Which ancestors? You have a lot of them." "Well, most of them." "There is a tradition that your Ridgeway ancestors--part of the English wave of conquest of Ireland under the Angevin kings--were descended from an illegitimate daughter of William the Conquerer, and that they were also descended from Earl Leofric and his wife Godiva, of the well-known story about Coventry. This was possibly the first supply-side tax policy in history: she asked her husband to lower taxes, and he replied 'Sure, I'll lower taxes when you ride naked through the streets on a white horse'." "Nobody knows that story, Dad." "No?" "I asked people in my class. They all agreed that Lady Godiva was the inventor of a kind of chocolate." "Oh." "But what was my typical ancestor like?" "Well, first let's calculate how many ancestors you had. You were born in the 1990s. Let's suppose that there is a generation born every twenty-five years. That means that a century ago you had 2 x 2 x 2 x 2 ancestors born in the 1890s....

Posted by DeLong at 07:52 PM

November 15, 2003
One Hundred Interesting Mathematical Calculations: Number 16: How Rich Is Fitzwilliam Darcy?

The mother of the bride-to-be says: Jane Austen: Pride and Prejudice, Chapter XVII of Volume III (Chap. 59): Good gracious! Lord bless me! only think! dear me! Mr. Darcy! Who would have thought it! And is it really true? Oh! my sweetest Lizzy! how rich and how great you will be! What pin-money, what jewels, what carriages you will have! Jane's is nothing to it -- nothing at all. I am so pleased -- so happy. Such a charming man! -- so handsome! so tall! -- Oh, my dear Lizzy! pray apologise for my having disliked him so much before. I hope he will overlook it. Dear, dear Lizzy. A house in town! Every thing that is charming! Three daughters married! Ten thousand a year! Oh, Lord! What will become of me. I shall go distracted.... My dearest child.... I can think of nothing else! Ten thousand a year, and very likely more! 'Tis as good as a Lord! And a special licence. You must and shall be married by a special licence. But my dearest love, tell me what dish Mr. Darcy is particularly fond of, that I may have it tomorrow. So how rich is Fitzwilliam Darcy, anyway?...

Posted by DeLong at 04:02 PM

October 16, 2003
One Hundred Interesting Mathematical Calculations, Puzzles, Diversions, and Amusements: Number 22: Strategy Secrets of ENRON

One Hundred Interesting Mathematical Calculations, Puzzles, Diversions, and Amusements: Number 22: Strategy Secrets of ENRON Megan McArdle uncovers one of the strategy secrets of ENRON: Asymmetrical Information: Question of the day: I've heard of a supposedly foolproof system of winning at roulette, as follows: 1) Put down your stake on either even, odd, black, or red 2) If you lose, double what you had on the table and bet again on the same thing. 3) If you lose again, double again. 4) Repeat until you win 5) Then stop. You can only win the amount of your original stake, of course. Can any budding mathematicians find the hole in this theory? Seriously, this is what ENRON did. Have a quarter in which prices moved against you and you would have to report a loss? Stuff the loss into a special-purpose entity (so you don't have to report it for a while) and place a bigger bet. Keep doing this. Sooner or later one of the bigger bets will come through, right? The example of ENRON tells you what's wrong with this strategy. If you look ahead five rounds, you see that you have a 31/32 chance of coming up ahead...

Posted by DeLong at 05:02 PM

October 01, 2003
One Hundred Interesting Mathematical Calculations, Puzzles, and Amusements: Number 20: The Federal Reserve Problem

The Federal Reserve Problem Index Page A man named Alan Greenspan is Chair of the Federal Reserve Board, and Chair of the Federal Reserve System's Open Market Committee. The Federal Reserve is our country's central bank: it is responsible for setting interest rates. The interest rate is set in percent per year. Currently, the interest rate on short-term U.S. Treasury bonds is one percent per year: buy $1,000 of Treasury bonds now, and the U.S. government will pay you $1,000(1 + 1%) = $1,000(1.01) = $1,010 back in a year. Interest rates matter because the higher the interest rate the less likely companies are to spend money building new buildings and factories and buying new machines. When interest rates are high, they say, "Couldn't we make more money by lending out our cash than by using it to build new buildings and factories and buy more machines to make more stuff to sell?" When interest rates are low, they say, "Couldn't we make more money by taking some of the cash we loaned out--the people who borrowed it are hardly paying us any interest, after all--and use it instead to build new buildings and factories and buy more machines to...

Posted by DeLong at 04:38 PM

September 03, 2003
One Hundred Interesting Mathematical Calculations, Puzzles, and Amusements: Number 19: The Distributive Law, or The Get-Out-of-the-Way Problem

Number 19: The Distributive Law, or the Get-Out-of-the-Way Problem "So do you understand why the distributive law of multiplication over addition is important?" "Yes. It is important because it provides math teachers with yet another way to torment Thirteen-Year-Olds." "That's the wrong answer." "It's a free country--or it's supposed to be." "The right answer is that the distributive law allows you to rewrite equations--to replace one equation with another that means the same thing--and that the main point of algebra is to keep rewriting and rewriting equations, expanding, factoring, and cancelling, until you get to a form where the answer is obvious." "Sure." "No. I mean it. Take... will you believe me if I say that if you fire a cannonball straight up with an initial upward velocity of 640 feet per second, then its height above the earth measured in feet t seconds after liftoff is given by the equation: h = 640t - 16t2 ?" "If you say so." "Well, now, let's factor this equation. Do the two terms on the right hand side share anything?" "They share a sixteen." "So we can rewrite the equation as: h = (16)(40t) - (16)(t2), right?" "I guess so." "And do...

Posted by DeLong at 08:22 PM

August 13, 2003
One Hundred Interesting Mathematical Calculations, Puzzles, and Amusements: Number 18: Sunscreen

The Freak Mutant Near-Albino Problem "OK. What do we do now?""We put on sunscreen!""Why do we put on sunscreen?""So we don't die of melanoma when we are fifty!""How much sunscreen do we put on?""Lots!""What kind of sunscreen?""Powerful hypoalergenic sunscreen!""Why do we have to worry about this?""Because we are mutants!""Freaks!""Near albinos!""Strange mutant freak near-albinos who would drive our remote ancestors away, gibbering in terror!""Why are we these strange near-albinos?""Because, 1000 generations ago, our ancestors moved out of Africa up onto the ice sheets and into the cold boreal forests of northern Europe!""Where it was cold!""And there was little sunlight to make vitamin D in our ancestors' skin!""So those with more melanin in their skin found that the melanin blocked the sunlight, and they didn't make vitamin D, and they got tickers!""Rickets!""Whatever, that disease where your bones don't grow straight and you can't hunt or gather and you die before you mate and your genes are excluded from the future of the river of evolution!""Suppose that 1/100 of the genes controlling our ancestors' skin color 1000 generations ago had been a gene for low melanin, and suppose that--for those of us of northern European descent--today 99/100 of our genes are genes for...

Posted by DeLong at 05:02 PM

April 17, 2003
One Hundred Interesting Mathematical Calculations, Puzzles, and Amusements: Number 17

One Hundred Interesting Mathematical Calculations, Puzzles, and Amusements: Number 17: The Clock Hands Puzzle At 12:00:00 the hour and the minute hands of a clock coincide. How much time passes before the next instant that the hour and minute hands coincide? After an hour the minute hand has gone all the way around the clock... but the hour hand has gone 1/12 of the way around the clock, so it is still ahead of the minute hand. So it is more than an hour. After 1 + 1/12 hours the minute hand has caught up to the previous position of the hour hand, but by that moment the hour hand has moved an additional 1/144 of the way around the clock. So it is more than 1 1/2 hour. Continuing the same chain of reasoning, we see that the minute hand catches up to the hour hand after the following amount of hours have elapsed: 1 + 1/12 + (1/12)^2 + (1/12)^3 + (1/12)^4 + (1/12)^5 + ... + (1/12)^n + ... You have to add up an infinite number of terms to get the answer! How can you add an infinite number of numbers? Isn't the answer infinity? No,...

Posted by DeLong at 11:19 AM

April 05, 2003
The Monty Hall Problem

Two posts down Robert Waldmann referred to the famous Monty Hall Problem. Here's an explanation of what it is: The Infamous Monty Hall Problem The Setup you are presented with 3 doors (A, B, C) only one of which has something valuable to you behind it (the others are bogus) you do not know what is behind any of the doors You choose a door Monty then counters by showing you what is behind one of the other doors (which is a bogus prize), and asks you if you would like to stick with the door you have, or switch to the other unknown door The question is should you switch? Another question is Does it matter? The answer lies behind this link Don't look until you've decided upon your answer....

Posted by DeLong at 02:12 PM

March 10, 2003
One Hundred Interesting Mathematical Calculations, Puzzles, and Amusements: Number 16

One Hundred Interesting Mathematical Calculations, Puzzles, and Amusements: Number 16: Relatives "Maddy and John are your cousins--your third cousins. Their father Tom is my cousin--my second cousin. Now what does it mean if somebody is your cousin?" "It means that you share two grandparents." "And what does it mean if somebody is your second cousin?" "It means that you share two great-grandparents." "And what does it mean if somebody is your third cousin?" "It means that you share two great-great-grandparents." "And how closely related does that make you? How many of your genes are copies of your ancestors' genes that were also copied into John and Maddy's cells?" [Silence] "Well, think of it this way... "You have two parents, four grandparents, eight great-grandparents, and sixteen great-great-grandparents. That means that one-sixteenth of your genes come from each of your great-great-grandparents--and conversely, that one-sixteenth of each of their genes were copied and now float in the nuclei of your cells..." "What if I'm a MUTANT?!?!" "Quiet. During this conversation, your lines are limited to, 'That must be true, Dad,' or 'How smart you are, Dad'. So one-sixteenth of your genes come from your great-great-grandfather John Anderson Lord, and one-sixteenth of your genes...

Posted by DeLong at 07:55 PM

February 23, 2003
One Hundred Interesting Mathematical Calculations, Puzzles, and Amusements: Number 15

One Hundred Interesting... Index Page Elementary Ballistics: What Goes Up Must Come Down The last problem established that if you dropped a cannonball down a very deep well, the distance it falls (with positive numbers being up, and negative numbers being down) could be described by the gravitational fall equation: yt = -(0.5)(g)(t2) That is, the vertical position of the cannonball t seconds after you drop it (that's the "yt" part) is equal to minus (that's because it falls--moves downward) 1/2 times the square of the number t of seconds since you dropped it. Moreover. on earth where the force of gravity g = 32 feet/second/second, this is the same as: yt = -16(t2) But usually we are not interested in cannon balls dropped down wells. Usually we are interested in cannonballs fired from cannon. And they have an initial upward velocity--call it vy, "v" for "velocity" and "y" because we are using the variable y to stand for how high above (or far below) our cannonball is from its launching point. What happens to a cannonball that is not dropped, but is instead initially launched with an upward velocity vy--say, 64 feet/second? Well, if there were no force of...

Posted by DeLong at 05:03 PM

One Hundred Interesting Mathematical Calculations, Puzzles, and Amusements: Number 14

One Hundred Interesting... Index Page Elementary Ballistics: The Kinematics of Falling Bodies Let's think about falling--the motion of a body dropped from the earth's surface into a very deep pit. Let's assume that what we are dropping is dense enough that we don't have to worry about air resistance (or, conversely, lift)--a cannonball, say. And let's assume that the pit is deep enough that we don't have to worry about it hitting anything and coming to a stop (at least not for a while). And assume that our distances are small enough that we don't have to worry about how the strength of gravity varies as we move closer to and further from the center of the earth. At the moment of its release, the cannonball has no upward or downward velocity at all. One second later, gravity--accelerating it at a rate of g feet per second per second--that is, every second gravity gives it an additional downward velocity of g feet per second, where on the earth's surface g = 32--has given it a downward velocity of 32 feet per second. Let's adopt the convention that upward numbers are positive and downward numbers are negative, so that after one...

Posted by DeLong at 07:29 AM

February 22, 2003
Notes: Interesting Math Calculations

A suggestion for an "Interesting Math Calculations" problem--something that I don't know the answer to offhand (both because I don't know how small semiconductor circuit features can be before quantum mechanics ceases to be your friend, and I don't know how large semiconductor circuit features are today). How long can Moore's Law go on? Starting from the average distance between atoms in a silicon crystal, find the time when chip features will be (supposedly) one atom wide... Suggestions for Entries Page Interesting Math Calculations Index page...

Posted by DeLong at 02:42 PM

100 Interesting Mathematical Calculations, Puzzles, and Amusements, Number 13: Introduction to Compound Growth

One Hundred Interesting... Index Page Introduction to Compound Growth "The problem is: 'Suppose there is a town with a population of 10,000. Its population grows at one percent per year. How large a population will it have in five years?' I guess that since .01 x 5 = 0.05, and since 0.05 x 10,000 = 500, that the answer is 10,500." "Ah. They've tricked you. Do it year by year. What's one percent of 10,000? And how large will its population be after one year?" "100, and 10,100." "Right. Now what's one percent of 10,100?" "101. Oh!" "Yes. The population grows by 100 during the first year. And the population grows by 101 during the second year. What's the population after two years?" "10,201." "And how much does the population grow during the third year? Round the population to the nearest whole person--because that's what it has to be." "102. So there are 10,303 people after three years?" "Yes. And 10,406 people after four years." "And 10,510 people after five years. Still, ten extra people doesn't seem all that many." "Ah. But as growth compounds the differences add up. How long would you say it would take the population to...

Posted by DeLong at 01:45 PM