Sequencing data

The New York Times recently posted a piece on problem solving which asked readers to first solve a problem:

"We’ve chosen a rule that some sequences of three numbers obey — and some do not. Your job is to guess what the rule is. We’ll start by telling you that the sequence 2, 4, 8 obeys the rule."

You can test your hypotheses by typing sequences into three boxes to see if they follow the unstated rule.  Once you think you know, you type in a description.  Most people it turns out, suggest an answer without ever trying a sequence that returns a firm "NO."  Psychologists interpret this as being evidence of confirmation bias: once we get a "yes" for our theory - we don't poke around trying to find a "no."

When I teach chemical kinetics, I point out to students that few experiments can prove a reaction goes in a particular sequence, only that the data is consistent with a proposed mechanism.  No answers can be as or more critical to problem solving as yes.

I failed to 'correctly' solve the puzzle, [SPOILER ALERT] though I did get several no answers.  One rule I tried was aⁿ: 2¹, 2², 2³ = 2, 4, 8.  The sequence 1, 1, 1 follows that rule (1¹, 1², 1³ are all one), but yielded a no.  The rule a_n = 2 x a_n-1: 2, 2x2=4, 2x4 worked for every sequence I tried, but is not 'the 'answer.  The answer is that correct sequences have each number larger than the last.

The study suggests I failed not only because of confirmation bias, but because I complicated the problem, assuming that there was some sort of trick to the rule. Actually, I assumed the technical mathematical meaning of sequence held, in that there was a rule that uniquely specified each number in the sequence given the starting value(s). An ordered list of numbers, each of which is larger than the previous value is not a sequence in the mathematical sense.

In retrospect, I should have tried the sequence 0, 0, 0. It follows the rule I proposed (a_n = 2 x a_n-1) as the correct one, but returns a "no." It would have ruled out my proposed rule, a useful "no".  (I might also have tried non-integer numbers.)  I failed in part because I didn't understand the question they were asking, we didn't have the same definition of "sequence."  In some sense I fell prey to the "when all you have is a hammer, everything looks like a nail" scheme.

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There are more than 2500 rules that would give you the mathematical sequence 2, 4, 8.  See Sloane's encyclopedia of integer sequences.  My first proposed sequence is A000079 in the collection.

For more about sequences and Sloane's encyclopedia, read this article at AT&T.

Chocolate Math Mystery

My youngest and I are heading into Philadelphia tonight for a chocolate dessert feast, so it seems apt that a friend sent me this bit of mathematical magic this morning - with a plea to explain how it works.

Chocolate Calculator:

This is pretty neat. Don’t say your age; you will probably lie anyway!

DON’T CHEAT BY SCROLLING DOWN FIRST

It takes less than a minute. Work this out as you read.

Be sure you don’t read the bottom until you’ve worked it out!

1. First of all, pick the number of times a week that you would like to have chocolate (more than once but less than 10)
2. Multiply this number by 2 (just to be bold)
3. Add 5
4. Multiply it by 50 — I’ll wait while you get the calculator
5. If you have already had your birthday this year add 1759. If you haven’t, add 1758.
6. Now subtract the four digit year that you were born.

You should have a three digit number

The first digit of this was your original number (i.e., how many times you want to have chocolate each week).

The next two numbers are YOUR AGE! (Oh YES, it is!!!!!)

THIS IS THE ONLY YEAR (2009) IT WILL EVER WORK, SO SPREAD IT AROUND WHILE IT LASTS!

So how does it work?
Expressed algebraically, the procedure if you have had your birthday can be written as:
50 (2n +5) + 1759 - y
where n is the number you chose and y the year you were born

The author asserts that this will produce a number where the digit in the 100's place is n and the remaining digits are your age or 100*n + age. If you have had your birthday this year, your age in 2009 can be written in terms of your birth year, y, as
age = 2009 - y
So the formula should produce 100*n + (2009 - y).

It is trivial (I love saying that) to show that

50 (2n +5) + 1759 - y = 100*n + (2009 - y)

This will not work if your age is greater than 99, but as long as you are younger than that, the last two digits will always be your age even if the number of times you want to eat chocolate in a week is greater than 10 -- so in either case eat all the chocolate you want!

Weird Words of Science: nonillion

“Do you know what a nonillion is?” queried my mathematician spouse as he plopped into the chair in front of our household computer, “Is it Latin or something?” “Something to do with nine I’m sure,” I offered from the sofa. “That’s OK, I can google it.” What’s the urgency I wonder? 1 vs. 100 is the issue. The mob won.

So what is a nonillion and does it have anything to do with nine? The short answers are: it depends and yes. Nonillion is a novelty number - a term I just coined for numbers that have names, but no uses. Like a googol. The early British usage of nonillion was for 10⁵⁴ - nine million millions. Americans used nonillion for 10³⁰ or 10^3+3x9. In other words, the result of multiplying a thousand (10³) by a thousand nine times.

The system of counting by thousands is sometimes called the “short scale” (from the French term echelle courte). The long scale (echelle longue) counts by millions. Most English speaking countries (both the US and UK included) use the short scale, while most of the rest of the world uses a version of the long scale.

It’s hard to get a sense of scale with these enormous numbers, but a nonillion (long scale) is (very) roughly the order of magnitude of the mass of the universe in kilograms. There are roughly 5 nonillion bacteria (short scale) on earth.

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Literary trivia: e.e. cummings used nonillion in the Enormous Room and in at least one poem.

Weird Words of Science: stochastic

I'm teaching a graduate course in mathematical modeling of natural processes. Many math modeling techniques rely on the random numbers and are more generally known as stochastic algorithms. A simple example is numerical integration. We used numerical integration techniques to the value of pi by (virtually) throwing darts at a circular target embedded in square (figure). The ratio of hits inside the circle to the total hits is pi/4. Stochastic comes from the Greek stochastikos "to take a guess", which itself derives from stochos - "target", so the target image above is apt.