7
Part of 2010 Tournament Of Towns
Problems(4)
Finding maximum number of fleas on a 10x10 chessboard.
Source:
2/8/2011
Several fleas sit on the squares of a chessboard (at most one fea per square). Every minute, all fleas simultaneously jump to adjacent squares. Each fea begins jumping in one of four directions (up, down, left, right), and keeps jumping in this direction while it is possible; otherwise, it reverses direction on the opposite. It happened that during one hour, no two fleas ever occupied the same square. Find the maximal possible number of fleas on the board.
geometrygeometric transformationreflectioncombinatorics unsolvedcombinatorics
A problem with a natural number
Source: Quite hard to describe
3/24/2010
A multi-digit number is written on the blackboard. Susan puts in a number of plus signs between some pairs of adjacent digits. The addition is performed and the process is repeated with the sum. Prove that regardless of what number was initially on the blackboard, Susan can always obtain a single-digit number in at most ten steps.
searchinvariantfunctionalgorithmnumber theory unsolvednumber theory
Seating in a round table for each day...
Source:
2/13/2011
Merlin summons the knights of Camelot for a conference. Each day, he assigns them to the seats at the Round Table. From the second day on, any two neighbours may interchange their seats if they were not neighbours on the first day. The knights try to sit in some cyclic order which has already occurred before on an earlier day. If they succeed, then the conference comes to an end when the day is over. What is the maximum number of days for which Merlin can guarantee that the conference will last?
combinatorics unsolvedcombinatorics
Prove that diagonal of square bisects total area.
Source:
2/19/2011
A square is divided into congruent rectangles with sides of integer lengths. A rectangle is important if it has at least one point in common with a given diagonal of the square. Prove that this diagonal bisects the total area of the important rectangles
geometryrectanglerotationinductioncombinatorics unsolvedcombinatorics