## Frog Leap Game

Posted on: January 21st, 2013 by

The puzzle with the image can be accessed here.

Start in this configuration with 7 rocks:

green frog - green frog - green frog - empty rock - brown frog - brown frog - brown frog

See how quickly you can switch the frogs to the opposite side of the pond.

Green frogs can only hop to the right. Brown frogs can only hop to the left. Frogs can only jump onto an empty rock. A frog can jump over one frog to an empty rock—but only over one.

Frogs cannot jump back, only forward.  It’s easy to tell which way they can go—they face the way they want to jump.

If you get stuck and your frogs can’t jump anywhere, move them back to where they started and try again.

When all the frogs have reached the other side of the pond, you win!

## prof. montanaris quals question

Posted on: January 20th, 2013 by
1

You are given an N x N board. Each cell can have at most 4 neighbors (left, right, up, down).  You are allowed to "infect" k cells on the board. At each time step:

1. An "infected" cell remains "infected".

2. A "clean" cell becomes "infected" if atleast two of its neighbors are "infected".

What is the minimum value of k and the initial infection pattern, so that eventually, the entire board becomes infected ?

---- prof. Montanari's quals question

## Third Child's Name

Posted on: January 20th, 2013 by
1. Before Mt. Everest was discovered, what was the highest mountain in the world?
2. Johnny’s mother had three children. The first child was named April. The second child was named May. What was the third child’s name?

via Forbes.

## Random point in an equilateral triangle

Posted on: January 20th, 2013 by

A point P is chosen at random inside an equilateral triangle of side length 1.  Find the expected value of the sum of the (perpendicular) distances from P to the three sides of the triangle.

## Powers of 2: deleted digit

Posted on: January 20th, 2013 by

Find all powers of 2 such that, after deleting the first digit, another power of 2 remains.  (For example, 25 = 32.  On deleting the initial 3, we are left with 2 = 21.)  Numbers are written in standard decimal notation, with no leading zeroes.

## Factorial plus one

Posted on: January 20th, 2013 by

Let n be a positive integer.  Prove that n! + 1 is composite for infinitely many values of n.

## Aces in Bridge Hands

Posted on: January 18th, 2013 by

Someone deals you a bridge hand 13 cards from a regular deck of 52 cards. You look at the hand and notice you have an Ace and say “I have an Ace”. What is the probability that you have another Ace?The cards are collected and different hand is dealt. This time you look at your hand and state “I have the Ace of Spades” which is true, what is the probability, this time, that you have another Ace?

Question: Is the probability in the second case the same as before, a lower probability, or a higher probability?

## Elastic collisions and $\pi$

Posted on: January 17th, 2013 by
1

Here is a surprising relation between elastic collisions and $\pi$:

Start with a large ball of mass $16*100^N$ kg. on the very left, a small ball of mass $1$ kg. to it's right and a wall to the very right. Send the large ball towards the small ball with an initial velocity $u_0$. The small ball will keep bouncing back and forth between the large ball and the wall. Assume all collisions are perfectly elastic. Eventually the large ball changes direction and starts moving away from the wall. What's surprising is that the number of collisions between the large ball and the small ball before the large ball changes direction is given by the first $N+1$ digits of $\pi$. As an example, if $N=2$, the large ball is $16*100^N=160,000 kg.$. In this case, the large ball undergoes 314 collisions before it changes direction. For $N=5$, it undergoes 314,159 collisions before changing directions. This result is exact (assuming perfectly elastic collisions and perfect mass without any errors). It's explained very well in the following youtube video:

## IIT-Madras ready for mass production of artificial blood - The Times of India

Posted on: January 16th, 2013 by

IIT-Madras scientists have blood on their hands — and nobody is complaining. A team of scientists from the department of engineering design has been successful in creating enough red blood cells from stem cells to be used as 'artificial blood' in people who need transfusion.

Having proved their oxygen-carrying capacity, the RBCs will now go into 'mass production' before starting human trials in three years, scientists said.

## Primate Intelligence

Posted on: January 16th, 2013 by
2

You walk into a room in which there are three primates: a chimpanzee, an orangutan, and a gorilla. The chimpanzee is holding a banana in each hand, the orangutan is holding a big stick, and the gorilla is holding nothing. Which primate in the room is the smartest?

Hint: View the comments for hints.

via Wu Riddles

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