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## Monthly Archives for March, 2013

## Does a number start with 9?

You know that is a 95424250944 digit number. Consider the set . 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

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

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

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

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

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:

- The room is connected: a fly can fly from one point inside the room to another part without hitting the walls.
- Part of the room is lighted and the …
**Continue reading the story**"Light and dark parts"

## Combo cube

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

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

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"