Friday, August 29, 2014

You Are Smarter than You Think

Originally published in
Northland Reader
now the
Reader Weekly
Duluth, Minnesota
February 3, 2000
Do you think in the same way all the time?  Or, do you look at things in different ways?  This article is a little test of your willingness to think creatively and to try different things.

I recently attended a dinner at which a speaker gave this exercise.
Pick any number from 1 to 9.  Multiply it by 9.  If the result is a two-digit number, add the two digits together.  Subtract 5 from the result.

Think of each letter of the alphabet as a number; that is A is 1, B is 2, and so on.  Match the number you had after subtracting 5 with the corresponding letter of the alphabet.

Think of a country name beginning with that letter.
Think of an animal name beginning with the last letter of the country’s name.
Think of a color name beginning with the last letter of the animal’s name.
Let me take out my magic wand, tap you on the head three times, and say that you thought of Denmark, kangaroo, and orange.

Now, let’s test your creativity.

First, did you ask yourself why the number part of the trick works?  Did you start juggling numbers in your head to prove it should work?  If you didn’t, your first lesson in creativity is to demonstrate why it works.  It may take several minutes of effort, but you can do it if you are willing to apply a little elementary arithmetic, a bit of abstraction to symbols, and a some small changes in viewpoint.

Secondly, did you ask yourself why I thought you would answer Denmark, kangaroo, and orange?  Are these the only country, animal, and color that begin with these letters?  Or are these more readily known than other examples?  If these aren’t the only country, animal, and color beginning with these letters, did you search your memory for any others?

Why not put the Northland Reader down right now, get out a piece of scratch paper and a pen, and work on these two problems?  First, prove that the number part of the exercise will work.  Second, list some other sequences of countries, animals, and colors in which the country name begins with D, the animal name begins with the last letter of the country name, and the color name begins with the last letter of the animal name.

OK, did you get anywhere?

The easy answer for the number question is the “rule of nine”.  Because we multiplied the original number by 9, then the second number is divisible by 9.  We will come back to 9 as our next result by applying the rule of nine.  That is, to see if any number is divisible by 9, add its digits together.  If the result is two or more digits long, add those digits together.  Repeat until you have a single digit.  If that digit is 9, then the number you applied the rule to is divisible by 9.  Since we started adding the digits together in the trick after we multiplied by 9, then the result of our addition has to be 9.

This method also applies to numbers divisible by 3, but the single digit may be 3, 6, or 9.  For now, we’ll stick with 9.

The harder part is now to explain why the rule of nine works.  Our first step is to introduce a bit of symbolism; let good old X stand for our original number.  Our first step is to multiply by 9, giving us 9X.  Now we have to make a little jump in viewpoint; 9 is 10-1, and so we have (10-1)X.  We can rewrite this as 10X - X.

That doesn’t look very promising.  Let’s try another jump in viewpoint; let Y be X - 1 or conversely, let X be Y+1.  We now have 10(Y+1) - X.  Why not replace both X’s with Y+1?  Because we need another jump in viewpoint.  First, rewrite our formula as 10Y + 10 - X.  Now let’s replace 10 - X with Z giving 10Y + Z.  If we add Y and Z using their equivalents expressed with X, we should get 9.  Here we go!  Y + Z is X - 1 + 10 - X which we can reorder as X - X + 10 - 1 which is 9!  Finally, subtracting 5 gets us 4, and D is the fourth letter of the alphabet.

To see it better, let’s put it in the compact but excruciating form we had to do in junior high algebra:

Assume we have a number 9X where X is any number from 1 to 9.
 9X = (10-1)X
 (10-1)X = 10X - X
 Let Y = X - 1
 X = Y + 1
 10X - X = 10(Y + 1) - X
 10(Y + 1) - X = 10Y + 10 - X
 Let Z = 10 - X
 10Y + 10 - X = 10Y + Z
 Y + Z = (X -1) + (10 - X)
 (X - 1) + (10 - X) = X - X - 1 + 10
 X - X - 1 + 10 = 9
Did you have difficulty following this?  Guess what?  I had difficulty coming up with it.  It took me several minutes of trial and error, spread over two days, to get all the steps right.  But Thomas Edison said that creativity is one percent inspiration and ninety-nine percent perspiration, and he went through a lot of glass to be the first to create a useful light bulb!

The second problem is a bit easier if you read a lot.  Did you also come with koala bear and red?  How about Dhubai, ibex, and hmm!  Oops!  Dubayy is a city in the United Arab Emirates.  Creativity can make errors.  How about Deutschland, deer, and red?  These were my thoughts at the dinner.

Curiosity, an important part of creativity, led me to the dictionary and atlas.  For colors other than orange we have olive and ochre.  Starting with countries, we have Dahomey, yak, and hmm?  Or, Dominican Republic, cat, and teal, or cougar and red, or chimpanzee and emerald, or …

Space doesn’t allow me to fill in my “hmm”s.

If you got this far, you know creativity is not that hard.  Unleash yours!  If you have a dull job, creativity may not get you a better job,  but it may make your job a bit more interesting.  If you have some tough fix-it problems at home, a fresh look at them may lead you to a solution or at least a workaround.  Finally, by looking past the figures and ideas in sound bites and headlines, you may gain a better perspective on many of the problems that confront us as a society.