## Three cans problem

Posted on: April 19th, 2013 by
You may have heard of the classic puzzle where you are asked to measure 4 liters of water, given a 8-liter can filled with water, and 2 empty cans: One with 5-liter capacity and the other with 3-liter capacity. Here is a generalization of the puzzle: You are given an empty A-liter can, an empty B-liter can and a filled C-liter can, where A and B Continue reading the story "Three cans problem"

## Number of unique distances

Posted on: April 18th, 2013 by
If (n+1)(n+4)/2 distinct points are chosen in n-dimensional space $\mathcal{R}^n$, show that there are at least 3 unique distances. The number of unique distances is the number of elements in the set $\{d_{ij}: \text{ i,j are 2 of the chosen points in }\mathcal{R}^n\}$. Here $d_{ij}$ denotes the euclidean distance between points i and j. Example: An equilateral triangle in 2-d or a regular tetrahedron in Continue reading the story "Number of unique distances"

## Butterfly theorem

Posted on: April 17th, 2013 by
Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD and BC intersect chord PQ at X and Y correspondingly. Then prove that M is the midpoint of XY.

## Ratio of 2 random numbers

Posted on: April 16th, 2013 by
2

Two real numbers X and Y are chosen at random in the interval (0,1). Compute the probability that the closest integer to X/Y is even. Express the answer in the form r + sπ, where r and s are rational numbers. via Putnam exam

## Four knights

Posted on: April 15th, 2013 by
Four knights are positioned on a 3x3 chessboard as shown on the ﬁrst chessboard below. Can they move to the positions shown on the second chessboard? via Washington math

## Find the error in the arithmetic

Posted on: April 14th, 2013 by
What's wrong with the following? $1+2+4+8+\dots \\ =(2-1)(1+2+4+8+\dots)\\ =(2+4+8+16+\dots)-(1+2+4+8+\dots)\\=-1+(2-2)+(4-4)+(8-8)+\dots\\=-1$

## Expected number of loops

Posted on: April 13th, 2013 by
3

Suppose you have n pieces of ropes, each of length 1m. Thus there are 2n ends. In each iteration, you randomly choose 2 ends and tie them together. After 2n-1 iterations, you have a bunch of loops. Find the expected number of loops.

## Elevator wait time

Posted on: April 12th, 2013 by
Suppose you are on the 13-th floor of a 15 floor office building. The elevator is programmed to go up and down continuously as 1,2,3,...14,15,14,13,....2,1,2,.... except that when a user presses a button the elevator stops momentarily for him/her. Assume the time to load/unload passengers is negligible. You complain that whenever an elevator approaches you, it is more likely to go up than to go Continue reading the story "Elevator wait time"

## Birthday problem

Posted on: April 10th, 2013 by