The cost of long life

We expect expensive things that we buy to have a moderately long life. For example, in 2 years:

• Car: tires shouldn't wear out, paint shouldn't peel, seats shouldn't tear
• Dress Clothing: Cuffs shouldn't fray, buttons shouldn't fall off
• House: roof shouldn't leak, windows shouldn't break, pavement shouldn't crack

Some thing have shorter lives, such as mobile phones or children's toys or laptop computers. We only worry if they fail too soon - in a matter of 2 months rather than 2 years. But what about ourselves? Are we ourselves lasting too long? Not making room for new models? In constant need of expensive repairs?

About 800 years ago, eyesight became one of the first age-related disabilities to be countered, with eyeglasses. Now we have contact lenses, implants and other corrective surgeries of various types.

Impaired hearing was tackled a few hundred years later, with the widespread use of ear wax cleaning spoons and ear trumpets. Now in this century we've developed hearing aids and cochlear implants. 

Older people have always faced stiffness and arthritis - and as our population exercises more, joints are wearing out even faster. But technology has come to our aid. Many people have new knees or artificial hips.

All of these things are beneficial for older folks who need relief, but no so good for the up-and-coming generations who are paying for our repairs. We are a burden to them, and the statistics are depressing!

Globally we face increasing health costs, pensions, housing congestion, senior care shortages, etc. because longer lives do not automatically translate into longer careers. There are penalties for retiring early but no great incentives for staying on the job years longer.

Although many jobs can be managed by a very-senior persons, in general companies like to move the old out and slip the young in, if they can. And they young need those jobs too, to support us.

Japan is leading the way, with the highest number of retired people relative to the lowest number of working people. Soon they'll have more retired folks than they have working people. When those retired Japanese (mostly men) leave their offices, they start using up their savings, drawing on their pensions, and also sit around moping and making their spouses miserable with RHS (retired husband syndrome).

I wouldn't be surprised if a few old complaining Japanese men get konked on the head with a cooking pot - but even without that potential threat we will have a surplus of elderly women. Men die far earlier than women from health-related causes (nearly 5 years earlier, on average) and many more men have died in wars. 

So. I had planned to support all these assertions with tables, graphs and charts - and maybe someday I will. But it's Friday and I have other things to do. Let me just conclude with this parting thought:

If we look forward in our crystal balls will we see a future filled with huge numbers of poor, elderly women wearing eyeglasses and shouting back and forth to each other about their former husbands and their new artificial knees?

Or will they all be suffering from Alzheimer's, dementia and other mental deterioration, and thus have forgotten all about us? (That's going to be my wife in the middle, in the track suit!)

If you don't want to guess, work it out in your head

Perhaps you have heard of the term math prodigy. Or math savant. Or mental calculator. Or mentat.

Are you one?  To find out, you could download the 2010 World Cup of Math material to practice on.

But first, I'll save you some effort. Get the material only if you can solve all these sorts of problems in your head:

1. calculate the number of days between each pair of 50 pairs of dates 
2. subtract 10 pairs of 8-digit numbers
3. add 10 sets of ten 10-digit numbers
4. calculate the square roots of ten 6-digit numbers
5. solve 27 multiplication and addition problems involving 8-digit numbers 
6. calculate cube root, fourth root and fifth root of three 6-digit numbers
7. solve some algebraic calculations
8. look at 20 different 5-digit numbers and identify two prime factors for each
9. solve a multiplication problem involving a fraction, pi and a decimal number cubed
10. look at 20 different addition or subtraction problems involving 2 or 3 digit fractions, then decide if the sum or remainder of the problem is greater than, equal to, or less than another integer

Yes or No? Are you going to the download? 

You may be able to do some of them in your head, some on paper, some with a calculator, etc. 

But to give you an example of a mental calculator's ability, let's take one of the 50 sets of dates in Example 1. 

I'll give you two dates and you tell me the number of days between those two dates. Ready? Go!

 23 July 1962 to 05 August 2026  

My Guess
Mentally I see it's 64 years, 13 days which is about 22,000 days. That's my guess and I'm sticking with it. I guess I'm too lazy to try harder.

The Answer
My calculator says the real answer is 23,389 = (((64 x 365 + (64/4)) + 13)  so I was only 10% off or so!

Mental Calculator
The mental calculator winner this year, Yusnier Viera, was able (in his head) to determine 48 of the 50 answers correctly in one (1) minute!

Momma mia!

For the rules on how to compete, read on:

Witnesses
Record attempts have to take place in public (university, school or museum). Two witnesses must confirm the record and at least one must have a professional background in mathematics.

Presentation of the Problem
Calculations can be done with the problem in sight. The mental calculator can dictate, write or type the answer but is not allowed to write down intermediate results.

Number of Attempts
Only 10 record attempts may be made within a period of 24 hours. The calculator should clearly say when he or she stops practicing. The witnesses may then signal the start of an official record attempt.

Timekeeping
Timing begins when the task is shown to the competitor and ends at the end of writing/dictating the answer. Two stop watches should be used. At the end of the attempt, average the two watches.

Computer Use
A computer program may show the tasks on the screen, let the calculator type an answer, compare the results and measure the time until the last keystroke. Correcting while still typing is allowed, but if the computer says a result is wrong, no correction is allowed. Attempts must still be in public and witnessed by two persons.

Are you ready?

If you've already decided this is your thing, go to Memory Camp and start working out!

To meet the mental calculators you'll be competing against, go here or go here.

If you know that you don't know, then guess

Guess - to form an opinion based on insufficient information; to arrive at a conclusion by conjecture, chance or intuition.

Guessing is often discouraged and downplayed as a means to determining a solution to a problem.

Personally, I think it's a legitimate option. Looking at test-taking strategy courses and websites, you will find plenty of math evidence to show that the odds are better if you guess than if you leave an answer blank.

In most cases, you get 0 points for leaving an answer blank, and you will definitely get more than 0 if you guess the right answer.

If there are 5 options, then you have a 1:5 chance (20%) of getting the answer if you guess. And you can increase your odds if you can eliminate several of the 5 options as impossible or unlikely.

Some tests penalize you for guessing and getting the wrong answer, but the penalties are unlikely to equal the value of getting the occasional answer correct from guessing. To determine the exact value, we'd need to know more information about the type of test and the penalty strategy. For example, guessing on an essay question is much more complex ...

But let's forget the "tests" and talk about real life instead. Do you guess very often?

I guess you do. For example, we guess about what to order in a restaurant - choosing something that sounds good. If you've had it before by the same chef at the same place, you might be making a an educated guess, but it's still a bit of a gamble compared to walking down a buffet line or sticking your finger into the cooking pot at home ... hence the plaintive cries of  "I don't know what to get!"

Let's take another term called guesstimate. Maybe using this word helps a bit. It certainly sounds better, with a dollop of estimate making the guess seem more credible. It allows us to use our previous experience, a bit of evidence and some reasoning to construct an educated guess.

I guess if you want to learn more about guesstimating, you could buy a book on the subject from Amazon. My guess is that it would cost you about $10-15 dollars and be delivered in 2-5 days, right to your home, or in a minute or two to your Kindle.

I found another book whose title can be roughly summarized as  "How Many Licks on an Ice Cream Cone, or how to estimate d*** near anything". That's guessing - number of licks on a cone...

After some consideration, I came up with an example where guessing causes nothing but grief ...

How much of what you know do you show?

Yesterday I wrote about how we all know much more than we can possibly share (or be tested on).

Today I'd like to ask - How much of what you know do you show? 

To what extent is the world aware of your mental prowess? Or to put it another way, do you Play dumb?

I want to distinguish this from the Sorry, I didn't hear you ask the question... ploy that we use with our moms or wives or husbands. Instead the Dumb technique is used when you don't want to get engaged in a subject, or you don't have any interest in learning something, so you just put on your  I Don't Get It and I Don't Want To Get It face.

We  know kids do this in the classroom. What's 2 + 2 =  you ask?  I dunno is the reply.

Are they not listening? Not thinking? Not caring? Not on the same astral plane?

Feigning ignorance is an effective tactic! says one article I read.

Another goes on: Let's face it, sometimes we just want to be lazy ... the easiest way to avoid responsibility is to pretend we are not as smart as we really are. People will be loathe to give you jobs to do, and won't be as mad if you make a mistake.

OK, I suppose it could be useful now and then - assume we all do a bit of this on rare occasions. It's a way to ease through a few tough spots in life. Not a responsible way, but an optional path. However, I am interested in the implications when it comes to math.

Do we say I don't get it when what we really mean is I don't feel like thinking hard today?

Would the IRS tolerate it if we said I don't think I really want to fill out my 1040 this year. It's too tiring.  I think we would quickly decide the penalties for playing dumb are too severe.

The dumb and dumber attitude goes beyond the classroom and extends to the testing room. In many states, students are asked to take standardized achievement tests at the end of the year. If they do well, (1) they get some useful credit in their personal account in life, (2) it helps generate a good reputation for the school and (3) there's a bonus of satisfaction for their teachers. 

Or so we are told. It's easy to make an argument that tests are just part of a giant churning mill that does no good for anyone except the companies who charge for writing and scoring them. But still - we are seeing more students blow off these tests, by not showing up at all, going through the motions and checking random answers, or actively boycotting the tests entirely.

Don't get me wrong, this isn't the case for the majority of students. Or is it? I dunno, it's too much work to find out the actual answer ...

I learned today when I searched for "refuse to take test", I got a list of stories about celebrities refusing to take breath tests for driving under the influence. If you want to read the 28-page NHTSA report to Congress on the issue, click here.  If you just want some easy math facts, read on:

• In 41 states, you have the choice of taking the test, or losing your license for one year.
• The average refusal rate is 22%
• In a 3-metro-area sample, refusals resulted in longer rather than shorter sentences
• California and Nevada allow police officers to draw your blood without a search warrant if you refuse the breath test. Refusals in California are only 6% ...

Let's hope this is one test NONE of us has to take.

How much do you know?

When we assess students' knowledge of math, we are trying to find out if they know certain things (things that we who give the test already know ourselves).

In a math test, we are not trying to discover how much you know. We are trying to determine if you know certain facts, various fact-finding strategies, and your response to specific math challenges.

Do you see the difference?

Each of us knows SO MUCH ABOUT SO MANY THINGS that no one else can measure the breadth and depth of our knowledge.

• How could they? 
• How would they know how to ask if they weren't you? 
• In what order and form could the questions be asked, or answered?
• Who could possibly grade the test of YOU?

These insights came to me today as I was driving to work. Think how many things you will never be asked on a test, yet you know them immediately. Think how smart you are! What is the sum of your knowledge?

Here are some things I know. Do you know them too?

1. Who is the man in the middle? Where he is seated? When was the photo taken?

2. Is this item animal, mineral or vegetable? Where does it come from and what is it worth?

3. Where is this place? It is dangerous to visit there? Do you need a special visa?

4. Who are these people and what are they pointing at?

5. What is the answer to this:  (2 x √(2 + 2) ) ÷ 2 =

The amount of knowledge we all possess is incredible. The amount of math we know is less than incredible but more than enough for most of us. The question to ask yourself is:

Do I know enough of the right stuff to get through the challenges of daily life?

The answer is probably no for most of us. We have more than enough knowledge and skills, in some areas, and less than we need in others. If we see areas where we are lacking, we can learn the skills if we want to, or we can ask someone who knows the answer.

Here are the answers to the test I just gave you:

1. Ronald Reagan, Coronado, California in 1958.
2. It's a black truffle, which is "fruit" of a fungus, it's from Italy and worth about $400-500. It's not quite a plant, not a mineral, and it's not an animal ...
3. It's a pile of sea salt sitting next to San Diego Bay. It's not dangerous and you don't need a visa.
4. This is a bronze relief map of the Scottish highlands. Tim, Jan and Lena are finding our campsite.
5. Two times two (square root of four equals two) equals four, divided by two equals two

Even if you couldn't answer all these questions correctly you still know plenty of stuff that I don't know! Give yourself an A on general knowledge of life!

Speaking of Cuboid Complex Polyhedrons

We start working with geometric shapes in elementary math, and their descriptions and dimensions are a bit simpler than those of a fat car (about which we spoke yesterday).

For example, here's a rectangular prism. Also known as a box to most of us. But to mathematicians, it's also a cuboid, or convex polyhedron, or right rectangular hexahedron, or rectangular parallelepiped. And if those names aren't enough to scare you, it's shown here in unfolded and 3-dimensional forms.

(I drew this manually with my Illustrator software so please excuse me if the shapes are not perfect!)

There are 6 faces; each one is a rectangle. (Ignore the folding tabs.)  It has 12 edges and 8 vertices.

The long side is called the length, the vertical side the height, and the other side the width or depth.

You can draw a diagonal between any two non-adjacent vertices in a single plane (as shown by the red line) or a space diagonal line between any two non-adjacent vertices in space (as shown by the blue line).

Looking at the diagram, the dimensions X, Y and Z 

 - can be multiplied like this to get the volume:  X x Y x Z = Volume (given in cubic units)

 - can be manipulated like this to get the surface area:  2XY + 2XZ + 2YZ = Surface Area (given in square units)

Take a whole bunch of little rectangular cuboid shapes like this, combine them together, and you can make almost anything, even a car with a wide track!

Not bad, eh?