Combinatorics Exam Paper

Junior Exam J4

Each question is worth 7 marks.

Time: 4 hours

No books, notes or calculators allowed.

Note: You must prove your answer.

Q1

Finn\(^{1}\) has made himself a \(2014 \times 2014\) solitaire chessboard. He also has 4 slippery rooks. A slippery rook moves as follows. It slips along its row or column until it is stopped by the edge of the board or by another piece. 3 of the slippery rooks are black and 1 is white. The 4 slippery rooks have been randomly placed the chessboard.

Prove that Finn will always find a sequence of moves to have the white rook be on a certain square.

Q2

Daniel has tiled an \(m \times n\) rectangular bathroom with blue and white tiles. He has tiled it such that the 4 squares defined by the intersections of 2 rows and 2 columns (they need not be adjacent rows or columns) are never all the same colour.

Find all \(m\) and \(n\) for which this can occur.

Q3

Calvin wants to pick a set of numbers from the set \({1, 2, 3, \ldots , 100}\) such that if \(x\) is in the set, \(3x\) is not.

(a) Find the maximum number of elements in Calvin's set.

(b) Find the number of sets for which the maximum number of elements are in the set.

Q4

Satvik and Krishna are playing a game on a strip \(10^{1000}\) squares long and 1 square wide. Every turn, Satvik colours 2 squares blue and then, Krishna colours a chain of blue squares (length of 1 square at least) black. The aim for Satvik is to colour a chain of 10 squares blue without Krishna being able to colour it black. Krishna wins if Satvik cannot do this.

Who has the winning strategy?

Q5

Sharky is tiling a \(13 \times 13\) room with 2 different types of tiles, a \(2 \times 2\) square tile and an 'L' shaped tile made by removing a corner square from a \(2 \times 2\) square tile.

Assuming that he does not cut any tiles, what is the minimum number of 'L' tiles required to tile the room.

1: Names have been changed for all problems.

Note by Sharky Kesa
3 years, 6 months ago

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I believe that in question \(1\), \(2\) black rooks and \(1\) white rook suffices, actually.

Daniel Liu - 3 years, 6 months ago

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Yes, that is all that's necessary. One of the senior questions for this was to find the minimum number of rooks needed to satisfy the condition. The answer was 3.

Sharky Kesa - 3 years, 6 months ago

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