December 8, 2020
March 17, 2007
February 9, 2007
Eggs Per Hen-Day
There’s an old brain teaser that is posed as a word problem, something like this:
If a hen and a half lays an egg and a half in a day and a half, how many days will it take three hens to lay three eggs?
or
If a hen and a half lays an egg in a half in a day and a half, how many hens will it take to lay six eggs in six days?
These are essentially the same problem and there is no significant difference to “three” or “six”. I’ve seen several internet sites that go through the math to calculate the rate of egg laying to come up with the answer, but that is not necessary.
With a little knowledge of Algebra, we realize that this is a rate problem: how many eggs are produced per hen-day? Mathematically, we would write that this way:
Eggs |
= |
Rate * Hens * Days |
and then solving for “Rate” we get this:
Rate |
= |
Eggs
(Hens)(Days) |
We then see immediately that we don’t even need to know Rate.
If we multiply Eggs and Hens by the same factor N:
Rate |
= |
N * Eggs
(N * Hens)(Days) |
that factor, being in both the numerator and denominator, will cancel and Days must remain unchanged.
In the same way if we multiply Eggs and Days by the same factor N:
Rate |
= |
N * Eggs
(Hens)(N * Days) |
again that factor, being in both the numerator and denominator, will cancel and Hens must remain unchanged.
So, we can change “one and a half” to X and three (or six) to Y:
If X hens lay X eggs in X days, how many days will it take for Y hens to lay Y eggs?
or
If X hens lay X eggs in X days, how many hens will it take to lay Y eggs in Y days?
For all values of X and Y, the answer will always be X.
January 23, 2007
Degrees of Separation
When I was in high school, I met my best friend’s father (not at all unusual, I hear). My best friend’s father was, and is, a cinematographer and worked on such films as Second Hand Lions, Space Cowboys, and Unforgiven. In fact, he was nominated for the 1992 Academy Award for Best Cinematography for Unforgiven. His name is Jack Green.
Jack Green has been the Director of Photography for many films, and he worked many times with world famous actor and director Clint Eastwood.
Clint Eastwood has acted in and directed many successful films, including Mystic River starring Kevin Bacon.
That’s three degrees of Kevin Bacon.
July 5, 2006
Love Thy Neighbor
Since I was a boy, I’ve heard it said that when Jesus preached, “Love thy neighbor as thyself”, He meant that we must first learn to love ourselves before we can love others. Thus, many have taken the teaching to mean that we must all become self-absorbed and focus on building our own self esteem. I lost a brother to a “Recovered Memory” cult because of this “love thyself” reversal of Jesus’ teaching.
In reality, Love thy neighbor as thyself is based on the premise that we all already do love ourselves. The trick is to figure out what that means. In what way do we all love ourselves?
We could probably think of others, but the primary way in which we all love ourselves is that we all seek the basic necessities of life for ourselves: food, water, warmth, shelter. We never thoughtlessly disregard our own hunger, for example. Even the most depressed person eats and drinks, and tries to stay warm and dry (even someone who commits suicide does so because they think it will benefit them).
Once we understand the ways we already love ourselves, then we can begin to understand what Jesus meant by “Love thy neighbor as thyself.”
March 1, 2006
Pulleys
When I was in college, many years ago, I studied physics. Somewhere along the way we learned about pulleys.
The purpose of a pulley is to change the direction of a force. Ideally it will do so with very little friction, but the purpose is still nothing more than changing the direction of a force. When you combine multiple pulleys in the right configuration, you effectively trade force for distance in the equation for Work: W = FD (Work equals the Force applied multiplied by the Distance over which the force is applied). The basic principle is that the necessary force goes down as the distance traveled goes up. A collection of pulleys increases the distance (you have to pull more rope) which reduces the required force by the same factor (i.e., four times more rope means 1/4 as much force).
In one pulley-related assignment, there was a problem where one pulley was replaced by a simple iron bar which was assumed to be frictionless. Instead of going around a pulley, the rope went around this frictionless iron bar. Since a pulley changes the direction of a force, and the bar changed the direction of the force, I treated the iron bar as if it where a pulley. The professor marked my answer wrong. When I discussed it with him, he said that since the bar was not a pulley, it did not add to the mechanical advantage of the system. He rejected my reasoning that since the bar changed the direction of the force just like a pulley, and since the bar increased the distance traveled by the rope just like a pulley, then the bar served exactly the same function as a pulley.
Although I got nowhere with that professor, I have since found that my analysis was in fact correct. You can replace a pulley with a metal bar, a metal ring, or even a loop of rope (you can actually make a “block and tackle” arrangement with nothing but rope). As long as the direction of the force is changed with relatively little friction added to the system, the physical principle is the same.
A doctorate does not prove that someone knows what he’s talking about.