Thursday, March 30, 2006

Adventures in Gubmint

This week, I went to the DC-area to participate in a panel evaluating proposals solicited by the Department of Energy. It was the first time I'd ever done anything like that, so I didn't really know what to expect. I learned a lot, most of it related to the proposal writing and evaluation process.

The first thing I learned was to never, ever connect through Newark. If you have to change terminals (as I did), you have to leave the secure area and go back through security again. I had an hour layover and thanks to the metal rivets on my clothes, I got personally searched and nearly missed my flight.

But I didn't miss the flight, and made it to the hotel where I was staying and where the panel was, by 8 p.m. Just in time to see my colleague, who was on the Tuesday panel (I was on the Wednesday panel), leaving for dinner. So I caught up with him and another colleague who turned out to also be on the Wednesday panel, and we all went out to dinner together.

The next morning was the panel. The first proposal on the table was one of the three that I reviewed. The DOE program manager knew that I'd never done this before, so instead of having me go first, he asked one of the other reviewers to talk about the proposal. After that I was able to chime in with my comments about the proposal. Basically there was fifteen minutes allotted to each proposal, and the three people who had reviewed that proposal gave overviews of the proposal. Everyone else was welcome to ask questions and make comments too. I didn't talk much at all, except during the time allotted to each of the proposals I reviewed. Mostly I just listened and tried to absorb what people were saying. Obviously the review is supposed to be confidential so I can't say much more than that about it. I can say, however, that they kept us well-fed throughout the day, which was nice. My friend on the Tuesday panel said that the lunch they served on Tuesday had shrimp, but ours didn't have anything nearly so fancy. Good thing we were the last panel of reviewers; otherwise, today's panel might have been served peanut butter and jelly!

After the panel was over, I went out to eat with a friend from our undergraduate days and his wife. This is the same friend who was here in February, interviewing. It turns out that he was offered a job, and they decided to accept it and move to Oak Ridge! And he starts at the beginning of May! I was really excited to hear the news, and didn't feel too bad when the evening was over, knowing that we'd soon be seeing them here!

I am insane, so I booked a flight leaving at 6:45 a.m. this morning. That meant I had to get up at 4 a.m. Luckily for me, the colleague who was also on the Wednesday panel decided to accompany me in my rental car back to the airport. This helped keep me awake, plus he was able to help with directions, so I didn't have to rely on my foggy, directionally-challenged brain. I made it back home without incident, this time going through Cleveland rather than Newark. Upon arriving home, I took a nap before eating some lunch and making my way to work.

At work we have a frantic schedule at the moment; we've suddenly learned that we are supposed to benchmark our code on one of the leadership-level supercomputers by Tuesday. This code is a work in progress and has not yet been ported to any large-scale machine, so this poses quite a challenge. At the moment we're having trouble getting it to compile on that machine, because the compiler doesn't like the way we use templates. Luckily that is not my part of the job at the moment; right now I'm supposed to be writing a benchmarking code that records the timings of each of the operators in the code (e.g. multiplication, addition, etc.). I still have to finish that. Unfortunately, by being on that panel, I've missed a day and a half of time that could have been spent working on this. But I think the knowledge I have gotten from serving on that panel made the trip worthwhile.

Thursday, March 23, 2006

Adventures in Houseselling

We are no longer homeowners twice over. Yesterday was the closing on our house in Illinois. It was successful for us and we are now quite a bit richer (at least until the money goes to the folks we owe it to!).

The buyer ran into a bit of bad luck, though. From what I understand, the inspector ran the dishwasher when he was inspecting the house, and forgot to go back and look at it, so he didn't notice that it hadn't drained. The water left standing in the bottom of the dishwasher ate away all the seals, and the water leaked out. When the buyer went for the final walkthrough yesterday morning, they discovered that the kitchen floor was wet and damaged. We had put down a new laminate floor before selling the house and apparently some of the tiles were warped. The buyer was (understandably) upset and tried to back out of the sale or at least renegotiate, but our lawyer and our realtor convinced her that she couldn't. Instead of going after us for the damage to the floor, she needs to go after the home inspector. Our realtor said he thought that once it dried, the floor would be just fine. Still, it's got to be upsetting to buy a house and have this be your first experience with the house. I think she's going to have to buy a new dishwasher. Maybe that will be covered by the home warranty that we bought for her at closing.

Anyhow, it's not our problem and we are no longer the owners of two houses! I am looking forward to having the money that we were paying for that mortgage. It will be nice to have that extra income.

Monday, March 20, 2006

Pareto-Optimal Healthcare

There are some things in life that don't mix: oil and water, Hatfields and McCoys, Pepsi and milk, my mother and my father, and profit and healthcare.

While the first pairs on my list are universally known to be incompatible, the last one may come as a shock to you. After all, don't we, citizens of the United States, land o' capitalism, have access to the best healthcare in the world?

Yes, yes we do! We have the best healthcare that money can buy. And therein lies the problem. Healthcare is expensive, and those who need it the most can generally least afford it.

Remember last week when I was talking about Pareto optimization, or trying to find the best solution to a problem with multiple conflicting objectives? The problem of healthcare in a capitalistic scenario is a Pareto optimization problem, with many complicated and conflicting objectives that can be boiled down to two main objectives: 1, minimize cost, and 2, maximize public health.

There are many different solutions to this problem, each with varying degrees of practicality. Of course we could choose a solution in which no money is spent but everyone is ill, or we could go to the opposite extreme and eradicate every single germ. The best solution lies somewhere in between.

If we say that minimizing ill-health is equivalent to maximizing health, then we could use the same graph as before, substituting "Ill-Health" for "Effort" on the y-axis. Once again we have a continuum of optimal solutions, and the final solution is based on the values of those who make the decision.

The problem is that the values of those who make the decision are skewed toward minimizing costs instead of minimizing ill-health. Today's healthcare landscape is run by corporations, with the objective of maximizing profit taking higher priority than maximizing public health.

In a for-profit insurance model, the underwriter agrees to insure a number of people for a certain rate per person. (Rates for different people might be different, depending on age, status, health history, and what they want covered.) The money collected from each individual goes into a pool that is used to pay the claims of the policyholders and the expenses of the underwriter. Any money left over is profit.

This is why if you try to apply for an individual health insurance policy, you are screened so carefully and (assuming you are approved) the cost is so high. The insurance company wants to make sure that you are a good investment; that statistically speaking, you are within a certain limit of acceptable risk. They hope that you pay your premium regularly and that you never make a claim.

Group policies, such as policies provided by employers, often accept anyone without any conditions on their prior health history. Insurers make money on these policies because they have a guaranteed pool size over which any risks can be amortized, and they charge more per head than they would charge a person whose health risks had been screened.

There is a law that requires that health insurance companies offer policies to anyone who has been previously covered for a period of two years or more. The loophole in that piece of legislation is that the cost of the premium is not regulated. The sickly sister of a friend of mine was approaching the birthday at which she would no longer be covered by her parents' health insurance policy. Sure, she could have health insurance, but only if she could afford a monthly premium of $2700!

Tragically, the requirements for obtaining health insurance, namely good health and the means to pay for it, preclude from obtaining health insurance those who by definition need it the most: the ill who cannot afford treatment. But even people who have access to health insurance are often denied the coverage they need.

True story #1: A mentally ill man with full health coverage is told that he must travel one hundred miles round trip to visit an in-network psychiatrist, despite the dozens of available psychiatrists in a ten-mile radius. If he chooses to see a local psychiatrist, he must pay for the appointment himself, at a cost of $200 or more.

True story #2: A five-year-old girl is in the hospital this past November through February because her transplanted liver is failing and she needs a new liver. The family of the five-year-old girl has comprehensive health care coverage with a yearly limit of $2 million. The plan year is the calendar year. The re-insurer (the people who insure the insurance company) decides that because the bills for the November and December hospitalization were not received until January, they counted for 2006 instead of 2005, and by mid-February, the family has exhausted the $2 million annual limit.

The behavioral profile illustrated by these true stories makes sense under the charters by which corporations operate, that the short-term bottom line takes first priority. It doesn't make sense under the charters of common sense or decency, however.

In the first case, the insurance company was trying to save themselves a buck now, but neglected to consider their later losses when the mentally ill man (who was incapable of driving 100 miles round trip on a weekly basis, and could not afford the cost of seeing a local doctor) stopped receiving psychiatric care and brought harm to himself or another of their clients. (Actually, the man's family appealed the decision to their state's regulatory agency, and it was reversed.)

In the second case, the insurance company was trying to get out of its obligations by bending its rules and looking for any possible violation of the rules by the little girl's family. This is a tactic used by the mentally ill man's insurance company too, and I anticipate that should they be forced to reverse their decision, the girl's family will have a hard time getting them to honor their obligations.

The problem is that with all the focus on profit, the insurance companies have forgotten about what they're really insuring: people. If I lose my house in a mudslide, it's traumatic, it's scary, it's unfortunate, but it's just things. If I get really sick, on the other hand, it's my life that could be lost if the money to pay for the care that I need doesn't come through.

I'm a numbers junkie (I was counting sheep by 17's last night!) but I know where the realm of numbers leaves off and the realm of real people takes priority. The well-being of my fellow human beings is a whole lot more important than the dividends I get to record on Schedule D of my 1040 form. But of course the dividends can be a big temptation for some folks, which makes it all the more important to remove the for-profit aspect from the equation.

What is the solution to this problem? Well, there is no single solution to this problem, as I have already indicated. I just think we need to move along the curve of Pareto solutions away from maximizing profits and towards maximizing public health. I think that doctors and nurses perform a valuable service to our society, and should be remunerated handsomely. I think that hospitals should be run efficiently and the managers of the hospitals should be sufficiently rewarded for their work. But they should operate clinics and hospitals under the concept that the care of patients is the foremost priority, and I strongly believe that all hospitals and clinics should be run as not-for-profit organizations.

I think that similarly, health insurance companies should be not-for-profit organizations, to eliminate any temptation to stiff real life human beings in exchange for a fatter bottom line. I would support a national health insurance program; for example, expanding Medicare/Medicaid to include premium-paying workers.

I know I've lost some people to the rhetoric against government programs by now, but quite honestly, if working people paid premiums to the government instead of to a private insurance company, there would be no change except that the premium would go down.

We already pay our money into a pool. (In fact, we pay twice: once for the indigent and elderly [Medicare tax] and once for ourselves.) My employer and I pay more than $900/month to a large health insurance company. Jeff and I don't use anywhere near that much in coverage every month. The money goes into a pool that pays for my colleague's asthma medication, or the neighbor kid's strep throat, or prenatal checkups for a woman I've never met. It also goes to pay the salaries of the insurance company's employees, but a big chunk of it goes for corporate overhead, such as shareholders' dividends, advertising, and nine-figure bonuses for the CEO, for example, overhead that would be eliminated from a government plan.

Some reports say that a single-payer Medicare system could be supported on a 9% payroll tax: 2% coming from the employee's pay, and 7% coming from the employer. If that were the case, my employer and I would be paying about half of what we're paying now for health insurance. Sounds like a deal to me!

People rail against the inefficiency of government (myself included), but Medicare is actually the most efficient health insurance company in the United States. Less than a nickel of every dollar is spent on overhead. In contrast, the best private insurance companies spend roughly twelve cents per dollar on overhead. According to some reports, my insurance company spends more than a quarter of every dollar on overhead! Wouldn't it be nice if that money went to saving lives instead of fighting against legitimate claims by mentally ill men or critically ill children?

I hate taxes too. I can count my money with the best of 'em. But I don't think that paying for the well-being of my fellow citizens is such a bad thing. (As we've seen, I'm already doing it when I pay my private health insurance premium.) The thing I don't understand is how we in this country can place such a high priority on the unborn (e.g. South Dakota's recent anti-abortion laws and my own state's proposed constitutional amendment) and such a low priority on the already born!

A man who is near and dear to the hearts of most Americans once said, "Therefore, whatever you want men to do to you, do also to them, for this is the Law and the Prophets." (Matthew 7:12, ESV) (This is not to say that the Golden Rule is unique to Christianity.) It bewilders me that so few people actually heed his words. If I were ill, I would like for others to help me get over my illness and get back on my feet. It's only fair that if I want others to do this for me, I should contribute my part in doing this for them. Let's take care of each other like we're supposed to!

There is a lot more to say about Pareto-optimal healthcare, but I had to Pareto-optimize this post for length, substance, and time spent researching and writing, so this is all I have to say for now. There are many interesting articles out there on the topic. Two comprehensive articles that I found were

and I encourage you to read them both.

Monday, March 13, 2006

Finally Feeling Healthy

Finally I feel a lot better. Looking back on it, I probably shouldn't have gone to work on Thursday. I came in for a half day on Friday but I couldn't last any longer than that so I left and went to the doctor. My husband also went to the doctor, so we had double walk-in appointments at the same time at this ambulatory care center that I went to for my arm one of the first weeks I was in town. The doctors gave us antibiotics for our secondary infections and sent us on our way. We had our prescriptions filled at Kroger while we stocked up on chicken soup, sparkling water, and juice. Then we went home and sat around trying to recover from our excursion.

On Saturday I felt much better, like I was finally on the mend, but we didn't do much of anything that day because I didn't want to push myself. That meant skipping out on our weekly role-playing game, which was unfortunate but had to be done. On Sunday I felt even better and I was able to do a few chores around the house without getting too exhausted.

The big problem has been that the weather has been absolutely gorgeous for the past week, and I've been too sick to go outside and enjoy it! Now, just as I'm getting well, it's supposed to storm and cool off again. That kind of sucks.

Springtime is definitely here, however. We've had daffodils in bloom for a couple of weeks here. I've noticed that they are pervasive: we have a big patch of them in our front lawn, I see them by the side of the road all the time, and they seem to grow like weeds. The flowering pears are out too, and I may have seen a redbud or two but I'm not sure. One thing I've noticed is the striking absence of tulips. I have yet to see a single tulip, in my yard, my neighbors' yards, or anywhere. So I don't know what to make of that.

Thursday, March 09, 2006

Wiped Out!

Whatever I was coming down with on Monday got progressively worse, until I ended up having to go home at about 2:00, because I just couldn't think or concentrate. Actually, I had to psych myself up to get myself feeling good enough to drive home. Driving requires a lot of concentration!

I stayed home all day Tuesday, too. Yesterday, I tried to get up and go to work, but to no avail. I slept very little on Tuesday night, and I felt worse on Wednesday morning than I'd felt all day Tuesday. So I spent yesterday recuperating too.

This morning, I wasn't at 100%, but I was feeling good enough to do a light day of work. Also, a classmate of mine is here today, giving a talk, and I wanted to see his talk. The nice thing is, going to a talk is an acceptable and low-effort way to spend my time.

I spent a lot of time watching daytime TV. I was too sick to do much more than be a spectator. My better half was also sick, which made it hard because neither of us could really wait upon the other. Mostly we just complained a lot to each other, and took turns getting food.

Also, I had read on the internet that a good way to get rid of a smoky smell is to dip a towel in a half vinegar-half water solution, wring it out, and then sling the towel over your head like a helicopter blade. On my way home from work on Monday, I stopped at the grocery store to buy some vinegar, because I figured that for $1.29 I wouldn't mind trying to see if this home remedy would help dispel this smoky smell. So on Monday evening, I gave it a try. It was a great effort to swing that towel around while at only about 40% power, but it was worth it, because it worked! It didn't get rid of all the smoke, but it definitely cut down on it. So this is a shout-out to my new friend vinegar, the odor neutralizer.

I'm at about 75% power right now, and this is a definite improvement. I think I'll last the whole day today.

Monday, March 06, 2006

Adventures in Sheer Stupidity

Is there a way to reload life? Because, if there is, I'd like to reload and replay the past twenty-four hours.

Yesterday I baked a delicious chicken. Rosemary-lemon chicken is my favorite baked chicken recipe, and I was having a craving for it so I cooked it, even though we had been out shopping all afternoon and it takes an hour and a half to bake. I didn't care. We ate after 8:00 p.m. but the chicken, along with freshly-baked seven grain French bread (thanks, Kroger!) and spinach salad really hit the spot.

At about ten, I went back into the kitchen and deboned the chicken, in preparation for making chicken soup. I love homemade chicken soup, at least the way we make it, plus I'm trying not to get the nasty cold that Jeff brought home from San Francisco. So I was really looking forward to this soup. I put the bones in a big pot, added enough water to cover the bones, added some more rosemary and a bay leaf, and put the pot on the burner to boil. I asked my husband to turn it off before he went to bed, which would be several hours later, and I went to get ready for bed. I retired with visions of chicken soup with kluski noodles and matzo balls dancing through my head.

I was rudely awakened by the sound of the smoke alarm, one hour later. I was confused, because at first I thought it was my alarm and the night had passed rather quickly. I heard Jeff bounding up the stairs from the basement. It turns out that all the water had boiled out of the pot, and the chicken carcass had caught on fire! My brave husband extinguished the fire, although there was still a lot of smoke, billowing from the kitchen. He turned off the central furnace system, opened all the windows and doors, and tried to let it all air out. But the damage was done, and the house still smells strongly of smoke, everywhere.

The pot is ruined. This morning I set it outside, because it smells really bad. There was a lot of splatter on the stovetop, but I managed to clean that up with my old buddy Softscrub. But I still felt really stupid, knowing that because I'd forgotten to go back to turn the burner down, I almost caught the house on fire. Can I give a big shout-out to my good friend the smoke detector? Thanks to its warnings, my husband was alerted to the fire before it really got out of control. I think the homeowner's insurance company is also grateful to the smoke detector.

Can I just take this time to say make sure that your smoke detectors are in working order? They might save your ass.

Unfortunately, I think that the excess smoke inhalation may have put me over the edge, so I woke up this morning with a tightness in my lungs. I'm sure that rushing into that smoke-filled room didn't help Jeff's lungs either.

Then, because life wasn't exciting enough for us yet, I discovered as I got into the car this morning that someone had broken into the car! It wasn't exactly a break-in, I guess, because the car was unlocked. Still, I was pretty upset to find that some handjob had helped himself to the cheap-ass CD player in our car. Thankfully, he was even more of a moron than I am, and didn't take the most valuable item in the whole car, namely our checkbook. I had seen it in there yesterday but had forgotten to take it inside. I'll have to do a more thorough search to see what, if anything, else is missing.

I was pretty surprised that this had happened. Adding to the irony is the fact that I had remarked to Jeff only yesterday, that I had never been robbed. But I was surprised because it seems like our neighborhood is a pretty safe place. The neighbors had said that the only crime that had happened in our neighborhood within the past year was somebody taking change out of unlocked cars parked on the road. But this particular person walked down our driveway and into the carport to check our car. I'm definitely going to ask the neighbors if they've noticed anything missing in their cars, and from now on I'm locking the car. I just didn't think I'd really need to lock the car in my own carport. I guess I was wrong.

Sunday, March 05, 2006

Adventures in Optimization

I think it's safe to say that optimization is my area of expertise. My dissertation was on the application of an optimization method to a problem arising in geophysics. My boss is putting me on all these proposals as the optimization expert. And the other day, I was asked to serve on a DOE review panel for a couple of proposals involving optimization.

This explains why optimization is always on my mind, then. And it explains why this blog entry is all about optimization! (Well, that and the fact that some people just come here for the math. Gotta know your audience!)

Once again, never fear, non-mathematicians! I'm not going to get all technical on you. Why would I want to relive all the technical details in my off time, after all?

Optimization is the process of finding the best possible configuration or solution to a problem. The best solution is called the optimal solution, or optimum, and the configuration at which we obtain the optimal solution is the optimizer. So, for example, if we want to buy at least three oranges from Kroger at the cheapest possible price, then the optimum might be $1.00, with the optimizer of the oranges that are 3 for a dollar.

Usually when we have an optimization problem, we also have a constraint. That is, we might want to buy no fewer than three oranges, but no more than a dozen, and minimize the price per orange. So maybe there is a bag of a dozen oranges that sells for $3, and a crate of a gross (144) selling for $5, in addition to the 3 for a dollar deal. Well, the crate price is definitely the cheapest per orange (about 3.5 cents per orange!) but it doesn't meet our constraints. So instead we must choose the bag of a dozen oranges for $3, which comes out to a quarter per orange.

Sometimes in an optimization problem, we have two conflicting objectives we wish to satisfy. For example, the other day I had a shopping list that was pretty much evenly divided between food items and drugstore items. If I wanted to minimize the dollar cost of buying everything on my list, I could buy all the drugstore items at Kmart for a pretty good price, and then head over to Kroger and buy all the food items. But I was also kind of tired (it was 6 p.m. on a Friday, after a long day of work!) so another, conflicting objective was to minimize my effort. The drugstore items were also available at Kroger, but they would cost slightly more. Which solution was better: going to two stores, or going to one store but paying more?

It would be nice if I could incorporate these conflicting objectives into a single problem. But how can I quantify "effort?" I could say that time is money, and express the problem like this:

minimize [(cost of shopping) + (my hourly pay rate) × (time spent shopping)].

The trouble with this approach is that the effort of shopping isn't always the same. For example, it takes much more effort to shop when you're hungry or sick with the flu than when you're well-rested and not in a hurry. So "effort" doesn't seem to readily translate into terms of money.

Optimization problems like these, with conflicting objectives that can't readily be combined, are called Pareto optimization problems. They are pervasive in the fields of economics, engineering, and mathematics.

Depending on how important each objective is, there can be a huge continuum of solutions. For example, if my only objective was to minimize the cost of the items on my list, I might hit several stores, compare all the prices, compile a list of the cheapest prices, and buy from a large number of stores. Of course, this process would take hours and I would have expended a great deal of effort. At the other extreme, I could pay someone to go to the store and buy all those items for me. This would require little effort but probably cost more than I would want it to.

Qualitatively, if we were to draw a graph of cost versus effort, it might look something like the following.

The x-axis (left to right) represents cost, and the y-axis (up and down) represents effort. The dotted lines represent the minimum possible cost and effort. The red dot where they intersect represents the (impossible) minimum combination of cost and effort. The blue curve represents all the possible minimum combinations of cost and effort; in other words, at any of the green dots on the curve, any decrease in cost will come at the expense of more effort, and vice versa.

You may be wondering how I could draw such a curve in real life. Well, I do have a means of converting effort to money (pay rate × time). I could quantify effort by weighting that quantity depending on how tired I am. I could set up the problem like this:

minimize (1- w) × (Cost) + w × (Effort),

where 0 < w < 1 is the weight I assign to effort. So on days when I am feeling chipper and adventurous, maybe I will assign w = 0.1, and on days when I am in bed with the flu, maybe w = 0.9. In other words, there are many Pareto optimal solutions to this problem, and I simply select the one that best suits my needs.

I mentioned before that Pareto optimization problems are central to economics and engineering. You can see from my example that the same sort of analysis could be applied to many economic issues.

As for engineering, hydrogeologists might want to determine the configuration of an underground aquifer using data from several different exploration methods, such as seismic data, borehole data, and electromagnetic data. All of these methods give different pictures of the aquifer. How do we combine the data to give us the most accurate picture?

Once we've figured out the configuration of the aquifer, we can now understand how the geology will affect the transport of a contaminant. What is the best way to clean up the contaminant, while minimizing environmental impact, cost, and time?

Suppose we want to route messages through a dynamically changing network where the cost of a path is determined by several non-additive cost functions. How do we find the best path?

Suppose we create a robot that needs to make sequential decisions in a changing environment. How can the robot make those choices?

These are all very difficult problems, and unfortunately, there are no unique solutions. The solutions generally depend on the priorities of the people posing the problems.

Friday, March 03, 2006

Adventures in Math

I know some folks come here just for the math, so here's another little piece of math to keep you coming back.

The other day I got an interesting e-mail from my fearless sister-in-law Ginger. (Ginger is a closet mathematician, but she just doesn't know it yet. I got her addicted to Sudoku, and now she's sending me math e-mails. It's only a matter of time before she starts doing calculus for fun.) Anyhow, here's what the message says:




4. ADD 1 




8. SUBTRACT 250 



I tried it on my old phone number in Illinois, 367-5384. (As an aside, that was the coolest phone number ever. It spelled DORKETH. One can be so fortunate only once in life, I suppose.)

3. 367 x 80 = 29,360
4. 29,360 + 1 = 29,361
5. 29,361 x 250 = 7,340,250
6. 7,340,250 + 5,384 = 7,345,634
7. 7,345,634 + 5,384 = 7,351,018
8. 7,351,018 – 250 = 7,350,768
9. 7,350,768/2 = 3,675,384

That is indeed a familiar number! It's the original phone number I started out with.

This trick will work with any phone number. I know you're all asking why. At least, the folks who are in this for the math are, and they're the ones who count (ha ha! a little math humor there!) so I'll now explain how this trick works.

First of all, remember from elementary school how starting from the decimal point and going to the leftt, we have the ones place, then the tens place, then the hundreds, etc.? In each of those slots, we can put any integer from 0 to 9, and this represents a number. The number is just the sum of each of those integers times the proper power of ten; for example,
436 = 6 x 1 + 3 x 10 + 4 x 100
= 6 + 30 + 400
= 436.

So, if we were to write out our phone number as an integer, we could write it as
abcdefg = g x 1 + f x 10 + e x 100 + d x 1000 + c x 10,000 + b x 100,000 + a x 1,000,000 = g + 10 f + 100 e + 1000 d + 10,000 c +100,000 b + 1,000,000 a
= g + f0 + e00 + d000 + c0,000 + b00,000 + a,000,000
= abcdefg.

So, following the above algorithm:
3. abc x 80 = (c + 10 b + 100 a) x 80 = 80 c + 800 b + 8000 a.

4. (80 c + 800 b + 8000 a) + 1 = 80 c + 800 b + 8000 a + 1.

5. (80 c + 800 b + 8000 a + 1) x 250 = 20,000 c + 200,000 b + 2,000,000 a + 250.

6. (20,000 c + 200,000 b + 2,000,000 a + 250) + defg = 20,000 c + 200,000 b + 2,000,000 a + 250 + g + 10 f + 100 e + 1000 d.

7. (20,000 c + 200,000 b + 2,000,000 a + 250 + g + 10 f + 100 e + 1000 d) + defg = 20, 000 c + 200,000 b + 2,000,000 a + 250 + 2 g + 20 f + 200 e + 2000 d
= 250 + 2 g + 20 f + 200 e + 2000 d + 20,000 c + 200,000 b + 2,000,000 a.

8. (250 + 2 g + 20 f + 200 e + 2000 d + 20,000 c + 200,000 b + 2,000,000 a) – 250 = 2 g + 20 f + 200 e + 2000 d + 20,000 c + 200,000 b + 2,000,000 a.

9. (2 g + 20 f + 200 e + 2000 d + 20,000 c + 200,000 b + 2,000,000 a)/2 = g + 10 f +100 e + 1000 d + 10,000 c + 100,000 b + 1,000,000 a
= g + f0 + e00 + d000 + c0,000 + b00,000 + a,000,000
= abcdefg (!)

I thought this was a cute trick. I wrote all the steps out there, but you can see, if you gather like terms, that after step 5, we have 2 x a,bc0,000 + 250. (The 250 is just there to throw you off the track.) After step 7, we have 2 x a,bcd,efg + 250, and after we take away the extraneous 250, we obtain 2 x a,bcd,efg. Finally, in the last step, we divide by two and get back the original number.

I have to admit that I did follow step one and pull out the old calculator when I first tried this trick. Although perhaps it would have been more fun to do it with my trusty slide rule or my new abacus that I bought in San Francisco. Until next time, math fans!

Wednesday, March 01, 2006

Adventures in San Francisco

San Francisco is one of my favorite cities in the world. I have fond memories of it as a young child, as a college student, and now as an adult.

When I was five, my family spent a year in Davis, California, close enough to San Francisco to visit the city fairly frequently. I have fond memories of the Exploratorium and Golden Gate Park from those days. As a college student, I spent a summer doing research at SRI International in Menlo Park, California, close enough to San Francisco to visit the city on several occasions. I have fond memories of Golden Gate Park, the Golden Gate Bridge, Chinatown, Cable Cars, and more, from those days.

On Friday night, Jeff and I rode the cable cars and it was a lot of fun. Just riding the cable cars is fun, but sharing the experience with him is what made it a wonderful experience. On Saturday morning, we took a ferry to Alcatraz. I had never been to Alcatraz before, so this was a new experience for both of us. The audio tour is narrated by a former correctional officer of the prison, and also features the points of view of another officer and two former inmates. It was just fascinating. Alcatraz was a very safe and efficiently run prison. The hardest thing the prisoners faced was the feeling of deprivation. The view from the island is just spectacular: San Francisco on one side, Marin County and the Northern side of the bay on the other. The punishment of seeing and sometimes even hearing all that you're being deprived of must have been severe.

On Saturday we also visited Ghirardelli Square and Fisherman's Wharf. At the wharf was a museum called musee mechanique, with a lot of old amusement machines, such as one that showed 3-D slides of the 1906 earthquake, and another that was a dummy that would just laugh and laugh.

On Sunday it rained all day, but we didn't let this stop us from having fun. We used our bus passes to go to the Golden Gate Bridge, where we took a lot of pictures and walked part of the way across the bridge. Because of the cold rain, we did stop touring a little earlier than we might have otherwise.

We woke up early on Monday morning, took the shuttle to the airport, and got home by 9 p.m. It was a fun trip, but we were glad to be home.