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Showing posts with label steps. Show all posts
Showing posts with label steps. Show all posts

Tuesday, March 1, 2011

Coins in equilateral lattice

$n(n+1)/2$ coins are placed on an equilateral lattice with side $n$, such that all but one coin are showing heads.

At each step, one is allowed to choose two adjacent coins $A$ and $B$, and then flip all coins on the line $AB$ (and its extension).

Characterize all starting configuration such that it's always possible to get all tails.

Saturday, September 25, 2010

Completable Paintings

An $n \times n$ checker board is painted all white except $m$ cells that are painted black. At each step, one is allowed to choose a rectangle that has three black corners, and paint the fourth corner black. A configuration is called completable if one can paint the entire board black using a sequence of steps described above.

First Question:
Find the smallest $m$ such that any configuration with $m$ black cells are always completable.

Second Question:
Find the smallest $m$ such that there is a completable configuration with $m$ black cells.

Wednesday, December 16, 2009

Stones and Marbles

There are 3 boxes, and 2010 stones and 2010 marbles are put arbitrarily inside those three boxes.

At each step, Bob is allowed to either:

1. take a stone from one box, a marble from another box, and put them on the third box

2. add or subtract all boxes by the same number of stones

3. add or subtract all boxes by the same number of marbles

Prove or disprove, that by repeating these steps, Bob can empty all three boxes.