Who to play first

Posted on: July 12th, 2013 by
You alternately play tennis against your beginner friend and a pro like Federer. You play a total of 5 games. You may either start playing with your friend, or you may start with Federer as an opponent. Your goal is to win 2 consecutive games. Who will you start playing with first? What if you want to win 3 consecutive games? What if the total number Continue reading the story "Who to play first"

Monochromatic rectangle in a colored plane

Posted on: July 11th, 2013 by
Each point in the plane is arbitrarily assigned one among a finite number of colors. Show that there exists a rectangle whose four corner vertices are of the same color. - via a student of Paul Cuff

Target 24

Posted on: July 9th, 2013 by
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Using numbers 1, 3, 4 and 6, obtain the number 24. All numbers must be used exactly once. You are allowed to use addition, subtraction, multiplication and division. Any number of parenthesis can be used. Example: 1+3*4+6=19.

Bridge crossing puzzle

Posted on: July 8th, 2013 by
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4 persons need to cross a bridge: An athlete, a student, an old man and an old woman. The crossing times for the people are 5 min, 10 min, 20 min, 25 min respectively. The bridge can support at most 2 people at a time. It is dark and they have only 1 torch amongst the four of them. They have to use the torch to cross Continue reading the story "Bridge crossing puzzle"

Prime power divisibility

Posted on: July 7th, 2013 by
Show that $p^4-1$ is divisible by 240 for p>7 and prime. It's very easy if you know congruences. - via Moscow mathematical olympiad

Another coin puzzle

Posted on: July 6th, 2013 by
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There are 4 coins, of which some may be forged. An original coin weighs 10 grams while a forged coin weighs only 9 grams. You have a spring balance with a single pan. How many weighings are needed to determine for sure which coins are forged? via Moscow mathematical olympiad

Posted on: July 5th, 2013 by
The diagonals of a convex quadrilateral divide it into four areas. Each area is an integer. Prove that the product of these 4 integers cannot end with the digits 1988. Via Moscow mathematical olympiad. P.S: I don't know the solution to this puzzle, yet.

Smallest number

Posted on: July 4th, 2013 by
Here is a problem that can be formally solved using Chinese remainder theorem: Find the smallest positive integer that: 1) Leaves a remainder 1 when divided by 2, 2) Leaves a remainder 2 when divided by 3, 3) Leaves a remainder 3 when divided by 4 and 4) Leaves a remainder 4 when divided by 5. There is a simple trick that helps to solve the problem faster! Problem source: SPOILER WARNING ... ... ... ... ... ... ... Aptitude Continue reading the story "Smallest number"

Man, Fox, Rabbit, Grass and Dog

Posted on: July 3rd, 2013 by