Number Series the GLAT and the SAT

Number series

At some time or other we were all challenged in grade school with an aptitude test that included a question requiring the next number in a series. You remember: “What’s the next number in the series 1, 2, 3, _?” “4.” Right? Wrong! It’s “2”! The numbers in the series are the largest primes dividing the natural numbers starting with one! Who knew?

At the risk of raising bad memories, especially after the school informed your parents, based on such ‘scientific’ testing, that you were a chronic underachiever I would like to submit that number series are interesting and not too terribly intellectually taxing by providing two examples and finally, at the end, show you a tool to crack any number series!

The Fibonacci numbers is a very simple series to generate. As with all number series one has to start somewhere. So start with the most basic numbers of all: 0 and 1. To find the third Fibonacci number, F3, simply add the preceding two numbers, 0+1, getting 1, boring, but it gets better. The fourth number, F4 is 2, then F5 is 3 and the series is off and running: 0,1,1,2,3,5,8,13,21…

Still, so what? Well, the usefulness of Fibonacci numbers almost exceeds the imagination! In third grade science we saw how flower petals and leaves and seed pods spiral out from the center in beautiful arcs to maximize accessibility to sunlight and rain. Maybe we counted the number of clockwise turning and counter-clockwise turning spirals while looking at the stem end of a closed pine cone or a pineapple and found 5 and 8 spirals; two adjacent Fibonacci numbers. Then in tenth grade art we saw how the beautiful Parthenon in Athens had linear height and width dimensions that were again a ratio of adjacent Fibonacci numbers. Finally in senior year algebra it was explained that the ratio of adjacent Fibonacci numbers was always the same number, the golden ratio, phi, 1.618… Wikipedia does a fine job with all the details and uses of these numbers (including stock trading).

But there is one thing they can do for you while you are driving that long stretch of I5 to San Francisco: amaze your family by converting miles to kilometers with ease. Here’s how. A moment of reflection shows that the numerical value for phi and the number of kilometers per mile are almost identical values, within half a percent! If you are going 65 miles per hour you decompose 65 into Fibonacci numbers: 55+8+2. Increase these to the next highest ones: 89+13+3 which adds to 105 kilometers per hour. To convert to miles, use the next lower numbers. I know, it’s a lame trick making minimal use of Fibonacci numbers, but still…well, I’ll bet you’ll try it anyway.

The next number series, with no particular usefulness what-so-ever, was made famous because it was problem number 20 in the GLAT (Google Labs Aptitude Test for perspective employees) booklet that Google inserted inside several popular magazines in 2004. Google “GLAT” for all the details and answers from Mathematica and others.

This integer series is interesting because it appears intimidating and shows how the human mind can be easily fooled.

Question 20. “What number comes next in the sequence: 10, 9, 60, 90, 70, 66, …?

A) 96
B) 1000000000000000000000000000000000\
0000000000000000000000000000000000\
000000000000000000000000000000000
C) Either of the above
D) None of the above”

Notice that answer “B)” is without a doubt intimidating and probably causes your subconscious inner-voice to say something like: “Well, let’s skip this one and come back later.” Or at least you realize there must be some kind of strange algorithm at work here to cause this number to be even considered as an answer.

But the key to this series is simply the largest natural number requiring exactly n letters in English. Since “10” is larger than 1, 2 or 6 and all four have three letters in their names the series starts with 10. So you just have to count the number of letters in the name of each number to figure out the next one. The only real mathematical knowledge you need is to recognize that answer B) has it’s own name, it’s called a “googolplex,” 10 letters! Ha, ha, fooled you! But the lesson to be learned, especially if you are a student interested in taking the SAT is: don’t panic! Look for some sort of conversion process going on like substituting names for numbers, etc.

Finally, the tool promised at the beginning is an absolute marvel of what the human brain can produce. The “On-line Encyclopedia of Integer Sequences,” http://oeis.org/, can without a doubt crack any sequence: simply type the first few numbers of interest in the opening page box and the Encyclopedia returns the full sequence with generating formula, historical notes and references to similar sequences. Take a look; show it to a student facing one of those SAT tests. What a great way to gain familiarity with and confidence in solving number series!

My favorite sequence is: “An optimal n-mark Golomb ruler [which] has the smallest possible length (distance between the two end marks) for an n-mark ruler.” OEIS reference number A003022.

Finally, Google has many entries concerning methods of solving for the next number in a series which can be of great help to the test taker.