When I was very young I watched a show called David the Gnome. As you might guess, the show was about David and he was a gnome. Not a terribly creative title, but when I was young I was captivated.
The Wikipedia article on David the Gnome says that the American version was actually a dubbed Spanish version that was based on a book written by a Dutch author. So apparently, in addition to being a gnome, David was an ambassador to the UN.
In one of David's adventures, he came across a chicken with six chicks. Tragically, the chicken could only count to three, and the chicks kept getting lost without the mother hen even knowing. Although a negligent parent, you have to give the bird bonus points for being able to count at all. [footnote 1]
David, recognizing the problem as a serious one, taught the mother that she should arrange her chickens in two groups of three. The mother could count to three, and she could do it twice, thus she could keep track of all her baby chicks. [footnote 2]
I think that the simple brilliance of this might have been lost on even the creators of the show. David’s poultry grouping insight alludes to many fundamental concepts in mathematics. The fact that every whole number can be represented with a unique product of its prime factors (like 6 = 2×3) is called the fundamental theorem of arithmetic. Had the mother had a prime number of chicks (like five or seven) David’s solution wouldn’t have been so simple because neither 5 nor 7 can be written as a product of smaller numbers.
The chicken could even keep track of 12 chicks, though it's a little trickier. You couldn't put the chicks in 2 rows of 6 (mama chicken can't count to 6), nor could you put them in 3 rows of 4 (4 is still too big), but you could put the chicks in three 2 by 2 groups:
You could also organize a peep of 18, or 27. If you're clever, you can do 16 too.
In addition to the fundamental theorem of arithmetic, David alluded to a tool often employed by mathematicians: when confronted with a difficult problem, reduce it to a problem that’s already been solved. Rather than teach the mama chicken how to count higher, most likely an impossible task, he had the chicken do the something simple twice.
David applied simple and elegant mathematical thinking to a life threatening situation and his solution is profound, yet straightforward enough for an animated chicken to understand. He's my hero.
-Nick
Footnotes
[1] There is actually some anecdotal evidence that some birds can count as high as three. (Chapter 1, fourth paragraph)
[2] I like to imagine that before David came across the chicken, there were at least a dozen more chicks, all dead now due to negligence.
As you look down the neck of a guitar, the frets get closer and closer. What is the rule for deciding how close the frets should be?
Preliminary Information:
A string halved in length vibrates at twice the frequency (physics).
An octave is the interval between one musical pitch and another with half or double its frequency (Wikipedia article on octaves).
There are 12 notes in an octave (music theory).
To go up one octave, you have to halve your string. To go up another octave you must quarter your string. In general, to go up t octaves you need to have a string of length
To have our length function, l, go up by twelfths of an octave (notes) rather than octaves, we need to adjust it to be:
I love Khan Academy, and I've been recommending it to the students I tutor since I first learned about it. It has the potential to revolutionize education, but not everyone agrees. I think it (or something like it) will be the future of math education.
The Khan Model, A Two Act Play:
Act I: Old Model In School
Teacher: Teach, teach, teach, teach, teach.
Student: Ms. Teacher, I don't quite understand.
Teacher: I'd love to individualize your education Sally, but it's time for you to go to your next class. At Home
Student: I hate math.
Act II: Khan Model At Home
Computer: Teach, teach, teach, teach, teach. In School
Student: Ms. Teacher, I don't quite understand.
Teacher: Well Sally, seeing how you already heard the bulk of the lesson at home, I have plenty of time to individualize your education.
Everyone: Yay!
Some time ago, a math student asked me and my fellow tutors a question from his class: how many diagonals are there in an n-sided regular polygon? We all went to work and came up with different solutions. When finished, the student asked which answer was the right one. Here are our solutions:
Counting
The student's solution was simple: draw a regular polygon and simply count how many diagonals. This works, and I've put together an applet that allows you to do exactly that (Java needs to be enabled):
Use "+" to increase the number of vertices and "-" to decrease them.
If you let n be 4, you can clearly see that there are 2 diagonals (in purple). Letting n equal 5, you can count 5 diagonals. The counting solution works, but with a catch. Let n be 10; can you still count how many diagonals there are? What if n is 15? At some point it just gets ridiculously hard to count the number of diagonals. This doesn't mean the solution is wrong (it's not, in principle you'll always be able to count how many diagonals), only that it's not a very useful solution.
Recursion
My first approach to a problem like this is usually looking for a recursive formula (this post for example). Recursive formulae are not very easy to work with, but recursive patterns are relatively easy to spot. Patterns are easier to see when the data is organized, so I'm going to arrange what I found by the counting solution:
n
3
4
5
6
7
d
0
2
5
9
14
For every n-gon, there are d diagonals. Can you see the pattern here? Look at how much the d row changes each step:
Each blue number is the difference between the two d values above it. As you can see, the blue numbers are always one less than the n numbers. This suggests the following formula:
With the initial condition d3 = 0. This predicts that the next d value should be 20 (14+6), which can be confirmed by counting. This is an improvement over the counting solution, it no longer means drawing time and space consuming pictures, but it's far from optimal. Recursion is messy and a bit of a pain. In order to find a particular d, you first have to find the d that comes before it. But to find that d, you first have to find the one that comes before it. And it goes on that way until you get to the beginning. It takes a lot of work:
It takes the above video 3 minutes to evaluate something as simple as d6, whereas the counting method would take only 30 seconds. So is this the better solution? It depends. This method is purely algebraic, which means it can be programmed [footnote 1]. It also has the advantage of being cumulative, if you have d6, it's really easy to find d7, just add (6-1). With the drawing method you would have to draw a whole new picture. Summation
If you expand the recursive formula from above, you'll start to see a pretty simple pattern:
This pattern shows that to find dn you just have to add all the numbers from 2 to n -2. This is just simple addition that can be quickly computed by hand, on a calculator, or in a spreadsheet. With sigma notation, this can be condensed to:
Is this the best answer? Well, it's certainly simpler than the recursive definition, but it's still a lot of computation.
Multiplication
We have the result
which can also be rewritten as
(Same thing as before, just with the terms reordered.) If we add the first to the second we get:
In the sum, n appears exactly (n-3) times, so we have the result
and with some algebra magic we get
This is a fantastic result, but is it a better result? All all of our previous results required a bunch of computation, this just requires three steps: subtract 3, multiply by n, divide by 2. But on the other hand, I think this ultra tidy result masks the meaning of the solution. It doesn't give any indication as to why that's the answer, it only gives you an answer. In other words, easy to use, difficult to understand.
Extrapolation
If we're technologically inclined, we can just take a few values (from the counting solution) and pop them into a spreadsheet application. From there we can extrapolate the data using a trend line. If you're used to this kind of thing, you might have a hunch that the data fits a quadratic, if not, you would have to try a few different options before determining which is best:
With an R squared value of 1, we can be pretty sure that this is the curve of best fit. Since 2E-14 = .00000000000002, I'm going to assume the curve of best fit is
which is equivalent to our other solutions.
Is this the best solution? It certainly is the least amount of work. But it doesn't come from any reasoning and doesn't offer any insights. In this case it works, but in general interpolation methods are an unreliable way to come to an answer.
Combinatorics
Different solutions provide different insights. In addition to what we've already done, this problem can be solved with combinatorics.
Each edge is just a connection between two vertices (it's a handshake problem!). Thus, asking how many edges there are is essentially the same thing as asking how many ways can you choose 2 things from a set of n. The answer to that is n choose 2, which can be written many different ways:
We don't want to count the n edges on an n sided polygon, so the the solution we're looking for is:
This is equivalent to our previous definition [footnote 2]. Is this a best solution? Stop asking that!. It does have a much shorter derivation (assuming you know a little bit of probability theory), but it's more computation (in its unsimplified form; see footnote 2). It certainly provides a different perspective, and having multiple vantage points is always a good thing.
Graph theory
A theorem in graph theory tells us that a complete graph of degree n has n(n-1)/2 edges. This number includes the outside edges, so we adjust for that by subtracting n:
Graph theory is definitely overkill for this problem, but I thought I'd include it to give a taste of what other approaches are out there.
Conclusion
So what is the right answer? An elementary student might say the counting solution. An algebra 1 student might give the summation, a bright algebra 2 student might give the algebraic solution. A college student might solve it with recursion or combinatorics [see footnote 3]. A Statistician may use a interpolation. A more advanced mathematician could give a more advanced solution like the graph theoretical one [see footnote 4].
Solution
Advantages
Disadvantages
Counting
Easy to understand
Absurdly difficult for large numbers
A new picture has to be drawn for every n
Recursion
Relatively simple
Cumulative answers
Computation intensive
Relies heavily on notation
Summation
Straight forward computation
Computation is tedious
Multiplication
Very simple computation
Solution is difficult to discover
May not be clear why it works
Extrapolation
Very easy to find solution
Extrapolation is often an unreliable
No way in advance to know what type of curve (linear, polynomial, exponential...)
Combinatorics
Simple solution
Easily computable when simplified
Relies on a branch of mathematic that many students may be unfamiliar with.
While these answers are all equivalent [see footnote 5], they are not all the same. There are plenty of wrong answers, there really is no one right answer.
Footnotes:
[1] For example, in Python:
def diagonals(n):
if n == 3:
return0else:
return diagonals(n-1) + n-2
[2]
[3] As part of my research for this post, I gave this problem to a small group of very bright 8th graders. I gave them a brief explanation about how to use subscript, and then gave them about a half hour to figure things out. I told them when they got a wrong answer, but other than that, no coaching. At the end of the half hour, they came up with both the recursive and summation solutions. To say the least, I'm impressed.
[4] A set theorist might do something like:
(set theorists hate you)
[5] They're all equivalent for natural numbers greater than three. For other values, some strange things happen:
Solution
n = 3
n = 2
n = 1
n =1/2
Counting
0
0
0
undefined
Recursion
0
undefined
undefined
undefined
Summation
undefined
undefined
undefined
undefined
Multiplication
0
-1
-1
-5/8
Extrapolation
0
-1
-1
-0.625
Combinatorics
0
undefined
undefined
-1/8*
* nCr is technically undefined for non-natural values of n, but you can use the gamma function to extend the factorial function and get the answer -1/8.
