Sudoku Logic II

This morning's Sudoku puzzle illustrates an interesting sequence of logic that arises in perhaps 1 puzzle in 10.  The partially completed puzzle is shown.  The logic concerns the contents of the squares A1, A3, E1, and E3.

By looking at the 3's and 7's in columns D and F, it is easy to see that the squares E1 and E3 must contain 3 and 7.  Row 3 is also missing 3 and 7, so that A3 must also contain 3 or 7.

Now think of E1, E3, and A3 as forming three corners of the rectangle A1-E1-E3-A3.  Whenever you have four squares in this arrangement, and three allow only the same two numbers (3 and 7 in this case), then the requirement that the puzzle must have a unique solution means that the 4th corner (A1) cannot be either of the two numbers.  For, if it did, then both (A1,E1,E3,A3)=(3,7,3,7) and (A1,E1,E3,A3)=(7,3,7,3) would solve the puzzle.  Caution: this rule only works if the 4 corners are contained within only 2 3X3 blocks of the puzzle.  If all 4 corners are in different 3X3 blocks, the logic is not correct.

So, A1 cannot be either 3 or 7.  Now look more closely at the A1-C3 block.  Neither B1 nor C2 can be 3 either.  Therefore the only place to place a 3 in the A1-C3 block is at A3.  It follows that E3=7, E1=3 as well.

Later in the same puzzle, a similar situation came up again (very unusual):

H1, H9, and I9 only allow 7 and 8.  What number goes in the fourth corner of the rectangle (I1)?

Flying Rebels from Addis Ababa

A couple of years ago, as Christmas was approaching, our youngest son couldn't figure out what he wanted for his main gift.  Suddenly, with about a week to go, he made up his mind.  A Packer's jersey, #12, would do just fine. The catch--his name on the back, instead of Aaron Rodgers'.  He scoured the internet, and found a site with a good price.  But of course it would not be available by Christmas.  Still, he preferred just the right jersey to having it on the Day, so he was willing to wait a couple of weeks more for his dream jersey.  

We got an email from "Eddie", explaining that since it was a custom jersey, it would be seven working days to get it finished, and then they would ship it. OK, fine. Then we noticed in the fine print that it would ship from Shanghai. Shanghai!  It'll be a month! Disappointed, we hankered down for the wait.  It was tough, because the Packers were doing well and we needed that jersey.

Imagine our surprise and joy when, just a couple of days after Christmas, the jersey was delivered at our doorstep.  From Shanghai!  How was it possible?  Only one way--they flew that jersey all the way, 11,000 km, from China.  

Today an email brought this to mind.  I have a delightful niece who some time ago pointed out an Ethiopian shoe company whose shoes are hand made by female Ethiopian artisans.  Fair trade, save the planet, recycled tires for soles, Abyssinian pure leathers--all that good stuff. Get some cool shoes, and do a good deed. Nice. Now, most of their shoes are too funky for a boring physics professor, but I found a pair that seemed not too outlandish and ordered them for $80, shipping included--not a bad price.  

I was a little disconcerted when the delivery time was quoted at one month, but, hey, it's got to get here from Addis Ababa, 12,000 km away. For $80 shoes, they're probably shipping them through some sane, but slow, method.  Right?

Wrong.  I just got the shipping notice. 3 business days delivery after shipping from Addis Ababa! They're sending them by air. So much for saving the planet! Those "renewable" Abyssinian hemp shoes are smokin' through non-renewable liquid fuels like crazy.

These two incidents illustrate a basic problem.  It is just too cheap, too easy, and downright nutty to waste the world's most valuable commodity, petroleum, sending non-perishable goods all over the globe. It comes so naturally to us, we hardly think about it.

We will one day rue such extravagant habits. We need a massive tax on oil to account for the future opportunity loss from our current wasteful practices.

The $30,000 Parking Space

The University of Wisconsin-Madison grew up in the center of the city, before the advent of the automobile and its requisite parking.  Until 20 or so years ago, parking was not much of an issue; students lived in the campus area (they still do) and most professors and support staff lived in Victorian homes within walking distance.  Those homes are now out of reach of most university employees, and most people now commute from the outskirts of town, or further.  There is now a dearth of parking spaces. Competition is ferocious to get them, and the cost to employees is significant, up to $1100/yr to park in a garage.  Another way for the university to make money, right?

Wrong.  I was astounded to learn that the cost of building a parking garage is $30,000 per space.  Amazing!  With that number, it is clear that the university is actually heavily subsidizing the parking.  At $1,000/yr for parking, it takes 30 years to recoup the capital cost of the garage.  I'll bet that the garage has to be rebuilt more often than that.

I admit to being, for a long time, dubious of the accuracy of $30,000/space.  Then a couple of years ago a famous Austrian physicist, Rainer Blatt, visited the UW physics department.  At lunch, we were asking him about the new institute they were building for him in Vienna, and he was describing the various travails of getting the building built, including the required parking.  I brought up the $30,000 number, which was pooh-poohed by several at the table.  Prof. Blatt thought quietly for a minute or two, then affirmed that that was close to the cost for the parking ramp he had to build for his new institute.

It is becoming increasingly clear that our customs of automobile usage are ferociously expensive, though often hidden from us, and are generally heavily subsidized.

Massive Education

I was cleaning my office this morning and came upon this standard introductory physics textbook:

It is a typical, massive, introductory physics text.  Its mass is somewhat more than a ream of copier paper.  It is 1088 pages long.   The current edition sells at Amazon for, brace yourself, a massive $268. The core content of this book is essentially identical to the book I learned freshman physics from in 1979.  

This book is a symbol of the massive problems facing modern university education.  First, in an attempt to be meaningful (whatever THAT means), it is filled with superfluous "applications", historical anecdotes, philosophical reflections, and photographs.  Here, for example, is the last paragraph of page 1088: 

It expresses some nice sentiments.  But one senses the emotion of the author, having made it so far, is being released in a final poetic burst to the line.  I doubt that the fatigued student, recognizing immediately that there is no content in this that will be on the test, will read it. For the student, the effective finish line was likely hundreds of pages earlier. Maybe, tragically, a thousand.

Second,  in my opinion the actual relevant content of this book, and its many twins, is about 200 pages worth.  The trouble facing the student is--which are the important 200 pages?  I confess to you that the instructor can't answer that question either.  I think I know the key ideas of freshman physics, but I can't FIND them in the morass.

What precipitated this post is the business card taped to the front cover of the book by the saleswoman (who I like a lot--this is not her fault):  

Did you notice--"Concise presentation"?!?  Are you kidding me?

A third problem the book serves as a symbol of, at my university at least, is the context in which this book is used: a massive course with hundreds of students.  We pack 200 students into the lecture hall two times a week and preach at them.  It is a very unsatisfying experience for everyone involved.  To our credit, we also break them up into small 25 student groups led by graduate students where the real learning occurs.  But it is a very difficult situation and most students transfer their hatred of the massive  course to the subject matter, which is a real tragedy.  We know what we should do--break up the class into a large number of smaller classes where the students get individual attention.  We've done this with our introductory course sequence for physics majors, and doubled our number of physics majors as a result (I consider this one of my major contributions to my university).  But we just don't have the money (facilities, people), to do this for the literally thousands of biology and engineering majors who are required to take introductory physics each year.

One irony of this situation is the current excitement about MOOCs--Massive Open Online Courses.  These promise to solve the money problem.  Students will, from the comfort of their living rooms, watch a "master" teacher present the material.  They will then log into the electronic homework system where their work will be automatically graded by computer.  Human interaction?  Electronic chat rooms will do!  

Education is an intensely human activity.  People need to interact.  Even at the most advanced levels of research, THE most effective learning occurs in small workshops where experts interact face-to-face.  Massive books and courses don't, and won't, get the job done.

Interesting Sudoku Logic

I enjoy working the Sudoku puzzles in the newspaper.  Starting on Monday, the difficulty increases from 1-star to 6-stars on Saturday, with a 5-star on Sunday.  I find I can nearly always work 1- through 4-star puzzles without making any marks on the paper except filling in the numbers.  Unless I want to put in 30 minutes or more of deep thought, 5- and 6-star puzzles usually require some guesswork that I write with small numbers in the corners of the unfilled squares.

Today's 5-star puzzle, however, I managed to work without guesswork, using a logic sequence that I cannot remember using before.  Here is the partially worked puzzle, with a few key squares shaded:

For you logic fans, where should the "2" go in the upper-left corner?

Here is the interesting logic sequence.  Notice that the green squares must contain "2" and "4".  In the upper left, the "2" can potentially go in either the blue or red boxes, and a "4" must go in one of the blue boxes.  But I happened to notice that if the red were not a "2", then both the blue and the green boxes would contain "2" and "4".  But this can't be, since then the solution would not be unique-- if the blue had the "2" on top of the "4" then the green would have the "4" on top of the "2".  But an equally valid solution would have the blue "2" and "4" switched, with the green also switched.  Thus there would be two valid solutions, which is not allowed.  The conclusion: the red box MUST contain a "2". Putting a "2" in the red box made the rest of the puzzle easy to solve.

Here is the full solution.

Green Bicycling

A few years ago I subscribed to Bicycling Times (BT) magazine.  I enjoy commuting to work by bike when it's warm and light enough, and the occasional day trip on the wonderful bike paths that criss-cross Wisconsin.  BT caters to people like me.  It has many articles on practical bikes and gear, and whimsical uses of bikes as well.  There's also a certain save-the-planet mentality that appeals to me.

One issue had a marvelous article on bamboo bikes.  If you live near New York City, you can go to the  Bamboo Bike Studio in Brooklyn and build your own!  Just the kind of thing I'd enjoy doing some day.

The article rejoiced in the renewable, practical aspects of bamboo.  It grows like a weed in New Jersey, and its strength makes excellent frames. 

In the next issue, there were a couple of letters to the editor which I unfortunately did not keep and cannot find on the BT website.  The letters argued that such bikes were not really "green" since they used carbon fiber on the joints.  I composed the following reply, which to my regret I never sent in.

"Most issues of Bicycling Times include one or more items that touch on “green” bicycling; the most memorable described some cool bamboo bicycles.  Amazingly, these precipitated complaints from some of your more earnest readers that joining the bamboo together with carbon fiber destroyed its green advantages.  Huh?  In comparison with the alternatives, riding bicycles is so green that it hardly matters what the bike is made of. Be yours carbon, magnesium, scandium, petroleum, whifnium or whafnium--as long as you ride it,  you’ll still be far greener than your fossil-incinerating neighbors.  Relax and enjoy the ride."

An early version of the above replaced "petroleum" with "plutonium"--tongue-in-cheek, of course.  The point is that the use of non-renewable, even exotic, materials on items such as bicycles that last a long time and, in particular, serve as a substitute for liquid-fuel transportation, makes ecological sense. The amount of petroleum that went into the above carbon fiber joint is probably comparable to a few block drive by the diminutive hospital worker who barreled past me the other day in her gigantic F-150 truck.

Until we manage to ween ourselves off unnecessarily using our valuable petroleum reserves to move ourselves effortlessly from one place to another, there is no point in obsessing over trivial matters such as carbon fiber joints on bamboo bicycle frames.

Is Abortion Rare?

From a 2012 New York Times editorial page blog , entitled "Is Abortion Rare?":

"In a given year, 2 percent of American women between the ages of 15 and 44 have an abortion. That means 98 percent of them do not."  To the blogger, this implied that abortion is rare.

Hmm--what kind of logic is that?  At 2 percent per year, over the 29 year span cited, this would imply that the probability of a given woman having an abortion in her lifetime would be 

 That is a remarkably high number.
$\mathrm{<br>}$ An even better perspective on the frequency of abortion is to simply compare annual abortion rates and birth rates.  In 2008, the latest year I found both numbers for, there were 4.2 million births in the U.S.  How many abortions?  According to the Guttmacher Institute, (and several right-to-life pages give similar numbers), 1.2 million.  That implies that 1.2/5.2=22% of all pregnancies are aborted.  That's 2 out of 9, almost 1 in 4.

An informal survey of my generally well-informed and well-educated friends, on both sides of the issue of whether or not abortion should be legal, finds that 100% of them are appalled by these numbers.

All agree: abortion is much too common in our society.  It is anything but rare.

Offshore Bank Accounts and the U.S. Debt

Did you notice this news item a few weeks ago?  The ICIJ obtained and released records of off-shore, secret, bank accounts of 250,000 organizations and individuals.  That there are substantial sums in such banks is known by anyone who has read spy novels.  But the real surprise is the amount of money represented:  at least 21 trillion dollars.

Numbers like $21 trillion are very hard to grasp without something to compare them to.  In this case, a good comparison is to the public debt of the United States.  For everyone knows that the U.S. is in hock for an unconscionable sum that will be impossible to pay off in anyone's lifetime.  What is the size of the U.S. debt?  As of April 2013, it is $11.9 trillion.

The staggering truth behind these numbers is that a relatively small number of people and organizations are so wealthy as to have (formerly) hidden money representing about twice the total public debt of the United States.

It is fashionable to present the public debt on a per-capita basis, $38,000 each.  This number is meant to frighten, but it is clearly deceptive.  For a relatively small number of people have more than enough hidden assets to repay that debt in full.  Any kind of progressive weighting of the debt will substantially lower the debt figure for the average person.

The final conclusions?  1) Our national debt, while serious, is a solvable problem.  2) We have an insanely inequitable distribution of wealth.

The Price of Gasoline

The fixation of modern American society on the price of gasoline at the pump is amusing.  The local newspaper  frequently runs front page stories when the price of a gallon of gas changes by more than a few cents. Gasbuddy.com tracks gas prices to 4 significant digits, as if my transportation decisions would be altered were the price to change from $3.362/gal to $3.363/gal.  Republicans and Democrats alike seem to agree-the price of gas is TOO HIGH!  The President should do something about it!

So, how expensive is gasoline?  One way to quantify this question is to compare the cost of gas to the total cost of operating a typical car, which is:   59.6 cents/mile as of 2012, according to the American Automobile Association.  That's a shocking number--can it be right?  Does it really cost me 6 bucks to drive across town?

Well, the AAA probably knows what it's talking about, but the AAA is not an unbiased observer so perhaps that number is exaggerated.  We can however check to see if that number is reasonable if we remember that the IRS is evil.  We can all agree on that.  Since the IRS is evil, and since automobile travel is deductible for business expenses, volunteer work, etc.,  it follows that the IRS would never exaggerate its reimbursement rate for automobile usage.  We can therefore take the IRS mileage reimbursement rate to be a lower limit on the average cost per mile of operating a typical vehicle.  What is the IRS rate?  56.5 cents per mile for business expenses! Surprisingly--suspiciously--close to the AAA number.

The next thing number we need in order to figure out how expensive gasoline is is the "gas mileage" of the average vehicle.  Data from the Federal Highway Administration for cars extrapolate to 23.3 mpg for 2013.  (23.0 mpg in 2010, with a trend of 0.1 mpg/year to get 23.3 mpg in 2013).

Now we can get an idea of what fraction of the cost of car ownership goes to gas.  At 23.3 mpg, and $3.36/gal, it costs 14 cents/mile for gas.  What!?!  Compared to the total cost of ~56 cents/mile, that means the cost of gas is only 25%,  1/4, of the cost of driving a car.  If the price of gas doubled--a hypothetical event that the newspapers would no doubt describe in the most apocalyptic of terms-- the cost of owning a car would increase by only 1/4.

Gasoline is dirt cheap.

This reminds me of an old Bloom County strip.  "Louise, dump the milk.  The cat drinks unleaded from now on!"