Hold on

A particle is in an upright circular path of radius r. At what speed, u, must it be injected (at the bottom) in order that it can go right round the track without falling off?

Neglect friction, drag and variation of g with height. Assume that the mass slides (it doesn't roll).

-

The time it takes to go round the loop involves non-analytic elliptic integrals. I can do it but only thanks to wolframalpha.com

Short changed

Three men go to a hotel. They pay for a $30 room to share, each giving $10. Later the manager realizes that the room was only $25 and so gives $5 to the bellboy to return to the men. The bellboy takes the $5 to the men. They give the bellboy $2 as a tip. Just after the bellboy left, one of the men says, "We each spent $9 just now, since we each got back $1, and we gave the bellboy $2, that's $29". Where's the missing $1?

This must have been a long time ago.

Isn't it marbleous!

You have 10 bags of marbles. Each bag contains 25 marbles. 9 of the bags have marbles that weigh 10 gm. The other bag has marbles that weigh 11 gm. The bags aren't labelled. You have a weighing machine that displays in grams, and is sensitive enough to reliably read to the nearest gram. Using only one weighing, how can you find which bag has the 11 gm marbles?

Mixing it

You have jugs A and B that contain the same amount of differing liquids. Take a small sample from A and mix it into B. Then take the same size sample from the A+B mix and mix it into A. What is the relative concentration of A in B to B in A?

There's more than one way of solving this.

xth root of x

Just how big can x^(1/x) be? Assume x is real number.

NEVER is prime

NEVER is prime,
NOW is too!
SHE is not,
Unlike YOU!

Where are we?

Who squares?

How many squares are there on a chessboard?

Assume a simple chess board with no margin.

Just making a point

Prove that the three perpendicular bisectors of a triangle always meet at a point.

Here's a link so you can see what I'm blathering on about:
http://www.analyzemath.com/Geometry/Circumcircle/Circumcircle.html

On that page, click the "click here to start button". Ignore the circle that you see. Drag the triangle around and see what happens.

Classical geometry on allows the use of a straightedge and compass. Rulers and protractors are not allowed. Trigonometry is not allowed.

However for this problem any reasonable proofs are OK with me.

Happy Birthday

I expect everyone knows this one, but it's better than nothing.

In a random group of 23 people, there is a slightly better than a 50% chance that at least two of them share the same birthday. How is that possible?

How big would the group need to be to get the chance of a shared birthday up to at least 95%?

Discard all leap year complications. Assume each birthday is equally likely. Only consider the day and month, not the year of birth. It is not a trick problem.

Treasure Revisited

The solution to the original post (yesterday) was to quick.
But we can hopefully find the numeric solution with some given data.

Let's say the Elm tree is at (30,35) and the Oak is at (80,44).
These are x,y coordinates.

Find the x,y of the treasure within a 1/2 unit.

Needle Mileage

A phonograph record has a total diameter of 12 inches. The recording itself leaves an outer margin of an inch; the diameter of the unused center of the record is 4 inches. There are an average of 90 grooves to the inch. How far the needle travel when the record is played ?

Treasure hunt

A man has a treasure map. It shows an island with a gallows, an elm tree and an oak tree. The map has instructions. They say go to the gallows and walk to the elm tree, counting your paces. Then turn 90 degrees right and walk the same number of paces as you had just counted. Mark that spot. Go back to the gallows. Now walk to the oak tree, counting your paces. Turn 90 degrees left and walk the same number of paces as you had just counted. Mark that spot. The treasure is buried halfway between the two marked spots.

When he got to the island, he could find the trees, but not the gallows. How can he find the treasure?

There are quite a few ways to answer this problem. One is partly outside the box. It is a famous problem.

Family Ties

Boy says: "I have as many brothers as sisters."

His sister says, "I have twice as many brothers as sisters."

How many of brothers and sisters are there in the family ?

Miller's Fee

In a Tennessee mountain community the miller retains as his fee one-tenth of the corn the mountaineer farmers deliver for grinding. How much corn must a farmer deliver to get 100 pounds of cornmeal back provided there is no loss ?

Bug on a Rubber Band

An infinitely long rubber band has one end nailed to a wall, while the other end is pulled away from the wall at the rate of 1 m/s; initially the band is 1 meter long. A bug on the rubber band, initially near the wall end, is crawling toward the other end at the rate of 0.001 cm/s. Will the bug ever reach the other end? If so, when?

Whirl pool action 2

First read the previous "Whirl pool action".

I put a six inch ball and the same six inch 2x4 into the pool and made a counter clockwise whirl pool.

The ball rotated about 10 times per revolution, and the 2*4 rotated about three time per revolution around the pool, both counter clockwise.

If this was a solar system, we could say the ball had 10 days per year, and the 2*4 had 3 days per year.

Both objects were placed at edge of the pool at the same place, about the same time. The ball made it to the center in about 5 years, the 2*4 made it to the center in about 15 years.

Both objects were about the same weight, why does the ball get the the center first.

Dishes and Ratings

A fast food place has N dishes on its menu that are rated from worst to best, 1 to N. You, however, don't know the ratings of the dishes, and when you try a new dish, all you learn is whether it is the best (highest rated) dish you have tried so far, or not. Each time you eat a meal at the restaurant you either order a new dish or you order the best dish you have tried so far.

How can you maximize the average total ratings of the dishes you eat in M meals (where M is less than or equal to N).

Graham's Number

Not a problem. Just for interest.
If you thought you knew what a big number is, then you may be in
for a surprise and a headache. Follow this link:
http://en.wikipedia.org/wiki/Graham%27s_number

Reversing a Number

You can get a reverse of number only by multiplying the number by X or Y ? Find X, Y ?
Assume the number is not a palindrome.

if you are wondering, reverse of ABC = CBA

Whirl pool action

In my small pool I often get a lot of leaves that settle to the bottom. I can make a "whirl pool" and the leaves all come to a small pile right in the middle of the bottom where I can scoop them out quit easily with a dip net.

The wet leaves are heavier than water because they sink. Then why don't they move outwards in the whirl pool? What scentific principals cause them to pile toward the middle?

The bare facts of the naked truth

I hope this one isn't too well known.

A man died and went to heaven. When he got there he found that everyone was naked. A man and a woman walked over to greet him. The (newly dead) man said to them, "Hello Adam, hello Eve".

How did he know who they were?

Do you Know Gambling

With two dice getting a total of X is more likely than Y.
Can you tell for what value of X and Y this is true ?

A Circle Mystery

From Harry Potter and the Deathly Hallows:
"Which came first, the phoenix or the flame?"
"...a circle has no beginning," Luna Lovegood once cleverly said.

But really, does a circle have no beginning?
I mean, there has to be something that begun the circle,
or else there would be no circle at all..........

What do you think?

Beam me up

You have two *square-section beams of wood. One has a density greater than half that of water, the other less than half that of water. You throw them in a pond. What do you notice about the orientation of the floating beams?

*In the original posting, I unintentionally wrote "rectangular".

You're so special

Numbers like pi and e are usually regarded as being special.
Are there any numbers that are not special?
Prove your assertion.

Three Magical Numbers

There are three simple numbers which has a mysterious property. Sum of any two numbers is a perfect square. Can you tell the numbers ?

one for all ...

What is 1/(1 + 1/(1 + 1/(1+ ..... ?

pie eyed

Just a sick and quilly.

What is e^(pi/2 e^(pi/2 e^(pi/2 e^(pi/2 e^(pi/2 e^(pi/2 e^(pi/2 e^(pi/2 e^(pi/2 e^(pi/2 e^(....e ^(pi/2 i)))...))) ?

Do you know Square Root

Leibniz , a German mathematician stated that

SQRT(1 + SQRT(-3)) + SQRT(1 - SQRT(-3)) = SQRT(6)

What do you think ?

Nobel Prize

Why there is no Nobel Prize for Mathematics ?

Hole in a Cube

Is it possible to cut a hole in a cube in such a way that the larger cube can be passed through the hole ?

Taking the pole into the room (in 3D)

Ok, no one has figured out the required hallway width.
Remember this...

"The pole from the previous flatland post, is now going
into a large room off a hallway. The walls are 6 inches thick,
the door is 30 inches wide. The pole is 24 feet long.
The hall is not very wide. How wide would it have to be for
the 4 inch diameter pole to be taken into the room without bending it?"

Now assume we are not in flatland. The door is 7 foot high, the ceiling
in the hall is 8 foot high. What is the hallway width required if we can tilt
the pole?

Do you know Addition

Which one is bigger ?

99⁵⁰ + 100⁵⁰ or 101⁵⁰

How ?

Can I say "annulus"?

Draw two concentric circles with different radii. Draw a chord
in the larger circle that is also a tangent to the smaller circle.
If the chord length is d, what is the area of the annulus (the
washer shape) between the two circles?

To enable consistent communication, use O for the centre of
the circle, T for the point at which the tangent touches the
inner circle, A and B for the ends of the chord and R and r for
the larger and smaller radii respectively.

aye aye

If i is the usual unit imaginary number (Sqrt(-1)), what is i^i ?

Take the pole into the room

The pole from the previous flatland post, is now going
into a large room off a hallway. The walls are 6 inches thick,
the door is 30 inches wide. The pole is 24 feet long.
The hall is not very wide. How wide would it have to be for
the 4 inch diameter pole to be taken into the room without bending it?

Proof that 1 = -1

I hope this hasn't been posted before

1 = sqrt(1)
=> 1 = sqrt( (-1) * (-1) )
=> 1 = sqrt(-1) * sqrt(-1)
=> 1 = (sqrt(-1))^2
=> 1 = -1

What!!!!

Hallway Pole

You are an architect. Your client, living in Flatland, wants a building designed with a long 3 foot wide corridor which opens into a larger hallway. You must design the hallway for the minimum width which will allow the inhabitants to move a 24 foot-long pole down the corridor and turn it into the hallway.

The corridor is perpendicular to the hallway. Since this is Flatland, the pole cannot be tilted up. The pole is rigid. How wide must the hallway be?

New Alphabet

If you had to make a new letter of the alphabet what would it be?

Algebra Trig ?

I have a problem I can solve but I need a better solution.

The equation is

y = r - r * COS( d / r )

Given d and y find r.

r= ?

I don't think a picture will make the equation solvable, but here's a picture.

http://1.bp.blogspot.com/_gQkeSWqb63Q/SoS1r1bjrCI/AAAAAAAAAFY/_aQzlEhMJY4/s1600-h/tom-Capture.JPG

One Half Pill Problem

You have a prescription to take one half of a pill per day for 12 days, but the pharmacist (who is too busy to divide pills for you) gives you 6 whole pills in a bottle. On day 1, you remove a pill from the bottle, break it into two half-pills, take one, and return the other half-pill to the bottle. On all subsequent days you shake the bottle thoroughly and pour something out - whatever comes out first - either a half pill or a whole pill; if it's a half pill you take it and you're done for that day; if it's a whole pill, you split it into two half-pills, take one, and put the other back in the bottle, exactly like you did on day 1.

On day 12 there can be only one half pill left in the bottle, but on day 11 there are two possibilities: either there is one whole pill or there are two half-pills left in the bottle. What is the probability that there are two half-pills in the bottle on day 11?

Weight Problem

A chain of length A cm and mass B kg is suspended vertically by one end with the other end immediately above a scale. The chain is released and falls. At the instant the entire chain has fallen onto the scale what does the scale read?

Black Holes and the Big Bounce

Big Bounce is a theory uniting two other theories, Big Bang and the Big Crunch.
The theory is that after the Big Bang, there will be a Big Crunch,
then after the Big Crunch, there will be another Big Bang and so on...

A Black Hole is what's left from a Supernova, an infinately small object with a giant mass of a supergiant that was massive enough to be a Supernova.

If a Black Hole is just a vast amount of mass all crushed together into an infinately tiny space,
then wouldn't the Big Crunch create a super, super, super....super, super massive Black Hole and also isn't the Big Bang a reverse of the Big Crunch, therefore "unwinding" the tiny volume of the supermassive Black Hole?

CAN THIS LEAD TO A NEW THEORY FOR THE CREATION AND DEATH OF THE UNIVERSE???? (sorry....got excited....don't answer this....just answer what I wrote above....)

Maximum Velocity

A shell flying with velocity 500 m/s bursts into three identical fragments so that the kinetic energy of the system increases 1.5 times. What maximum velocity can one of the fragments obtain?

Mass at Planet51

A space traveler about to leave for Planet 51 has a spring balance and a 1.0 kg mass A, which when hung on the balance on the Earth gives the reading of 9.8 newtons.
Arriving at the Planet51 at a place where the acceleration of gravity is not known exactly but has a value of about 1/6 the acceleration of gravity at the Earth's surface, he picks up a stone B which gives a reading of 9.8 newtons when weighed on the spring balance. He then hangs A and B over a pulley and observes that B falls with an acceleration of 1.2 m s–2 .

What is the mass of stone B?

Patient, Pills, and Probability

A Patient is taking one each of 5 different types of pills every day but he don’t like having to open and close 5 different bottles, so at the beginning of each (30-day) month he put 30 of each type of pill into one big bottle. When it is time to take your pills, he draw them out of the big bottle one at a time until he have (at least) one of each type.

On the last day of the month he will draw exactly 5 pills and they will all be different (because that’s all that’s left in the bottle), but on other days he will generally have to draw more than 5 pills in order to have (at least) one of each type. So, the question is: On the first day of each month (when there are 150 pills in the bottle), how many pills, on average, must he draw from the bottle in order to have (at least) one of each?

Archimedes Principal

A fisherman rowing his boat on a very small lake throws his anchor into the water. Does the water level of the lake rise, fall, or stay the same?

Do you know Gravity ?

If you throw a small ball vertically upward in real air with drag, does it take longer to go up or come down?

Beetle on Sphere

A Beetle starts from rest at the top of a sphere of radius R and slides* on the sphere under the force of gravity. How far below its starting point does it get before flying off the sphere?

* assume sphere is frictionless.

Pile of Bricks

A uniform brick of length L is laid on a smooth horizontal surface. Other equal bricks are now piled on like a ladder , so that the sides form continuous planes, but the ends are offset at each brick by a distance L/a, where a is an integer. How many bricks n can be used in this manner before the pile topples over?

Palindromic Fun

1. Mischievous children of celebrities.

2. What did I see ? was it a rodent ?

3. Mother is as self - sacrificing as myself

4. All inhabitants of ancient Rome were smart ; not stupid

5. All chieftains moan

Card Staircase

If you take a pack of cards, you can slide them over to make
a kind staircase. What is the maximum horizontal distance the
top card can be from the bottom card?

The staircase will not have a constant slope. Use obvious
idealisations, this isn't a real engineering problem. You can
use as many cards as you want.

- Photino

Time Turn

At this moment the time is 9:00 PM. Can you tell me what time it will turn, 23999 999 992 hours later ?