xyfactor

The Monty Hall

2011-08-06T07:28:00.000-07:00

If you have ever struggled with the Monty Hall, check this out:

http://www.blog.assignmenthelpsite.com/probability-the-monty-hall-paradox/

Prepare to square off

2009-12-19T10:05:00.000-08:00

Consider an 8x8 chess board as shown in the adjacent figure. How many squares can you count without repetition?

Now consider the infinite chess board shown in the next figure. What is the inverse of the sum of the squares of all the distinct areas of squares that can formed from this board? What is the inverse of the sum of squares of all the square areas that can be formed from this board?

Playing with dominoes - II

2009-12-19T09:51:00.000-08:00

This one is often attributed to Martin Gardner.

How many dominoes would you require to fill up a 8x8 chess board?

Now, let us give vent to the suppressed child in us, and mangle the chess board by removing the opposite corner squares (see picture). How many dominoes would you require now?

Playing with dominoes

2009-12-18T08:50:00.000-08:00

A domino is a 2x1 piece of rectangle (see picture). Now consider a rectangular area of size nx2. The question is simple: given a large number of identical dominoes, how many different ways of arrangements of the 2x1 dominoes into the nx2 area are possible? Assume all the dominoes are identical. Note that as always, key ideas are more important in this blog than complete solutions.

DARPA mathematical challenges

2008-06-19T22:56:00.000-07:00

Alright dudes. Its time to show you some real $$$. Check out the DARPA mathematical challenge:

https://www.fbo.gov/utils/view?id=4a43eacdb1c247bc9809e581987b8f31

As the one who told you about the problems, I get a 15% cut. Thanks.

People on a plane!

2008-02-22T14:10:00.000-08:00

(Motivated by snakes on a plane) Here are two of the standard people on plane problem just to get things rolling again:
1) There are (say) 100 seats on a plane and since people with disabilities board first, the blind person boards first. Let us say this isnt Southwest and that each person is assigned a fixed seat (even the blind person is assigned a seat) and if his seat is occupied, he will sit in any seat that is free. Let us further say that the blind person picks his seat randomly of all the ones on the plane. The question is: What is the probability that the last (100th) person gets to sit in his assigned seat?
2) (Easier) Say there are n seats and n people. What are the number of ways in which each person sits in a different seat?

Queens!

2007-11-13T08:59:00.000-08:00

Apologies for the long absence. Lets try to solve the problem of the queens. This one is derived from the question asked by Vichu.

We know that the problem of placing 8 queens on a 8x8 chessboard without any queen cutting any other is quite simple. The question to be solved now is: How many such unique arrangements are possible?

Bit by Bit...

2007-04-26T19:13:00.000-07:00

What is the smartest way of counting the number of bits in a byte? (psuedocode please)

Catching a cop

2007-04-25T19:20:00.000-07:00

Say X has not received a speeding ticket in the last year.

Do the odds that X gets caught speeding tomorrow increase because the event has not occurred during the last year?

Alternately, can A confidently speed tomorrow if he received a ticket today?

Jump or Stay?

2007-03-16T19:46:00.000-07:00

I have this seemingly innocous habit of taking the ladder up to my attic window.

However, on this sunny summer afternoon things take an unpleasant turn. My ladder starts falling away from the wall just as I am about to step into my window. Would you recommend that I stay with the ladder and fall to the ground, or should I jump off the ladder?

An Uncomplicated Kill

2006-10-03T01:50:00.000-07:00

On a top secret assignment in Azerbaijan, you and your partners are rather bored, and Vodka is in short supply. Being the daredevils that you all are, you decide to play a modified game of Russian roulette to eliminate one of you. A six shooter revolver is purchased from the neighborhood arms dealer and three bullets are loaded in a row in three consecutive chambers. The barrel is spun only once. Then, each player points the gun at their head and pulls the trigger. The game stops when one person is dead.

Would you rather be first or second to shoot?

Two real problems

2006-09-28T19:51:00.000-07:00

I pick two reals x and y. I show one of them to you randomly by tossing a fair coin. You have to tell me if the number you see is the larger or the smaller of the two. How do you do this so that for any choice of x and y, the probability that you are correct is strictly larger than .5?

On a sold out flight, 100 people line up to board the plane. The first passenger in the line has lost his boarding pass, but was allowed in, regardless. He takes a random seat. Each subsequent passenger takes his or her assigned seat if available, or a random unoccupied seat, otherwise. What is the probability that the last passenger to board the plane finds his seat unoccupied?

Stringer Verlag

2006-09-27T20:54:00.000-07:00

What is the longest string you can make using only C's and G's within the following constraints:

1. No GG can occur within the string,
2. No sub-string can repeat itself 3 times consecutively eg: CGCGCG is not allowed.

Why?

Suspension bridges

2006-09-23T21:05:00.000-07:00

You must have seen the Golden Gate bridge or one of the other suspension bridges during your travels. What curve do the cables follow?

Admittedly, this is a bit mathematical. Hence, I shall abstain from giving points if someone complains. Otherwise, 1 point!

It gets somewhat interesting after you solve the problem. Shown below is a stolen picture of the chelsea bridge. In red diamond symbols are second order polynomial fits to the main cable. I simply digitized the middle cable and fitted it in matlab. While the good fit makes it very tempting to conclude that its a proof for a parabolic curve, the catenoid can also be series expanded to a parabola accurate to 4th order! Any small error in curve fit could push it this way or that. So, I guess the fitting does not say anything useful other than the fact that the cable is roughly a parabola.

ps: Check out Manoj's angle trisection comments at http://xyfactor.blogspot.com/2006/05/trisect-angle.html

Angles of a Quadrilateral

2006-07-18T13:11:00.000-07:00

Simple Question

a) What is the minimum number of acute angles in a convex quadrilateral? Prove.

That may be too easy. So, I'll give a point to (or split point between) the most creative and entertaining answer(s).

b) What is the minimum number of acute angles in a concave quadrilateral? Prove.

Clarification:

By acute, I mean angles that are 90 deg or lower. Acute = Non-obtuse.

Trisect an Angle

2006-05-23T20:15:00.000-07:00

Ok, here is a slightly more interesting one to further the progression of puzzles we have had here. Given a pencil and a straight edge, trisect a given (acute) angle. This is a classic geometry problem that has resisted the advances of the scale toting geometers for quite some time. Of late, a new and interesting way has been proposed to solve this problem which approaches geometry in a very nice and innovative way. I will post a hint if you guys have much trouble (though going by the other problems I have posted, this seems unlikely..)
Macaulay

A deadly gamble

2006-05-23T13:51:00.000-07:00

You are sentenced to be hanged by the neck till death. Arnie (the governor) agrees to pardon you if you can solve this puzzle for him (since he is thickheaded as we all know). You are given a deck of 52 cards in a dark room (so you cannot see the cards). You are told from the get go that exactly 17 of these cards are face up (and the tactile feel of the cards is the same on both sides). The question then is to split the deck into two stacks which have the same number of face up cards.

- Satish

Fair toss

2006-05-23T10:01:00.000-07:00

As you know, most coins are not perfect (i.e. the probability of heads is not equal to the probability of tails). So here is a simple probability exercise. Given a biased coin with P(heads)=p (obviously p is not a half), how would you simulate a fair coin?

-Satish

Call me!

2006-05-22T21:34:00.000-07:00

My phone number has all the integers from 1 to 7 except one number. The missing digit is the number of times the 1's appear in the number.

The 1's always immediately follow the 2's, and are, very conveniently, equispaced.

If you remove the 1's and the 2's, the rest of the digits form a number that is a product of three prime numbers. The first prime has a single digit, the second has two digits and the third has three.

Find it!

ps: Don't friggin look at your address book...
pps: If you do figure it out, you'd better give me a holler, feller. :)

ppps: Its a US number with 10 digits.

Stability of a train

2006-05-19T20:50:00.000-07:00

Ok, this one does not carry any points. It is only to feed your intellect; maybe even that little kid that lurks inside you and thinks about a train once in a while. With fascination and passion.

What keeps a train on the track?!

Sylvester's theorem

2006-04-07T13:00:00.000-07:00

There are a finite number of points on a plane. They are situated such that any line through any 2 of the points have at least one other of these points lying on it. Prove that all points are collinear.

From prior experience of hearing proof attempts, if you are giving a proof by induction, please consider your proof carefully before taking the effort to type it up.

General background: Sylvester was the first person to prove this, but his proof was a 7 page long proof! However, the current best known proof is very elegant.

~Grenade

Sudoku

2006-04-07T07:53:00.000-07:00

You must definitely know about sudoku,which has become a runaway hit all over the world.Exactly how many sudoku puzzles can be constructed?

~Sai

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

Angles without the angles...

2006-04-07T07:51:00.000-07:00

I saw this in a Martin Gardner book a while ago. Shown in the figure are 3 squares, and 3 angles in the bottom left. Prove, without any trigonometry (or equivalent), that the sum of the two smaller angles equals the largest.

~ Grenade

Seven Eleven

2006-04-02T12:17:00.000-07:00

Given a seven-digit number,whose digits add up to 59.Find the probability that the number is divisible by 11.

PS: Would this qualify as a puzzle?? And sorry for any bitter exam-related memories I may have inadvertently invoked in some of you...:)

~sai

Hydrocarbon based bipeds - Solve this!

2006-03-31T20:19:00.000-08:00

In a hasty attempt to remove comments about me from the top of the page, I pose this puzzle(?)based on something I did about 10 years ago:

A lot of us may know/remember that the bond angle of methane - CH_4 is approximately 109 degrees 28 minutes. This is created by 4 H atoms in 3d space repelling each other to produce a symmetric arrangement about the central C atom. Now tell me what is the bond angle of a hypothetical CH_(n +1)molecule in n dimensional space?
(Note for those who don't like chemistry: The chemical notation is used only to simplify description of what I would have to otherwise state as the angle subtended by the edge of a regular n dimensional hypertetrahedron at its center).

~ grenade
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In general, a hyper- prefix stands for a something with a generalised number of dimensions. ie a 2 dimensional (hyper)cube would be a square. A 3 dimensional (hyper)cube is a cube. The hyper prefix, however begins to become non-trivial when go into a larger number of dimensions. Just as a 3D cube is made by using 2 parallel 2D cubes that are offset in the 3rd D, and then by joining pairs of corresponding vertices by edges, the 4D hypercube is made by 2 parallel 3D ubes that are offset along an imaginary 4th axis (that's perpendicular to x, y and z) and then joining the corresponding vertices of the cubes.

A regular tetrahedron is a little more simple. In n dimensions, the regular n-hypertetrahedron consists of n+1 points that are equidistant from each other. So a 3D regular tetrahedron is a normal regular tetrahedron. A 2D regular tetrahedron is an equilateral triangle. A 1D regular tetrahedrom is a line segment (which is also a 1D cube). A 0D tetrahedron is a point (which is pretty much a 0D anything - well, anything that's not nothing, anyway).

I am afraid I am unable to give a better explanation within the confines of a blog editor, but I would recommend reading Martin Gardner's Mathematical circus, one of whose chapters has an introduction to hyperspaces.
~grenade
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From the lack of responses, I assume that the jump into hyperspace was made too far. So I will pose a simpler(?) question from the aforementioned Martin Gardner book:
Imagine an infinite checkerboard. Circumscribe the black squares with circles. These circles will intersect the white squares, and it is a matter of simple arithmetic to check what fraction of the white squares' areas are left unoccupied by the circles (or discs, if you would prefer).

Now extend it to 3D. We have an infinite "3d chessboard" where black and white cubes repeat alternatingly in all 3 directions. The black cubes are circumscribed by spheres whcih overlap the white cubes. It is a pretty hard exercise to calculate what volumes of the white hypercubes are left unoverlapped by the spheres, because the spheres intersect each other, and the math becomes messy.

My question is this: In 4D, what fraction of the hypervolume of the white hypercubes is left unoverlapped by the hyperspheres circumscribing the black hypercubes in an infinite 4d chessboard (alternating black and white 4d hypercubes along 4 mutually orthogonal directions)?

How about n-dimensions? n>4?
~grenade

A construction problem...

2006-02-24T10:09:00.000-08:00

The ancient temples in the kindgom of Pantheia are built with dimensions that are exact multiples of a unitum. Further, the dimensions of each room must be all different. For example, a temple with rooms of dimensions 5x11 and 13x5 is not allowed and neither is one with a square room.

You are the architect and are asked to build a new rectangular temple at the top of the flat topped mountain, Polypheme with atleast 2 rooms. The limited space on the mountain top means that you have to minimize the area of the temple. What is the temple with the smallest area that you can construct under the above constraints?

Explan clearly.

Unfork your way through...

2006-02-15T19:31:00.000-08:00

Two tribes live in a jungle - the truth tellers (who always speak the truth) and the liars (who always lie). A traveler is walking along a path when he comes to an unmarked fork in the road. One path goes to his hometown. And he does not know which. Just then, a native comes along belonging to one of the two tribes mentioned above. What is one question he could pose to the native so that he finds his way home? (0.5 points)

Food for thought...

2006-01-24T17:38:00.000-08:00

A crazy (suspected Brit?) explorer wishes to cross a desert on foot. It takes six days to cross the desert. One man can carry enough food for only 4 days. Under this scenario, using trial and error, you can find the fewest number of men required so that the explorer crosses the desert and the coolie(s) return(s) alive. What is this minimum number? (0.1 point - sorry, this is easy!)

The real question (1.0 point) is to find an elegant logic to generalize this problem. That is, if it takes N days for an explorer to cross the desert and each man can carry enough food for k days, what algorithm should be employed to attain the objective?

Does it ring a bell?

2006-01-20T00:38:00.000-08:00

Can a wind chime be viewed from an angle so that the tips of the chime rods fall in a single plane? (Non-trivial solutions only allowed)

Even-steven

2006-01-13T10:11:00.000-08:00

Twelve iron canon balls have the same weight, but for one that has a different weight. Find the odd ball, using only three weighings on a balance scale and determine whether it is light or heavy.

Vowel talk...

2006-01-13T10:08:00.000-08:00

Find three English words that have the vowels a,e,i,o and u in order.

0.2 points for each word upto a maximum of 1.0. No googling allowed.

The attack of the academics...

2006-01-04T17:02:00.000-08:00

Here is a rather academic problem which nevertheless reduces to some other interesting problems:

What is the expected value of the maximum of N random variables, each of which is between 0 and 1?

What is the expected maximum value of 2 variables under the constraint x+y=1?

If you have figured each of these out, you have passed the qualifying round for this month's IBM ponder this!

The cook in the noodle shop

2005-12-29T11:21:00.000-08:00

After a tiresome day boiling vegetables, the cook sits down in the corner of his dingy kitchen for dinner. As is his quaint habit, he puts exactly 100 noodles in his soup and begins his ritual:

He extracts two noodle ends from the bowl, ties a knot and slips it back into the soup. He repeats this 100 times until all the ends are tied. Now, he reaches his hand into the soup.

What is the probability that he would extract a garland containing all the 100 noodles?

credits: tuku

A number born of 3 digits

2005-12-21T22:53:00.000-08:00

ok, all this fence planning has prompted me to offer a simpler puzzle for those who would like to relax a little: what is the largest number that can be created out of 3 digits and any operators ? no googling please.

Fencing without fights...

2005-12-17T21:04:00.000-08:00

A farmer, one Mr. S, owns an uncultivated square field of green grass surrounded on all sides by pastures owned by his friends. Annoyingly for Mr. S, the cows grazing in the surrounding pastures often travel across Mr. S 's land to one of the other pastures.

Mr. S decides to stop this nonsense and considers isolating his field with a fence, but then a brainwave strikes him: "Instead of making the field impassible for these cows, what if I simply fool them into thinking it's impassible? I could build the fencing in such a way that they cannot see the other pastures and therefore won't try to walk across my field!"

At first, Mr. S considers building two fences in the shape of an 'X' across his field, which would block the view from any side of the square field to any other, thereby preventing the cows from trying to cross while saving on the amount of fencing needed.

But then he realizes that he could accomplish the same goal with even less fencing...

Can you determine the minimum amount of fencing required to block the view from any side of Mr. S's field to any of the other three sides?

This one is a big guy and carries 2.0 points!

Can you sum me out?

2005-12-14T10:41:00.000-08:00

The rules of this game are simple:

1. Numbers 1, 2, 3... 9 are arranged in a row. At every turn, each player chooses a number and adds it to all the previously chosen numbers.
2. The player who gets to 15, by summing any 3 of his chosen numbers, wins.

For example

Turn 1:
Me 5
You 9
Me 8
You 2
Me 3
You 4

You win (9+2+4=15)!

If you win, I will give you triple the money. If I win, I keep your money. If neither wins, the pot is split. To give you the advantage of winning, I will let you choose your turn - that is, you can choose to play first or second!

Would you want to play with me?

Clue: This problem can be reduced to a certain well-known game played by us all during boring classes!

The figures of the Hindus...

2005-12-07T20:13:00.000-08:00

This one is from the book Liber Abaci, published in 1202. Authored by Leonardo Picano, as a compilation of all that he learned from his extensive travels, the book played a pivotal role in introducing Eastern knowledge into the (then) developing West. Because of the reputation of his father Bonacci (who was a state official), he sometimes took on the name of Leonardo Fibonacci (Fibonacci, from Latin Filius bonacci meaning the progeny of Bonacci). Eight hundred years later, only his last name stuck, and today he is known as Fibonacci.

The following puzzle from Fibonacci's book is a play on the "Hindu" decimal number system that he learnt in India:

"A group of people are seated in a row, and one of them is wearing a ring on a certain joint of a certain finger. One person, who knows the whereabouts of the ring, computes a number as follows: He finds the wearer's position in the row, multiplies it by x, adds 5, multiplies that by z, and adds p. Then, he adds a number indicating the finger, and multiplies this by q. Then, he adds a number indicating the joint. When this number is announced, it is easy to pinpoint the ring. "

Find x, z, p, q and tell me how to nail Frodo Baggins!

(*) The problem is not that hard. You can assume that the joints in each finger are numbered 1 to 4. Further, a particular solution will suffice but make sure that you state the crucial logic. As usual, the first solution gets a point. But there is a bonus for the most elegant solution!

Squarely solvable?

2005-12-02T14:12:00.000-08:00

Consider N squares of sides 1, 2, 3...N. What is the minimum length of the square that will circumscribe all of these smaller squares?

If you find that to be too easy, extend it to a circle!

(No overlaps allowed)

An ancient Egyptian puzzle

2005-11-27T16:49:00.000-08:00

An Arabic merchant willed that his 17 camels be divided amongst his 3 sons in the proportions 1/2, 1/3, and 1/9. You are a clever lawyer who is trusted to execute the will. How will you do it without slaughtering any animal?!

(This one carries 1.5 points)

Strange matrices from a dark auditorium...

2005-11-16T14:44:00.000-08:00

Consider a matrix M, such that the sum of every row is the same as the sum of every column. That is, if you sum any row or column, it amounts to the same number. For example, with a sum of 18, M could be

9 2 7
4 6 8
5 10 3

Is it possible to construct a matrix M such that the sum of all its elements = 2^99 - 1?

Original problem courtesy six hours of trecherous pain during a training session a la the movie Office Space!

Note: Please include sisyphus' valid cribs in the comments section and the corrections concerning them...

Belt the answer out, if you can!

2005-11-13T18:29:00.000-08:00

If a belt were placed around the Earth's equator, and then had six meters of length added to it, and you grabbed it at a point and lifted it until all the slack was gone, how high above Earth's surface would you be?

Note: This might sound like a very simple problem but... :-)

This is especially to counteract the disappointment of last problem!

When do I take off?

2005-11-12T16:53:00.000-08:00

You made millions in GOOG shares and is now the proud owner of a light aircraft to make your lunch trips from office to home (of course, home is situated on a hill overlooking the oceans in the distance and near a forest with ferns). While you are rich and smart, you still carry a peculiar grad student attribute much to your wife's annoyance: you dont find the necessity to lunch at any fixed time in the afternoon.

It takes 15 minutes one way on a calm day without any wind and you spend 30 minutes at home to eat. Your house is due so
uth of your airport (which adjoins your office). You notice that the wind always blows along the north-south axis throughout the year.

All things being the same, the wind being the only variable, what should your strategy be to minimize the time of flight?

ps: Wild shots are now accepted!

A Lewis Carrol puzzle!

2005-11-11T22:16:00.000-08:00

The White Queen tells the following riddle to Alice in Through the Looking Glass. Lewis Carroll never answers it in the book. Let us see if you can!

'First, the fish must be caught.'
That is easy: a baby, I think, could have
caught it.
'Next, the fish must be bought.' That is easy: a penny, I think, would have
bought it.

'Now cook me the fish!'
That is easy, and will not take more than a minute.
'Let it lie in a dish!'
That is easy, because it already is in it.

'Bring it here! Let me sup!'
It is easy to set such a dish on the table.
'Take the dish-cover up!'
Ah, that is so hard that I fear I'm unable!

For it holds like glue-
Holds the lid to the dish, while it lies in the
middle:
Which is easiest to do,
Un-dish-cover the fish, or dishcover the
riddle?

Cold Calculations

2005-11-01T13:27:00.000-08:00

Winter approaches and its time to face the snow in Minneapolis. As you turn on the morning news on your dark green Jaguar's radio as you leave for office (with a voice command of course), you hear the weather report.

The snow begins to fall before 12:00pm and falls at a constant rate. Minneapolis sends out its first snow plow truck at noon and it begins removing snow at a constant rate (in cft/min). At 1.00 pm, it has gone 2 miles. At 2.00 pm, it has gone 3 miles, and is still not retracing its path. At what time did the snow start to fall?

Party Shakers-2

2005-10-28T10:56:00.000-07:00

Four couples attended a party. One couple was you and your spouse; another was Rama and Sita. Some people shook hands with one another, but nobody shook hands with oneself or with one's own spouse.

After the party, you asked each person (except yourself) how many different people the person had shaken hands with. Each person replied with a different number--and each person's reply was truthful.

You shook hands with Sita. Did your spouse shake hands with Sita? With Rama?

Bring a chessboard again..

2005-10-19T21:04:00.000-07:00

Two small problems here...

1. There are a number of squares on a chessboard. For instance, any four or eight squares joined together form another big square. Count the total number of squares (all sizes).

2. Considering boundaries of all 64 squares to be the roads, count the total number of routes you can take to traverse from one corner of a chess board to the diagonally opposite corner.

Lets play chess...

2005-10-17T04:10:00.000-07:00

A knight is placed in one corner (the corner square, that is) of a chessboard. It moves in its own peculiar way and has to visit each square once and only once. Can it reach the diagonally opposite corner of the chessboard (again, the corner square at the diagonally opposite end)? If so, in how many moves?

Humble Bicycle and friction

2005-10-16T08:31:00.000-07:00

All of us have rode the bicylce at some point of time...and we all know that friction always opposes the motion.
So, what is the direction of frictional forces on the front and rear wheel?
a) When pedalling
b) When idling (not pedalling).

Source: This was a question from Resonance (an IISc publication for undegrads and grads) may be in he year 1998/97.

Hey Water, Can I drink you?

2005-10-13T10:50:00.000-07:00

The way we pollute water, me thinks thatone day we will all have to go to Mars to get some drinking water... So, NASA decides to send a spaceship to Mars to explore this possibility... The spaceship breaks the law of relativity using a loophole and reaches Mars within an Earth day. They reach Mars... find the frozen water in the lakes, take it to a region of 5 degrees C in Mars, melt it and then filter it. finally, to check safety of drinking it, he finds the pH which is close to 8. He goes on to check the chemical components of water thinking that there must be something basic in the water but finds it to be pure water. Explain the paradox.

Of sand mixtures and thermodynamics!

2005-10-09T11:50:00.000-07:00

Hope you all are familiar with the common experience with haldiram bhujia...The big chunks of the yummy stuff is always on the top and separates out from the rest of the smaller sized bhujia. I say this because, we always had a fight at home as to who gets to open the pack! Similarly...if you shake a mixture of sand, you will find the bigger sized stones always on top.

Now the question is: Since shaking is a random process, we should expect the particles to be randomly distributed (to have more disorder, increase in entropy) but we have an apparent order (separation of particles into different sizes). Isn't it a against our second law of thermodynamics? Where is the catch? :)
@

Party shakers...

2005-10-07T07:49:00.000-07:00

In any party, there is a lot of handshaking. Show that there are atleast two people who have shaken hands with an equal number of people.

(credits:tuku)

The clank of the little kids...

2005-10-05T21:12:00.000-07:00

You walk into a school with 100 kids all of which are hyperactive. The school has 100 lockers one for each numbered from 1 to 100. The first kid walks into the locker room and opens all of them. The second kid zooms past and changes the status of every even numbered locker. The third kid waltzes through the room changing the status of lockers divisible by 3 (3,6,9 etc.), the fourth one does the same with all that is divisible by 4 and they do the closing and opening alternately. Which lockers are open at the end of the day? And what's your logic?

Making roads

2005-09-30T17:46:00.000-07:00

You are an engineer in the middle of a small and beautiful village with mountains looming in the background. They have a mud road and insist on building another thats perfectly perpendicular to the first one. Skanda's house is on the original road and the new road intersects in front of his house. Like all spirited people, they want to test your college education and give you a long rope. How do you find the right angle?

In other words, given a long rope, how can you fold it into a right-angled triangle?

I am tempted to give you a hint but GOD, someone please attempt a solution... :-)

Power balance

2005-09-17T10:09:00.000-07:00

A chess board has 64 squares. Pick numbers between 1 and 64 and place these numbers on each of the squares of the chess board such that the value in every square is the average of the values of its neighboring squares.

Prove that the only way to make such an arrangement is when all squares have equal numbers.

(http://www.cisl.columbia.edu/grads/tuku/brainteasers.html)