Jumping Off a Circle

Where is the best point to jump off a circle? Someone posed this question a few months back — I'm not sure who — and I've been able to resist thinking about it too deeply until this morning. As so often is the case with interesting questions, there is a bit of room for interpretation. Here's how I've chosen to interpret the question:

Let's say an object is launched from a circle with radius 1 meter, centered on the ground, and the object is launched tangentially to the circle (see the red flight path shown above, for example). What should the angle (shown in green above) be so that the object travels as far as possible before landing?

If you think about this question a little bit — no calculation required — I bet you could come up with a pretty good rough answer. If you think about it a lot more and do quite a bit of calculation and/or simulation, you could refine it some more.

Middles and halves and alternative definitions

My almost-7-year-old son Blaise and I recently had the following enlightening conversation.

Blaise: (approaching me out of the blue) Did you know that every number has a middle or a half?
Me: Huh. What about 7?
Blaise: Yeah, its middle is 4.
Me: Ohh, I see what you mean. You know, there's a name for numbers with a middle. They're called "odd." And numbers with a half are called "even."
Blaise: Oh, so 1 isn't even or odd!
Me: (alarmed pause) Aw, crud. (longer pause) Wait a minute! Isn't 1 the middle of 1?
Blaise: Oh yeah! So 1 is odd. (pause) Hey, I know why they call even numbers "even." (holding up two fingers on each hand) Like the number 4 — you can split it into two even parts.

Predicting the future

Every ten years, the U.S. Census counts the approximate number of people in the United States. The table below shows the number of people in the years 1800 through 1900. Imagine you were alive just after 1900, and you wanted to predict how many people there would be in 1920. How would you do it?

What if you wanted to figure out how many people there were back in 1883? What about this year? If you were alive just after 1900, and you saw this list of numbers, how many people would you predict for 2013?

Year   People
1800 5 million  
1810 7 million
1820 10 million
1830 13 million
1840 17 million
1850 23 million
1860 31 million
1870 39 million
1880 49 million
1890 63 million
1900 76 million

Colorful toys

Let’s say I want to make seven plastic toys (an airplane, a boat, a camera, a dinosaur, an elephant, a firetruck, and a gun), and I can make the toys in three different colors: red, green, and blue. How many different ways can I pick the colors for the toys? 

Now, what if I want to use each color at least once? How many different ways can I pick the colors for the toys then?

The speed of an approaching object

The visual angle (AKA angular size) of an object depends on how far away the object is. The passenger jet shown below has a wingspan of 200 feet. Click the image for an interactive version, where you can move the jet around.
Suppose a jet with a wingspan of 200 feet is approaching you. Its visual angle is 1.5°, and the angle is increasing by 0.1° per second. How far away is the jet, and how fast is it moving?

Squares and cubes and primes

There's something a little strange about the square numbers (e.g. 1, 4, 9, 16, 25, 36, etc.) — the only time a square number comes right after a prime number is when the square number is 4, which follows the prime number 3. If you test this out for a few other square numbers, you may begin to suspect that square numbers (other than 4) rarely, if ever, come right after a prime number.

What I'm saying, though, is that no square number aside from 4 will ever come immediately after a prime number. It can never happen. Ever. What an all-encompassing claim! How can I be so sure of this? Well, let's say I pick some square number. Let's call the number "n", just for the sake of having a name for it. What does it mean when I say n is square? It means n is the square of some other number, like 1, 2, 3, 4, etc. In other words, let's say n is the square of some positive whole number, which we might refer to as "m" (or whatever letter suits you). So n = m2. Now, what is the number that comes right before n? It would be the number n - 1, which is really m2 - 1.

Now, why on Earth couldn't m2 - 1 be a prime number? Well, another way to write the number m2 - 1 is this: (- 1)(+ 1). And unless I'm mistaken, - 1 and + 1 are both whole numbers, so my previous sentence means is that no matter what m is, m2 - 1 can always be "factored" into two whole numbers (specifically, whatever the numbers m - 1 and m + 1 are).

Since m2 - 1 can always be written as the product of the numbers m - 1 and m + 1, the only way m2 - 1 might be prime is if one of those two numbers turns out the be 1. But since m is a positive whole number, m + 1 has to be at least 2. Can m - 1 be equal to 1? Sure it can! But only if m is 2. In short, the only way m2 - 1 might be prime is if m is 2. In other words, the only way a prime number could come right before a square is if the square number is 4.

Isn't it astounding, and beautiful, to realize that human beings can know something so infinite in scope with such absolute certainty? No matter how many square numbers I check, I know that the only one that can follow a prime number is 4.

Can a similar statement be made about the cube numbers (e.g. 1, 8, 27, 64, etc.)? If so, can you prove the statement is definitely true, no matter what cube number I might check?

What about the fourth-power integers (e.g. 1, 16, 81, 256, etc.)?

Working together

My garden hose can fill my kids' wading pool in 14 minutes. My neighbor's garden hose can fill it in only 10 minutes. If we use both hoses at once, how long should it take to fill the pool?
I posed this question to some students the other day, and I was taken aback at an approach that one student came up with, seemingly spontaneously: Find the average of the filling times, and divide it by 2 (since the two hoses are working together).

In the situation at hand, the student's approach gives a nice round answer of 6 minutes. Unfortunately, this isn't quite right. But doesn't the student's approach seem reasonable? What's the easiest way to see that the approach can't be right?

Cooperation and small probabilities

Watch this clip from the TV show Friends (a popular late-90's sitcom). Joey, one of the six "friends," announces his intention to enter the Connecticut Powerball lottery. Most of the rest of the group decide that it would be a good idea to pool their money to buy many tickets and split the payoff if any of the tickets wins. However, Ross (a college professor in a scientific discipline, I might add) opts out, citing the incredibly small probability (or "odds") of winning. Chandler jokingly replies that with six times as many tickets, the group will have six times the probability of winning.

Is Chandler right about that?

Hide and Peek

There is a game for the Nintendo Wii called Wii Party (see also Wikipedia), which largely consists of a set of four-player "minigames." In one of the minigames, shown below, three players hide on a playground and a fourth player tries to find them.

Hide-and-Peek minigame (see the Wii Party website, under "All Minigames: 1 vs 3," for a video)
The non-hiding player has five chances to find all three of the others. There are seven hiding places, and any number of players may hide in each one. However, strangely enough, anyone who chooses the small hiding place in the center, behind the blue and pink spring horses, will actually be visible there (so unless the non-hiding player is completely inexperienced, he or she will definitely find that person).

The non-hiding player wins if he or she finds all three of the others. If not, all of the others win (regardless of which of them were found). Should someone hide behind the spring horses?

Geometry of the sextant

As I mentioned before, I've been working with a sextant lately. Not surprisingly, Wikipedia has an informative article on sextants, with many details about their operation. The diagram below shows the most basic aspects of the geometry by which a sextant works.
(Click for the interactive version, in which the red dots can be dragged.)
The fundamental idea is that you look directly at one object while adjusting the angle between the sextant's two mirrors so that your view of a second object coincides precisely with your view of the first object. That is, the path of light from object 1 to your eye overlaps exactly the path of light from object 2 to your eye after reflecting off the two mirrors, as shown above. When this is accomplished, the scale at the bottom of the sextant shows the visual angle between the two objects.

The sextant derives its name from the fact that its scale forms a physical arc of about 60° — or one sixth of a full circle. But one of the first things that struck me about the scale is that its marks run from 0° to about 120°. So, since the sextant's mirrors are parallel when the sextant is set to 0°, it seems that the scale shows two times the angle between the sextant's mirrors.

So here are my questions: Is the visual angle between object 1 and object 2 really two times the angle between the mirrors? Can we prove it?