**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.

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## Comments

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

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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.

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