I've just had my first puzzle published on Tanga.com. It was an Easy One Word Wonder. Can you figure it out? The solution to this puzzle is a single word. Let me know if you need any help.

Any ideas?

Here's an old, but still tricky question of probability.  You will find it discussed many places on the Net, but give yourself some time to think about it before looking elsewhere.  Be warned that looking elsewhere may only lead you to differing conclusions anyway...

Imagine you are a contestant on a game show. You have reached the final round and have to choose one of three closed boxes to claim your prize. There is only one prize, so two of the boxes will be empty. The show's host knows which box contains the prize, but isn't telling.

You pick a box. Now, before you open it, the host opens one of the other boxes and shows you that it is empty. He then asks you if you want to switch your choice to the remaining box or stick with your first choice.

What should you do in order to maximize your chances of winning the prize? Stick or switch?

If you want to look this up, it is commonly referred to as the "Monty Hall problem".

For an easier explanation, with a clear example, see the extended entry below.

I just bought this.  Can you guess what it is?  Earn prestige and honor by flaunting your superior knowledge.

Does anybody know what the crayon was used for in the original Dungeons & Dragons set by TSR in 1981?  You know, I'm talking about the white, unmarked crayon that came in the little bag of dice.  I've been wondering about this for 25 years and it's about time I found out.

I recently sold this old set on eBay and the fact that I don't know what the crayon is for is still bugging me.  Before I let it go, I looked through a lot of the material in the set and they don't mention it anywhere.  The TSR catalog description says that the set comes with 6 dice, but says nothing about the crayon.

I seem to recall a friend of mine stating a long time ago that the crayon should be used to "color in" the numbers on the dice.  Like you would rub the crayon vigorously on the sides of the dice and then wipe off the excess... leaving the white wax in the crevice of the numbers.  While this seems plausible, it is seriously ultra-low-tech, even by 1980s standards.

Here's a fairly easy riddle for you:

There are three light bulbs in a room, and three light switches outside the room. You are outside and want to match which switch goes with which light bulb. You can only travel into the room once, and cannot come back in again. You can do anything you want upon entering the room. How can you set the situation so that you will know which switch goes with which light bulb?
View the "extended entry" link below for the answer.  Don't be tempted to click too early.

I have been doing a lot of thinking and research into the "Airplane on a Conveyor Belt" conundrum.  For some reason, I find this thought experiment to be extremely compelling.  You can tell a lot about a person's thought process (and background) by the way they defend their position.  It really is amazing.

As I've said previously, this problem tends to divide people radically.  There are some people (very few) who just "get it" right from the start.  Then there are those who believe that their initial, intuitive answer must be the right one... there's just no other way.  I must admit that I was in the camp that immediately said "no way, that thing can't possibly fly... it's obvious!"  Eventually, after giving the problem at lot of though, I finally understood what was going on.

The Problem:

An aircraft is standing on a runway that can move (a conveyor belt). The aircraft moves in one direction, while the conveyor moves in the opposite direction. This conveyor has a control system that tracks the aircraft's speed and tunes the speed of the conveyor to be exactly the same, but in the opposite direction.  There is no wind.  The pilot begins to add thrust to the engines...

The question is:

Will the plane take off or not?
The Result:

Yes, the aircraft will proceed down the conveyor-belt runway in exactly the same way it would down an asphalt runway.  It will take off as normal.  If you were watching from the sidelines, the take-off roll would look identical to any other that you may have seen.  The only difference being the rate at which the wheels are spinning.  They would be spinning approximately twice as fast.

The aircraft does not suddenly lift vertically (like an elevator) as some proponents of the "No, it won't fly" camp seem to think we are proposing.  That would be physically impossible, assuming no wind.


The Assumptions:

In order to avoid any messiness, let's make one basic assumption.

1. Let's assume no friction at the wheel hubs.  ie... the aircraft wheels can spin as fast as they want.

This isn't a deal breaker or a shortcut, in fact, it's not really even necessary.  It is merely a way to dismiss the completely irrelevant argument that "there's no way an aircraft's wheels could spin 500mph without burning up."  Yeah, that's fine... but it has no relevance in our thought experiment.  Nobody is going to build a giant runway conveyor belt either.


The Solution:

Here are the keys to understanding this problem:

1. The wheels of an aircraft are "free-wheeling".  They do not provide propulsion, and therefore do not "push against" the action of the conveyor belt.
2. The thrust for aircraft movement comes from jet engines or propellors... not the wheels.  Therefore, the thrust being applied to the aircraft body is completely decoupled from how fast the wheels happen to be spinning.
3. Thrust acts according to Newtons Third Law of Motion - every action has an equal and opposite reaction.  The thrust of the engines is acting against the air.
   

Because the wheels are free-wheeling and we have assumed zero friction at the hub, it follows that the conveyor belt, no matter how fast it is moving, CANNOT EXERT ANY FORCE on the aircraft with respect to forward motion!  There is no force in our experiment that can oppose the thrust vector of the aircraft.

If the conveyor belt cannot exert any relevant force on the aircraft, you can completely ignore it.  Ergo, the aircraft takes off as if nothing unusual is happening.


Addressing Some Common Arguments:

1) "The conveyor belt will cancel out any forward motion of the aircraft.  The plane will not move at all."

Short Answer: The belt has no way to exert force with respect to the forward motion of the aircraft. All it can do is make the wheels spin faster or slower.

Long Answer: Your conditions are illogical.

• You claim that the aircraft will not move
  
• If the plane doesn't move, then the conveyor belt doesn't move either
  
• If the belt is not moving, then how is it cancelling any forward motion of the aircraft?

2) "The plane will remain stationary, but will lift into the air... thus taking off."

Answer: This is an aerodynamic impossibility (assuming no wind), which should be obvious.


3) "But if you said the the conveyor belt matches the speed of the WHEELS, it wouldn't be able to take off."

Short Answer: See Argument #1.

Long Answer: Once again, your conditions are illogical.  The conveyor belt can never "match" the speed of the wheels unless the aircraft does not move.  With a tremendous thrust vector behind it with no opposing force, the aircraft will move.  Once the aircraft begins to move, we enter into a paradoxical situation.

• X = Wheel Rotational Speed
  
• X = Conveyor Belt Speed, as per your conditions
  
• Z = Speed of Aircraft = Some non-zero positive number
  

The equation is: X = X + Z, which is illogical.

Example.  The aircraft is moving 10mph (X = X + 10) with the wheels rotating at 10mph.  Therefore, the belt must react and accelerate to 10mph.  But now the wheels are rotating at 20mph... and so on to infinity.


4) "You can't just ignore the conveyor belt as you claim.  Take this situation for example..."

A guy is standing on the conveyor. He sees a plane moving forwards away from him at 10mph on the conveyor. He also knows that the conveyor itself is also moving at 10mph in a forwards direction. The total velocity of the plane in relation to the ground must be 20mph.

If the conveyor can be ignored then why is the plane's total velocity twice what it would normally be if it was moving along on tarmac.

The fact that you cannot explain this indicates that you've either ignored or overlooked some of the forces at play between the conveyor and the plane.
I don't see any violations.

You state that the aircraft is moving 10mph relative to the conveyor (perhaps as measured by a speedometer on the wheels). The conveyor itself is moving 10mph relative to the ground in the same direction. The total speed of the aircraft relative to the ground (tarmac) is 20mph.

I don't see any problem with this. Ignore the conveyor by making it pop out of existence and you suddenly have an aircraft traveling down the tarmac at 20mph.

You can change your point of view as much as you like, but you still end up with an aircraft traveling 20mph with respect to the ground.


References and Further Investigation:

The Straight Dope - "An airplane taxies in one direction on a moving conveyor belt going the opposite direction. Can the plane take off?"

AVWeb.com - Pilots Lounge #94 - "Conveyor-Belt Runway"

Airliners.net Tech Ops - "If A Plane Took Off A Conveyor Belt..."

Tempus Fugit Blog - "Airplane on a Conveyor Belt"

PhysOrgForums - 275 pages of discussion!!!

I've come across a thought experiment that SHARPLY divides people. The funny thing is that there is only one right answer.  Physics dictates this to be so.  Here it is:

A plane is standing on a runway that can move (a conveyor belt). The plane moves in one direction, while the conveyor moves in the opposite direction. This conveyor has a control system that tracks the plane's speed and tunes the speed of the conveyor to be exactly the same, but in the opposite direction.  The pilot begins to add thrust to the engines...

The question is:

Will the plane take off or not?
Please answer via comments.

It is a mathmatical certainty that at least two people on the Earth have the exact same number of hairs on their body.

First find an acceptably larger number.  ie.. overestimate.

Nobody on Earth is 100 inches (8.3ft) high and nobody is 100 inches around.  100 * 100 = 10,000 square inches of skin.  Overestimate the number of hairs per square inch... say 10,000 hairs per square inch.  10,000 * 10,000 = 100,000,000 hairs.  Assume that we have a room for each one of those hairs... 100 million rooms.  There are 8.6 billion people on the planet.  Everybody lines up and each person enters the room that marks the exact number of hairs on their body.  Let's say that after a while, every possible room has a person in it.  We've still got 8.5 billion people waiting to enter rooms... we are forced to allow them to double up... hence we have proven the theory.