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Monthly Archives for March, 2013

Interesting applications of pigeon hole principle

Posted on: March 31st, 2013 by
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Go here to see interesting applications!

Does a number start with 9?

Posted on: March 30th, 2013 by

You know that 9^{100000000000} is a 95424250944 digit number. Consider the set \{9^1,9^2,9^3,\dots 9^{100000000000}\}. How many of these numbers start with a 9? (Example: 91 starts with a 9, but 729 does not start with a 9). (Obviously you are not allowed to use a computer. That's why the number is bloated up by so much that even a computer can't feasibly solve it). Source: One Continue reading the story "Does a number start with 9?"

Star Gazing

Posted on: March 29th, 2013 by

An astronomer observed 20 stars with his telescope. When he added up all the pairwise distances between the stars, the result was X. Suddenly a cloud obscured 10 of the stars. Prove that the sum of the pairwise distances between the 10 re- maining stars is less than 1/2 X. Bonus: Can you improve the bound? What is the smallest real number r such that the Continue reading the story "Star Gazing"

1 Gold Coin

Posted on: March 28th, 2013 by
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Six pirates plunder a ship with only one gold coin. After venting some of their frustration by killing all on board the ship, they now need to divvy up the one coin. They are so angry, they now value in priority order: 1. Their lives 2. Getting money 3. Seeing other pirates die. So if given the choice between two outcomes, in which they get the same amount of Continue reading the story "1 Gold Coin"

The Greek Philosophers

Posted on: March 27th, 2013 by

One day three Greek philosophers settled under the shade of an olive tree, opened a bottle of Retsina, and began a lengthy discussion of the Fundamental Ontological Question: Why does anything exist? After a while, they began to ramble. Then, one by one, they fell asleep. While the men slept, three owls, one above each philosopher, completed their digestive process, dropped a present on each philosopher's forehead, Continue reading the story "The Greek Philosophers"

Smallest distance

Posted on: March 26th, 2013 by
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There are 6 points in a rectangle with the sides, 3 and 4. Prove that the distance between at least two of these points is smaller than the square root of 5. via Utah Math puzzles

Light and dark parts

Posted on: March 25th, 2013 by
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You want to build a room with perfect mirrors for the walls, ceiling and floor. You are free to choose the room. However, you must build the room and place a light source in the room such that:

Combo cube

Posted on: March 24th, 2013 by
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Eight cubes are marked with one dot on two opposite faces, two dots on two opposite faces and three dots on two opposite faces. The eight cubes are put together to form a 2 × 2 × 2 cube. The dots on each face of the large cube are counted, to get six totals. Can the cubes be arranged so that the six totals are consecutive integers? via Continue reading the story "Combo cube"

Lazy fly

Posted on: March 23rd, 2013 by
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A rectangular room is 30 feet long, 12 feet in height, and 12 feet wide. A lazy fly is resting in the middle of a wall at the end of the room, at a point one foot down from the ceiling. In the middle of the opposite wall, one foot up from the floor is a tiny scrap of food. The fly is too lazy to fly and Continue reading the story "Lazy fly"

Prisoners and boxes

Posted on: March 22nd, 2013 by

This is the classic prisoners and boxes puzzle: There are 100 prisoners numbered 1,2,... 100 and 100 boxes. Each box is numbered and the number is stored inside the box as a label. The boxes are identical, there's no way to tell them apart without inspecting the labels inside. The boxes are placed in a row. The jailer announces that each prisoner will be given Continue reading the story "Prisoners and boxes"

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