Suppose M is a matrix with real entries.
Suppose that each of it's rows sum to an integer, and so does each column.
Can you always find a matrix with integer entries that has the same sum along each row and column as M?
--Via Kartik Venkat.

## Monthly Archives for March, 2013

## Largest network of cities

Suppose that there are n cities. Suppose that between any 2 cities, there exists a road with probability c/n. Thus each city is on an average connected to c other cities (for large n). This road network forms a random graph. A connected component of this city network is a set of cities such that one can travel from one city in the set to …

**Continue reading the story**"Largest network of cities"## Black and white socks

You have 12 black socks and 12 white socks mixed up in a drawer. You're up very early and it's too dark to tell them apart. What's the smallest number of socks you need to take out (blindly) to be sure of having a matching pair?
via Brain Teasers

## Read More About this Post Here

A farmer challenges an engineer, a physicist, and a mathematician to fence off the largest amount of area using the least amount of fence.
The engineer made his fence in a circle and said it was the most efficient.
The physicist made a long line and said that the length was infinite. Then he said that fencing half of the Earth was the best.
The mathematician laughed at …

**Continue reading the story**"Read More About this Post Here"## Water in the Cup

A man in a restaurant asked a waiter for a juice glass, a dinner plate, water, a match, and a lemon wedge. The man poured enough water onto the plate to cover it.
"If you can get the water on the plate into this glass without touching or moving this plate, I will give you $100," the man said. "You can use the match and lemon …

**Continue reading the story**"Water in the Cup"## How do you empty a bathtub?

During a visit to a mental asylum, a visitor asked the Director what the criteria is that defines if a patient should be institutionalized.
"Well," said the Director, "we fill up a bathtub. Then we offer a teaspoon, a teacup, and a bucket to the patient and ask the patient to empty the bathtub."
Okay, here's your test:
1. Would you use the spoon?
2. Would you use …

**Continue reading the story**"How do you empty a bathtub?"## Monty hall?

You are in a game show. The host shows you 3 doors, leading to 3 different rooms. Two of the rooms contain a goat and one contains a car. Obviously, your goal is to find the door containing the car. You randomly choose one of the doors. Then the host opens an other door to a room that contains a goat. Now he gives you …

**Continue reading the story**"Monty hall?"## Million dollar question

In a particular nation, the currency consists of notes
of four diļ¬erent values: $1, $10, $100, and $1000.
Can one have exactly half a million notes with a
total value of exactly one million dollars?
via Australian Mathematical Society