Running order

Say there are 333 marathon runners. Each is randomly handed a number (1 to 333) to wear. What is the probability that the first three to finish will be wearing numbers that are in decreasing (not necessarily sequential) order?

Coin tossing (yet another)

You flip two coins simultaneously. If one coin is heads, what is the probability that the other coin is heads?

Curious Thing

Can you prove that a triangle with sides that can be written in the form n²+1, n²-1 and 2n ( where n>1) is right angled ?

What a shot!

A young man stands on the top tier in a football stadium
(no, not soccer). He throws a basketball to the football field
below (let's assume it's 500 feet horizontal and 50 foot lower)
to a bastetball goal, and makes it. What was the probability of
making this basket?

Assuming that he has tried many, many times, and 85%
of the time he can come within a 20 foot radius of the
center of goal.

You can see this video at http://RagKnot.blogspot.com

(Texas A&M University is where I went to school.)

To save time: diameter of goal 17 inches. diameter of ball 9 inches.
The shot was what we call a bank shot, not a "net only".
(An only net would be harder to make)

Cross that bridge

Three people (A, B and C) need to cross a bridge. A can cross the bridge in 10 minutes, B can cross in 5 minutes, and C can cross in 2 minutes. They have a bicycle and any person can cross the bridge in 1 minute using it. All three men and the bicycle start together on one side of the bridge. Only one person can use the bicycle at a time. The bicycle can be mounted and dismounted anywhere. All three men can be crossing together. What's the shortest time that all three men can cross the bridge in?

Turning heads

You are sitting at a table. On the table are 887 coins. 113 of the coins are heads up, the rest are tails up. You have to create two sets of coins. Each set must end up with the same number of heads facing up. That sounds easy. Unfortunately, it is totally dark and you can't feel which side of a coin is which. How can you do it?

Sick or not?

10% of people have a particular disease. A test for the disease gives the correct result 90% of the time (i.e that they do or do not have the disease). If a person is picked at random and the test result is positive, what is the probability that that person has the disease?

Prime time

Show that any odd prime can be written as the difference of the squares of two integers.

Paint balling

You have a bag with four balls, each is a different colour. You draw two balls from the bag, one at a time. You paint the first ball to match the colour of the second ball, you then put both balls back in the bag. What is the expected number of drawings before all four balls are the same colour?

Assume very quick drying paint.

More Prisoner Problems

Out of three prisoners scheduled to be put to death, Tom, Dick, and Harry, one of them will be pardoned. Tom asks the warden to tell him the name of one of the others who will be executed. As the question is not directly about Tom's fate, the warden obliges — either naming the other prisoner to be executed, in case Tom was too, or secretly flipping a coin to decide which of the remaining names to give Tom if Tom is the one being pardoned. Assuming the warden's truthfulness, there are now only two possibilities for who will be pardoned: Tom, and whichever of Dick or Harry the warden did not name. Did Tom gain any information as to his own fate, that is, does he change his estimate of the chances he will be pardoned?

If the warden says "Dick will be executed" and Tom could switch fates with Harry, should he?

Prisoner Problem

Consider a prisoner who’s about to be put to death but is promised his freedom if he draws a silver ball from one of two identical vase. He’s allowed to put 50 silver and 50 gold balls among the two vase any way he likes. The vase are then going to be shuffled around out of his sight and he has to pick one vase and draw one ball at random from that vase.

So how does the prisoner maximize his chances of survival? If he puts an equal number of silver and gold balls into one of the urns, the other urn would also contain an equal number of silver and gold balls, and thus the probability of his drawing a silver would be one in two. Can these chances be improved? If so, how?

A flipping fortune

You begin flipping a coin. Each time it comes up heads you double your money . You stop when the coin lands tails up. How much money do you expect to make before you throw a tails? You start with $1. I've no idea where the money is coming from.


Addendum, 10/10/09. The expected value is the same as the average value. To measure it, you would have to observe an infinite number of trials.

You cannot lose in this game. If you threw 10 heads, then a tail, you'd walk away with $1024. The probability of doing that is 1/2048. If you threw a tail on your first flip, you'd still have $1. The probability of doing that is 1/2.

Marbleous 2

You have a collection of bags of marbles. Some of the bags contain marbles of weight 10 g, the other bags contain marbles of weight 11 g. Your mission is to determine which bags are which in one weighing. You have a suitably accurate weighing machine. I won't state the other assumption, in case I give too much away.

This is probably another familiar golden oldie.

Prime Density

Primes numbers become less frequent as the numbers get larger. For example between 0 and 100 there are 25 primes. From 100 to 200 there are only 21 prime numbers.

If we use this step size of 100 (numbers that end with a double zero), there's a step where there is only one prime number in that range of 100 numbers. Actually there's an infinite number. But what step is the lowest that has only 1 prime?

Sample answer = “1000 to 1100" (that step has 16 primes)

Tape Recorder Trouble

Have you ever had a tape recorder and wondered, why it doesn't increase the speed and pitch of the sound while playing. if you are thinking because it's speed is constant, think again, while playing, one wheel becomes bigger and the other one smaller. Is it constant ?

Cash cows

Two men have equal shares in a herd of N cows. They sell all the cows for $N each. The money they get is in $10 bills plus less than ten $1 bills. One at a time they take a $10 bill. The man who takes the first $10 bill also takes the last one. The second man then takes all the $1 bills. How much does the first man owe the second?

Galileo's Principle

Air is a vapor (gas), which means that its atoms and molecules are not in such close proximity to each other as in a solid and liquid. So why don't air molecules just fall to the ground? After all, Galileo's principle states that objects should fall to the ground with equal acceleration independently of their size and mass.

Smallest Integer #223

What is the smallest integer greater than 0 that can be written entirely with 0s and 1s and is exactly divisible by 223?


I don't know how difficult this will be for others, but it took me 4 seconds.

No binary numbers

Another sequence?

87
104
97
116
32
105
115
32
116
104
101
32
110
117
109
98
101
114
32
116
104
97
116
32
98
101
103
105
110
115
32
116
104
105
115
32
115
101
113
117
101
110
99
101
63

Series Story

Till how many terms can you carry on the following series:

The series starts with A3, B1, C1, continues with G2, H2, J2, K2, and so on . . .?

Smallest integer

What is the smallest integer greater than 0 that can be written entirely with 0s and 1s and is exactly divisible by 225?

This one's fairly easy.

Floating hour glass

This is an old mystery. People have argued about the solution. It was published in Omni Magazine back in the '60. You can see it and read about it. But there's little argeement about the "why".

An hour glass containing air and sand is in a cylinder of water. While sand is flowing from the top to the bottom compartment, the hour glass remains submerged at the bottom of the cylinder, but when the sand has completed its downward journey, the hour glass floats to the top. Why?

Opposite of itself

here are few words which also means opposite of itself !

LEFT: 'remaining' as well as 'having gone'
CITATION: Award for good behavior as well as penalty for bad behavior
FAST: 'moving rapidly'; also 'fixed in position'
OVERLOOK: fail to notice; as also disregard deliberately or indulgently.

What kind of contranyms you come across everyday and understand it without any confusion?

Two abreast

This is probably my last dabble with relativity (for a while).

Prove that if in a given inertial frame, two photons are travelling in parallel and abreast, at a distance r apart, then they do so in all inertial frames.

Hints: Consider the case when they are not travelling abreast and then when they are abreast. There is a nice way to do this.

Link to the reference book for this problem:

here

Warren Buffet and Bill Gates

There is a round table conference. Warren Buffet is the richest person while Bill Gates sitting on his right has one dollar less then him. And the person on Bill’s right has two dollars less than Warren. And so on. Warren feels game and gives 1 dollar to Bill. He in turn gives 2 dollars to the person on his right. And so on.

But at one stage some one says, "Now I don’t have 1 dollar extra to give to the person on my right." However, he also says that he has 101 dollar more than him. How many dollars did Warren have initially and how many people were there?

Elementary Arithmetic

In the introduction to an early literary work on elementary arithmetic, the author looking back over his life, muses: "I was once x years old in the year x cube. I am now x square years of age and in another x years my only son will be y years of age in the year y square."

In what year was the author of the manuscript born?

Two trouble

Prove that 2/10=2 !

- Kushal chandel

Star Trek

I was researching for a new problem, but decided that the following link was more interesting: http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

OK, I've thought of a question: how fast must the rocket be going in order that the microwave background temperature rises to 10000 Kelvin? Ignore GR considerations, treat it as a SR doppler shift problem, use simple black-body radiation concepts only (unless you can do better).

Return to WowDVDFilm

A few days ago, there was a ToM that presented a problem where 8 variables could be valued as single digits 0 to 9, but there could NOT be a repeating value. Such as W could not equal D. The eight variables were W,O,D,V,F,I,L, and M.
My question is how many combination would there be? Review the WOW ToM - and figure out the number of combinations are possible with this set of eight variables, and is there a trick here? If so what?

Favorable Bet

Let’s say you get into an quarrel with a friend as to who owns the more expensive canine and decide to settle the spat by checking with the pet store where the two whelps were bought.

Meanwhile you also agree on a bet whereby the person who wins (owns the more expensive one) will give it to the loser. Thereafter you’re both tickled pink with this arrangement, because you’re both thinking, "If I win I’ll lose my hound, but if I lose I get a pricier pooch. And as the chances are obviously equal, the bet’s clearly to my advantage."

The problem is, how can a bet be favorable to both parties?

Heads or Tails?

A box contains two coins. One coin is heads on both sides and the other is heads on one side and tails on the other. One coin is selected from the box at random and the face of one side is observed. If the face is heads what is the probability that the other side is heads?

Phasers on stun

Sorry, this is a bit advanced - I won't do too many like this.

In the inertial rest frame of a dust cloud of identical particles, at some time t'=0, all the dust particles spontaneously and simultaneously begin to emit light.

What physically happens in an inertial frame where the dust is moving with speed v?

Compare this with the phase velocity of the associated de Broglie wave.

17 (camels)

A father dies leaving instructions that his 17 camels are to be split up between his 3 sons as follows:
Half the camels are to go to the eldest son
a third of the camels are to go to the middle son
and a ninth of the camels are to go to the youngest son

Failing to think of a way of carrying out split, they sought help from their wise but poor uncle. Their uncle arrived on his tatty old camel. He said, "I'll lend you my camel, then you'll have 18, and you should be able to divide them up without difficulty." So the eldest son chose his 9 camels, the middle son chose his 6, and the youngest chose his 2 camels. The uncle then got back on his camel (which no-one wanted) and went home.

DUH!! and DUH!! ... or is it? ^^

You're kidding!

A and B have the following conversation:

A: How old are your three kids?
B: The product of their ages is 36.
A: I still don't know their ages.
B: The sum of their ages is the same as your house number.
A: I still don't know their ages.
B: The oldest one has red hair.
A: Ah, now I know their ages.

How old are the kids?

Assume the kids ages are an exact number of years.

Wow, DVD Film

Solve: WOW/DVD=.FILMFILMFILMFILM ...

Each letter is a unique digit.

Humphrey Thou Sand

A camel has to transport as many as possible of an available 3000 bananas. For every comlete mile it walks, it must consume 1 banana (regardless of its load). It can only carry up to 1000 bananas at a time. What is the maximum number of bananas it can transport to a place a 1000 miles away?

Even upside down?

What word, when written in capital letters, is the same forwards, backwards and upsidedown?

So long since anyone made a post, I thought I would add this one.

Any way

Solve for A, B ,C ,D and E, the palindromic number equation:
AA + BCB = DEED

On the rebound

A cubic block of metal of mass M is hanging from a long thread. You throw a perfectly elastic rubber ball of mass m at speed u, perpendicularly to one of the block's vertical faces. Immediately after the ball bounces off the block, what is the speed of the ball and the block?

17 again?

1. Find a six digit number that has a 1 and a 7 in it. Call it A
2. Reverse the digits, Call it B
3. Divide A by B and the result is 17 (rounded to two decimals)

What is A?

Beady Eyed

If you drill a diametrical hole of final length L through a sphere, the volume of the remainder is the same as the volume of a sphere of diameter L. How come?

Literary Digest 2

A bookworm eats it way through a ten volume encyclopedia set on a bookshelf. Including the covers each of which are 2 mm thick, each volume is 5 cm thick. The bookworm starts at page 1 of volume 1 and makes its way through to the higher numbered volumes until it emerges from volume 10. Whilst partaking of this edible edification, how far did the bookworm travel?

Assume the bookworm goes in a straight line perpendicular to the covers and pages. Don't sweat it as to how the bookworm got to the starting point.

6

Ok this one's a bit tougher than the "17" problem.

Given an integer that ends with 6 (units column), when you move the 6 from the end to the beginning, the new number is 4 times the original number. What is the smallest number that does the trick?

Pretending that the first number is 3216, we want 4*3216 = 6321.
Obviously this example fails.

The best thing since...

Three men are about to eat. Fred has five small loaves, Tom has three small loaves and Ed only has $8. Fred and Tom agree to share their loaves equally in exchange for the cash. So Ed gives Fred $5 and Tom $3. Fred and Tom, being smarter than Ed start laughing. Why?

Easy - Just add 'em up

A= each non prime numbers between 1 and 1000
B= the lowest divisor (integer) for A
C= A * B

Example: If A = 10 B= 2 Then C = 20
And the sum of the C's up to A=10 is 83

What is the sum of all the C values for A up to 1000?

Literary Digest

Here’s a poem that was published by Adam C Orn in The Literary Digest of Chicago in 1906:

“Now I – even I – would celebrate,
In rhymes unapt the great,
Immortal Syracusan rivalled nevermore,
Who in his wondrous lore,
Passed on before,
Left men his guidance,
How to circles mensurate!”

What on earth does it mean?

17

How many three digit numbers are divisible by 17?

Life of a Battery

The users guide of my scientific calculator says its battery life is “approximately 9000 hours continuous display of flashing cursor OR approximately three years when left with power turned off.” I use the calculator on an average of 30 minutes daily. Approximately how long will the battery last?

Red faced children

A teacher tells a group of children that at least one of them has a red face. ‘Step forward if you have a red face’, he says at intervals, until... What happens next?

Assume that the children are clever enough to behave logically, they aren't disobedient and they can see each others faces, but not their own.

Unbalanced balance

You have a balance whose fulcrum is not at the centre of the pans. When you put a mass m on the left hand pan, it is balanced with a mass m1 on the right pan. When you put the mass m in the right hand pan, it is balanced with a mass m2 on the left hand pan. What is m?

Rain with an Angle

Why is there less rain per unit area in 45degree tilt?

Conveyor belt

A conveyor belt has speed v m/s. A stream of goods is being loaded onto it via a hopper, at a steady rate of m kg/sec. How much power has to be supplied to the conveyor belt to compensate for the load?

Neglect any unnecessary complications.

Sorry, a bit boring.

Stiff Problem

A stiff which was discovered in a park in Chicago in the middle of summer. It had a fractured skull and many other broken bones, but the cause of death was determined to be hypothermia.

Pass The Exam

Basically the problem concerned a student who didn’t stop writing his exam paper when the bell rang. So when he went to place his test booklet on top of the rest of the pile a good five minutes after everyone else had handed theirs in, the professor in charge said he needn’t bother since he had automatically failed. “Do you have any idea who I am?” sneered the student. “No and I couldn’t care less,” replied the prof. So how can the student pass – assuming of course that s/he’s not the failing type?

Three Gods

Three gods A , B , and C are called, in some order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A , B , and C by asking three yes-no questions; each question must be put to exactly one god.

The gods understand English, but will answer in their own language, in which the words for yes and no are “da” and “ja”, in some order. You do not know which word means which.

- hamujemy

Carry on towering

What is the range of convergence of x^(x^(x^(x^(x... ?
What are the corresponding minimum and maximum values of the expression?

You will find the following very useful:
http://en.wikipedia.org/wiki/E_%28mathematical_constant%29
Go half way down the page to x^(1/x). Have a glance, then follow the tetration link.

This is slightly modified from the original post in view of the info provided.

Only basic differential calculus is required.

Twoering inferno

What is the limit value of:

sqrt(2)^(sqrt(2)^(sqrt(2)^(sqrt(2)^(... ?

That's an infinite power tower of sqrt(2)s.