The lure of mathematics...

I am by no means any kind of mathematician of the caliber of my erstwhile colleague Thomas O., who used his undergraduate degree in English Lit to enroll in what turned out eventually to be a doctorate program in math (I still fondly recall him explaining to me the ramifications of various kinds of infinity, though I do not recall many of the details of the explanation). On the other hand, there are moments when one can derive some pleasure from learning of some little twist that turns a seemingly unsolvable problem into something fairly straightforward.

Take for example, the problem of determining the sum of a series of positive integers, such as 1+2+3+4+5+6+7.

The intuitive way of doing this is sequential, the way a computer might do it. Out loud, the procedure would sound like: "One plus two is three; three plus three is six; six plus four is ten..." and so on.

But what if you want to know the sum of the numbers (integers) between 1 and, say, 243?

You could do that the same way, but it'd take a while, even with a calculator.

So here's the twist. Given any series of integers, for example, 1, 2, and 3, start by duplicating the series in reverse (3, 2, and 1). Then add the first two numbers of each list (1+3=4), add the second two numbers (2+2=4), and so on, until you get to the last two numbers. (In this simple example, there's only one pair of numbers left: 3+1=4).

Notice that each pair adds to the same number. If the list were 243 integers long, then after reversing and adding all the integer pairs, you'd get the number 244 each time (for example, 1+243, 2+242, 3+241, etc.).

How many times? Well, there's 243 numbers in the series, so we get the sum that many times.

So if we multiply the sum of each pair, 244, times the number of pairs, 243, we get 59,292. Now remember that to get this number we started out with our original set of integers, duplicated it, reversed the duplicate list, etc. That means that the number 59,292 is equal to just double the sum of the integers between 1 and 243. Therefore, if we divide 59,292 by two, we get 29,646, which turns out to be the sum of all the integers between 1 and 243.

Notice that however many integers there are in the series, the sum of any pair is always going one more than the number of integers, which you have to admit is a pretty cumbersome way of describing that state of affairs, which is why mathematicians like to use letters. So, going over to math-speak, we can say that if the number of integers is n, then the sum of each pair of numbers is going to be n+1. Turning the whole thing into a formula, we can describe the result of summing the integers between 1 and some number n, by the formula nx(n+1)/2.

You're asking: "Yeah, right,... whatever. So, can you think of a practical application of this knowledge?"

The question begs two answers: one, philosophical, on the value of "useless" knowledge; the second, practical, on how something you think is useless can actually be useful.

Both are tempting subjects of contemplation on a snowy February day in the Colorado Rockies, but for now, I must chase paper.

Cheers...

Dropping a shoe...

From time to time, someone will come into the store and ask about buying bubble wrap in lengths longer than the six-foot sections we have out on sale (we create such sections by cutting appropriate lengths from what starts out as a 125-foot bulk roll). Often, said someone will spy some part such a roll and ask: "Say, how much do you want for that?"

It's hard to tell just by looking how much bubble wrap is left, and if you assume that a roll that is half the diameter of a full roll contains half the amount of bubble wrap, well... you're going to be dramatically wrong.

So the question is: how can you get a good estimate of how much of a product there is in a package where the goods are wrapped around a core? Would you believe the little math exercise illustrated in the previous post helps? Follow:

First, we must recall the formula relating the diameter of a circle (d) to its circumference (c): c = πd. Since the diameter is just twice the radius, the formula can be rewritten c = π2r (or, more conventionally, c = 2πr) and that's where we'll start.

To make things simple, imagine that, instead of forming a spiral around the core, the product forms a set of concentric circles. Now what remains is to add up the individual circumferences of all these circles.

The innermost circle of product has a length 2πr, where r is the radius of that innermost circle. The next circle out has a length 2π(r+t), where r+t is the radius of that next circle, and is equal to the radius of the first plus the thickness of the product. The third circle from the core has a length 2π(r+2t), where r+2t is the radius of the third circle (which again, is the radius of the previous circle, r+t, plus another thickness of product, t).

The length of the fourth circle?  2π(r+3t)
Fifth? 2π(r+4t)
Sixth? 2π(r+5t)

See the pattern?

For the n-th circle, the length is going to be 2π(r+(n-1)t).

So, to figure out the overall length of n concentric circles whose radius increases by thickness t in each ring, we add up an equation that looks like:

2π(r)+2π(r+t)+2π(r+2t)+...+2π(r+(n-1)t)

rearranging, we get:

n x 2π + 2π x (t+2t+...+(n-1)t)

after one more simplification, we have:

n x 2π + 2π x (1+2+...+(n-1))t

Does that series look familiar? (The sum of digits between 1 and n-1? It's (n-1)(n)/2.) So we can rewrite the equation as:

n x 2π + 2π x ((n-1)(n)/2)t

So if the bubble wrap is 1/2-inch thick, if the radius of the cardboard core is 2 inches, and there are 14 layers of product, how much bubble wrap is on the roll? Substituting, we get

14x2x3.14 + 2x3.14x((13)(14)/2)x0.5 = 87.92 + 45.5 = 133.42 inches or a little over 11 feet.

Remember what I said at the beginning about guessing how much might be in a roll having twice the diameter?

Let's calculate how much product is in a roll with 28 layers of bubble wrap (14 as in the first example + 14 to double that number, which will come to a little less than double the diameter, since we're not accounting for the core):

28x2x3.14 + 2x3.14x((27)(28)/2)x0.5 = 175.84 + 189.00 = 364.84 inches or a little over 30 feet.

Surprise! A roll twice the diameter contains about three times as much product!

Simplifying the above formula gives us the formula:

   Length on roll = 6.28 x n + 3.14 x (n-1)(n)/2

So now it's easy to determine the length of bubble wrap on the roll: count the number of layers, plug that number in instead of n, and do the calculation.

The result is, BTW, very close to the real length.

Cheers...