Monday, March 14, 2011

Exceptional or Ordinary? You decide . . .




How do you solve math problems in your head? Perhaps a better question is, do you solve math problems in your head? With the availability of electronic devices to do it for us, I would not be surprised to learn that many people never try.

I was reading Darold Treffert’s book on savants, and I was intrigued by a few examples of savant thinking. I tried solving some of the problems in his book to get a feel for how “comprehensible” they might be to me, with no recent practice calculating. Here is a simple example:

You have a carriage with a wheel that’s six yards in circumference.How many revolutions will the wheel make while traveling two hundred twenty miles?

This is how I answer that question in my head. I’d be interested in how you might do it:

Six yards is eighteen feet. I see that as a short line.

So one hundred revolutions of a six yard wheel would take me 1,800 feet. That’s a much longer line in my head, one that curves.

Three hundred revolutions would take me 5,400 feet – more than a mile. Now the line has curved back unto itself, making a circle.

How many rotations are there to a mile? Less than three hundred. A mile is a smaller circle. I can see those circles, on inside the other.They do not quite match.

I adjust the length of the longer line that forms the big circle. Try 290 . . . that’s 5,400 less 180, or 5,220. A mile is 5,280. Now I see the line laid flat, like a straight stretch of highway. Two hundred ninety revolutions leaves us sixty feet short of a mile marker. So what’s the fraction?

Three eighteens go into that sixty-foot remainder with the same six remainder. Adding that to the 290, I see the answer is 293 and a third. The six-yard wheel does not fit a one mile line, but it fits perfectly into a three-mile ring. If you put a mark on the wagon wheel, and mark any point where it touches the big circle, those points will touch every time the wheel rolls past. I like that.

If you roll the same wheel around a one-mile ring the points will only touch every third trip around, which is unsettling to me. I like smooth fits, so I will solve the next step using three-mile units.

I can now see the answer: 880 revolutions. A perfect fit. Six yards, three miles, and eight hundred eighty turns.

How many three-mile eight-hundred-eighty revolution units are there in 220 miles? My mind visualizes stacks or piles for this next step.Seventy units reach two hundred ten miles. I quickly see how seventy-three and a third are needed to reach the two-twenty goal.

Stacking seventy-three piles of 880 in my mind takes a little time.Eventually, the stacks add up and I see the result is 64,240. Now I just have to add the third (of 880) and I’m done. To do that, I add three hundred to the pile, making 64,540, and then take back six and two-thirds.

64,533 and 1/3 is the answer to the question.

As a further experiment, I scaled up the distance, to 2450 miles and then 20,315 miles to see if I could keep scaling up the numbers. There must be some limit to that, and it certainly took me longer, but I solved those bigger problems in a few more minutes. Solving the longer distance problems involved one and then two more levels of “stacking” in my mind.

It does not seem that hard to me. I often did similar calculations as a kid, for fun. I’m sure I could do it again, pretty quickly, with some practice.

I test my answer with a calculator. The process to do that is considerably simpler.

I multiply 220 (miles) by 5,280 (feet per mile) to get 1,161,600 – the total distance in feet.

I divide that by 18 (the wheel circumference) to get 64,533.333 – the revolutions turned.

It’s a lot faster to get this answer with a calculator, for sure. But is the ability to figure this out in one’s head really exceptional? In today’s world, I would not be surprised if kids never develop these skills. When I grew up, though, pocket calculators did not yet exist and I had to know how solve problems like this. I suspect many people of my generation could solve a problem like this in their heads, but perhaps I am wrong. What do you say?

Thursday, March 3, 2011

Some thoughts on spark plugs



Advances in technology have lengthened the maintenance intervals for many pieces on our cars. One of those pieces – the one I am writing about today – is the spark plug. All gas engine cars have them.

When I started in the car business, it was common for plugs to need cleaning every year, and replacement by 15,000 miles. As technology improved and engines got cleaner, the plugs started lasting longer. First it was 30,000 miles, then 50,000, 60,000 miles.

Today, many of the new cars we service have 100,000-mile spark plugs installed at the factory. That 100,000-mile rating was derived by installing the plugs in test vehicles, and then driving them hard and fast to pile on the miles. At various intervals the plugs were removed and inspected for wear. After a number of engineering tweaks, Bosch, Beru, NGK, and other spark plug manufacturers came up with a plug that would last the 100,000 miles and still perform acceptably. Based on that, the carmakers established the current change interval for spark plugs.

So the question today is: How often should you change your plugs, if you have a car with the 100,000-mile change interval? Should you follow the manufacturer’s recommendation, or do something else? Why?

The first thing I point out to new clients when we discuss maintenance is that there is a time component to service as well. Spark plugs may hold up fine for 100,000 miles if driven on the highway every day, but short trips and occasional use will wear them out a lot faster. A person who drives 10-12,000 miles per year may not hit 100,000 miles for almost ten years. That’s way to long to leave a set of plugs in the car.

Carmakers recognize that. If you look in most owners manuals you will see a time specification for plugs. They’ll say something like five years or 100,000 miles. I strongly suggest you pay attention to this time limit when considering long-life wear items like spark plugs.

I have read of spark plugs breaking off in the cylinder head when removed after many years. I’ve never experienced that on a five year old car, but there may well be parts of the world where corrosion is worse (near the ocean, as an example) and it you live in such a place, you’d be wise to consider that fact and adjust your service intervals accordingly.

When changing plugs, the next decision a motorist faces is what plug to buy. If you are at the dealer, the decision is simple: you’ll get original equipment plugs. If you’re at a Bosch Car Care Center, you should get the correct Bosch plugs, and if you’re at an independent or chain store, you best find out what they propose to install to be sure you are comfortable.

Thirty years ago, selection of spark plugs was simple. A dozen part numbers would service a majority of the cars on the road. Today many cars have special plugs and it’s important to install the right one. Every parts store has cross-reference catalogs, all of which lead you to believe any “crossover” plug will work. In my experience, that’s often true for older cars but often wrong on newer vehicles. We’ve seen Land Rover, Mercedes, and BMW cars with ignition misfires that were ultimately traced to “supposedly correct” but off-brand spark plugs.

If you have a late model car, be sure you fit the right plugs.

The last point I’d like to address with spark plugs is what happens if you don’t change them in time. As plugs age, the voltage to fire them increases. A plug that needs 20,000 volts to fire when new may need 80,000 by the time it’s used up. If you go beyond that, the voltage may rise to 100,000 volts or higher. This increased voltage puts much greater stress on ignition coils and wires. Premature ignition failure is the usual result of running plugs too long.

Six spark plugs might cost $80 for your BMW or Mercedes. Six spark plugs and six coils (because you waited too long) could cost $700, maybe more. As you can see, changing plugs before the ignition fails makes very good economic sense!