Friday, January 11, 2008

Two Roads to the Castle

You are walking ing the woods on a road to the castle, when all of the sudden, the road forks into two dirctions. Standing in front of each fork is a Knight. One Knight always tells the truth. and the other Knight always lies. The problem is that you don't know which Knight is which.

If you had only one question to ask either of the Knights to figure out which fork in the road leads to the castle, what question would you ask?

Tip: If should not matter whether you are askin the Knight that lies or tells the truth.

Answer is below in white on white text. Highlight over text with your mouse to view.

The question you would ask is "If I were to ask you 'is this the path to the castle, what would you: say Yes or No?"

If you happen to be asking the Knight who tells the truth, he'll tell you in the path he is in front of is the right one or wrong one.

If you happen to be asking the Knight who always lies, he would always typically give you the wrong answer. However, the question is worded as a double negative. By asking "If I were to ask you", he would typically have told you a lie, but since you are asking him to tell you what he would say if he were asked he would have to lie about what his answer would be; therefore making his answer also always the truth.

Symbol Sequence

What is the next symbol in the sequence?

Answer is below in white on white text. Highlight over text with your mouse to view.
The answer is two eights "88" right next to each other. The sequence is the numbers 1 through 7, each with its mirrow image.

Take a look again!

Sunday, February 18, 2007

The Circle Game

Ok, so this is really not a riddle or puzzle. It's just a fact. In fact, one of the facts of nature I find the absolutely most un-intuititive.

So here is the question: Imagine if you tied a string around the world and stretched it tight till you circled the Earth once, so it was tight against the surface of the planet. Then take that same string and added 3 feet to its length. If you stretched it around the Earth again, there would be some slack it in this time. If you were to space that slack evenly, how far off the surface of the planet would the string be?

The answer is 6 inches. Yes, adding 3 feet of string to the string actually makes the string raise 6" off the surface of the planet all around the entire planet. Hard to believe huh?

I think so. Well here's the proof.

The string without the 3 feet added represents the circumference of the Earth, let's call it X. We are now changing X to be X + 3. If the string was initially stretched tight to the surface, then its distance from the center of the planet is basically the radius. Let's call it R. The question is really how does the radius from the center of the planet change if the circumference is changed from X to X + 3?

Doing the geometry, initially the radius was r and the circumference of the planet was 2 * Pi * r. Once we add 3 feet to the circumference, it becomes = (2 * Pi * r) + 3. To generically solve for radius, we divide circumference by 2Pi. Dividing the new circumference of (2 * Pi * r) + 3 by 2Pi will give us the new radius of r'. Solving this equation gives us:
(2 Pi r)/(2Pi) + 3/(2Pi) = r + 3/(2Pi) where r = the original radius, or roughly r + 3/6. This means the new radius after adding 3 feet to the circumference results in a new radius from the center of the circle of the original radius plus 3/6 or 1/2 of a foot or 6 inches.

Yes, it is hard to believe, but adding 3 feet to the circumference of ANY circle results in a radius 6" large. Even if you wrap a string a 3 foot string around a pin point, adding 3 more feet to the string will increase the radius by 6". The same is true if you due it to the circumference of a basketball or the planet Earth. How can this be?

It seems so unintuitive, though the truth is in the proof above. One way to think about it to make it a bit more intuititve is that while 6" is always the absolute change in the length of the radius regardless of the size of the object, the 6" gain in radius length is a different percentage change in the radius depending on the object size. To a 3 feet total string length (eg. a basketball), adding 3 feet to the string will add 6" inches to the radius - which almost doubles the length of the radius. Adding 3 feet to the circumference of planet Earth (24901 miles) also increases the radius by 6", but relative to the original length of the radius (3963 miles) this is a miniscual relative increase.

The Trip - Averaging 60mph

I love this puzzle since you can have arguments for hours with folks who just refuse to believe it.
If you are driving up a mountain that is 2 miles up and 2 miles down, and averaged 30mph for the first 2 miles, how fast would you need to go to for the remaining 2 miles to average 60mph for the entire trip?

(Hint: Avg miles per hour for the entire trip must look at total miles / total time.)

Answer is below in white on white text. Highlight over text with your mouse to view.
You can't! To average 30mph for the first 2 miles means it took you 4 minutes. To average 60mph for the entire trip....a total of 4 miles, you need to complete the entire trip in 4 minutes. You've already used 4 minutes up. So no matter how fast you go, you can no longer averae 60mph for the entire trip.

Try the same problem with a trip 30 miles up and 30 miles back. It's much easier to see then. 30miles at 30mph would take you 1 hour. To complete 60 miles at 60mph you would need 1 hour. Any amount of time past 1 hour for driving 60 miles will result in an average less than 60mph.

Three Men and a Hotel

Three men walk into a hotel, looking for a room for the night. They ask the Hotel Manager at the front desk, "do you have a room available"? He says "yes, but only one". They talk amongst themselves, and decide to share it.

They ask "how much is it"? The Hotel Manager answers $30. Each of the three men pull out $10 bills, pay the Hotel Manager and head up to their room.

A little while later, the manager realizes there is a promotion going on, and the room is really on $25. He calls over the Bell Hop, and says here's $5. Please give it to the men in Room 328 as they have overpaid.

In the elevator, the Bell Hop realizes the 3 men have no idea they are expecting to get change back. As a result, he pockets $2. He knocks at Rm 328. When the door is answer, he tells the men they overpaid, and gives them $3 back.

NOW.....if the three men initially paid $10 each and each get $1 back, then they have each paid $9. $9 times 3 = $27. Plus the $2 in the Bell Hops pocket is $29. Where is the 30th dollar?

Answer is below in white on white text. Highlight over text with your mouse to view.

The trick to this riddle is a slight of hand. There are three different amounts you can solve for.

You can solve for the total amount of money available, or $30. This is $3 returned to the three men plus $2 in the bell boy's pocket plus $25 behind the front desk. $3 + $2 + $25 = $30.

You can also solve for the amount of money the three men have out of pocket, or 3 * $9 = $27. This would be $2 in the bell boy's pocket plus $25 behind the front desk. $2 + $25 = $27.

Or alternatively, you can solve for the amount of money paid to the hotel, or $25. This would be the monet paid by the three men or 3 * $9 = $27 MINUS the $2 in the bell boy's pocket, or $27 - $2 = $25. Not PLUS. Switching the MINUS to PLUS when the riddle is told is the trick. You are making people think you are solving for the totla amount of money available when you are really solving for the amount paid to the hotel. A hard to catch 'slight of hand!'

Three Light Switches

A hard to get one....

You have a set of 3 light switches outside a closed door. One of them controls the light inside the room. With the door closed from outside the room, you can turn the light switches on or off as many times as you would like.

You can go into the room - one time only - to see the light. You cannot see the whether the light is on or off from outside the room, nor can you change the light switches while inside the room.

No one else is in the room to help you. The room has no windows.

Based on the information above, how would you determine which of the three light switches controls the light inside the room?

HINT: Be sure to use all of your senses!!!

Answer is below in white on white text. Highlight over text with your mouse to view.

No, there is no melting ice cube as in other 'closed room' stories nor any real trick other than thinking out of the box. Here's the answer. Turn two switches ON, and left one switch OFF. Wait 10 minutes. Then shut one of the ON switches OFF. One switch is ON and two are now OFF. Immediately go into the room. If the light is on, you know it's the switch that is still ON. If the light is off, feel the light bulb. If the bulb is cold, it's the light switch that you never turned ON. If the bulb is still warm but the light is off, it's the switch you turned ON but then turned OFF prior to entering the room.

See it's more than visual. You have to use your sense of touch....not intuitive with a light bulb riddle!