3D Object

* Draw Square 1 with sides of length x. Draw a smaller square, inside square 1, with sides of length x/2. Align the smaller square such that one of it's sides lies on top of, and in the center of, the lower side of square 1. (lower side of square 1 refers to the side closest to the bottom edge of the paper upon which you are drawing)

* Repeat the above instructions and label the second drawing as Square 2

* Square 1 and Square 2 are identical.

* Square 1 is a plan view (top view) of a 3 dimensional object.

* Square 2 is the elevation (front view) of the same object.

* Draw a 3 dimensional representation of the object.

* DESCRIBE the OBJECT.

-- Zaux

Peer to Peer

Suppose you are just starting to download a 4.2 GB file on a peer to peer network. There are 20 other people also up and/or downloading this same file. Nineteen have 20% and one has 100%, and you of course have 0%. (Assume each 20% is of random sections... none are the exact same sections.)

Everyone can download up to 500 k/sec and will upload 10 k/sec. When they get 100% each will begin uploading at 50k/sec.

You must understand that on peer to peer networks you request a random section of data you need and the request will be put on hold by others who have the data. Later someone will ask if you still need it and if you answer yes, it will be sent.

Assuming everyone stays active till everone reaches 100%, how long will it take for you to reach 100%?

Candle Problem

You are in a dark room and seeing a lighted candle at 6-to-8 meters distance. Slowly move towards it till the candle is right in front of your eyes (a mm or two away). As you move towards the candle the eye stars accommodating to keep it in focus. This it can do to a certain maximum point beyond which the flame goes out of focus. Since the candle is still moving closer to the eye, its focus which till now was on the retina will keep going backwards, so much so that an inverted image formed on the retina should at one point become upright. Then why don’t we see objects which are very close to the eye, upside down?

WARNING: DON'T TRY THIS AT HOME, YOU CAN BURN YOUR EYE BROWS OR WORSE LOSE YOUR SIGHT

Dice Drama

There are six squares marked 1, 2, 3, 4, 5, 6. You are invited to place your money as you wish on any one square. Three dice are then thrown.

The Rules
-------------
If your number appears on one dice only, you get your money back plus the same amount.
If two dice show your number, you get your money back plus twice the amount you placed on the square.
If your number appears on all three dice, you get your money back plus three times the amount. Of course if the number is not on any of the dice, the operator gets your money.

Now, you might reason: the chance of my number showing on one dice is 1 : 6, but since there are three dice, the chances must be 3 : 6 or 1 :2, therefore the game is a fair one. Of course this is the way the operator of the game wants everyone to reason, because it’s quite fallacious. But is the game favourable to the operator or the player, and, in any case, just how favourable is it ?

Parallel Lines

How will you find the mid points of two parallel lines. You have a ruler without the scale. (Mid points of each of the lines)

Galileo’s Twist

Climb up the Eiffel tower, a rod made of one metre iron and one metre wood. Holding it horizontal drop it. It will reach the ground with the heavier iron side touching down first. Is this a violation of Galileo’s: time of fall or acceleration is independent of mass?

Unique Poem

This poem is actually very gimmicky.
It is an attempted attempt at fusion.
Cause and result have been made to occupy a common plane.
Enough to drive me insane.

The author has tried to analyse
Some not yet existing lines

In a vaguely sonnet form.
But in that case result has pre-
Ceded cause because the poem

Only comes into existence on comple
Tion of the analysis. What then
Was the poet analysing?

Yet the poem
And its analysis
Exist occupying the same frame. That is,
The content and its criticism
Are actually one and the same.

What so unique about this poem ?

Wise Men

A king wanted to kick out all the wise men in his land but there was an old law he could not disobey which said there should always be

"Seven blind of both eyes:
Ten blind of one eye:
Five that see with both eyes:
Nine that see with one eye."

So how many wise men did he keep?

25 lockers

A length of 25 lockers are in the hallway. The doors are all closed when guy #1 begins to walk by. He changes the state of each door - he opens all 25 locker doors.

Guy #2 walks by and changes the state of every second door. (He closes doors number 2,4,6 etc. Remember they were all open)

Guy #3 walks by and changes the state of each third door. (He closes open doors and opens closed doors. (doors 3,6,9 etc).

This continues with more and more guys. Each one changing the state of doors according to their number.

At some point, I took this picture of the lockers. Some open, some closed (the blue locker doors are closed).

How many guys had done their job at this point?

Shakespeare Quote

"Strange weather! What could equal it? Yesterday sunshine and soft breezes; today a summer cyclone raging noisily; then other changes, as floods of the fiercest rain eddy beneath the blast."

Concealed in it is a familiar quotation from Shakespeare, each word being buried in its proper order. Can you find it?

Circles

You have 2009 concentric circles, with radii 1 to 2009.
a point is taken on the largest circle (the one with radius 2009)
from here, tangents are drawn to all the other 2008 circles.

How many of these tangents have a length that is a whole number?

Compute area between two curves

Given two simple equations y=sqrt(x) and y = -(x^2)/5

What is the area between the curves from x=0 to x=6?

Collatz Series 2

If you don't know what the Collatz series is, see the previous post.

We learned lots of the initial numbers give a series that contain 40.

Give two initial numbers between 100 and 200 that are ODD and don't contain a 40 in their series.

Collatz Series

A strange series of numbers exists where the next number in the sequence depends on the previous one being either even or odd. If it was even, the next is one half of the previous, if odd the next one is 3 times the previous, plus one. The last number of the series is always one. The number of steps to get to one is sometimes called the halting point. (Even the next is N/2; if Odd the next is 3*N+1)

Example, Intital number is 6 then the steps are... 3,10,5,16,8,4,2,1
This has 8 steps and the high number is 16.

(If you plot the initial number vs. the high, you get a neat picture)

There once was a reward of $500.00 for solving various aspects of this series. See

http://en.wikipedia.org/wiki/Collatz_conjecture.

Your question is this. There are six different initial numbers between 3 and 100
that end with these nine steps... 40,20,10,5,16,8,4,2,1 ... what are the six initial numbers?

As a hint, each of these six initial number has 17 steps to reach 1.
So as you can see the first 8 steps vary, but the step nine is 40.

Please give as many of the six you can solve.

Trio is Back

Tom, Dick and Harry went fishing, having agreed to share the catch equally. After a successful day, they put the catch on the boat deck to be divided up later, had a big party, and went to sleep. In the middle of the night the Tom wakes up, feeling ill, and decides to go home with his share. But he sees that the catch can't be divided evenly into three because there is one fish too many. So he throws the extra fish into the lake, takes his share, and goes home. A little later the Dick wakes up, also feeling sick, and decides to go home with his share. Not realizing that Tom has already left, he also sees that the catch can't be divided equally because of an extra fish. He throws the extra fish into the water and departs. Finally Harry wakes up, ill as well, and unaware that his friends have gone, counts the fish, finds one extra, throws it away and takes his share.

The question is, What is the smallest number of fish that the fishermen could have caught during the day?

Space Elevators

Armstrong says space elevators must be built. He’s planning to build a carbon nanotube ribbon that is three feet wide, anchored somewhere in Ecuador, stretching a mind-numbing 62,000 miles upwards into space. The centripetal force of Earth’s rotation will keep the ribbon taut. The problem is – if I sneak on to the platform in Ecuador and snip off the ribbon at its base, what will happen?

Circular Coverage

Consider a disk with a radius of one unit. We need to cover the disk’s surface with some smaller disks so that none of the larger disk can be seen. These smaller disks have a diameter of one unit. How many (the least number) will it take to completely cover the surface?

On Logic

1) All of the humankind, other than my footmen, have a certain amount of common-sense;
2) No one, who lives on barley-sugar, can be anything but a mere baby;
3) None but a hopscotch player knows what real happiness is;
4) No mere baby has a grain of common-sense;
5) No engine-driver ever plays hopscotch;
6) No footman of mine is ignorant of what true happiness is.

What logical conclusions you can make from here ?

Six and Seven

How do you proof, for the fact that if any set of integers is repeated six times to form another integer it must be divisible by seven? (eg. 111111)

Sum Problem

In a group of 10 people, each one is asked to write sum of the ages of the other 9. The sums form the 9 element-set (82, 83, 84, 85, 87, 89, 90, 91, 92). Find the ages of the youngest and the oldest in the group and the ages of the two persons with the same age.

Mirror oh Mirror

How can you differentiate between a normal mirror and a 1-way mirror (used by spy and police interrogation work by CIA etc, where you can see through from the back) without turning the mirror over?