# Sunday Afternoon Maths XV

## Tennis

What is the minimum number of times a player has to hit the ball in a set of tennis and win a standard set (the set is not ended by injury, disqualification, etc.)?

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A set can be won by hitting the ball 12 times: three service games can be won with four aces, the opponent loses their service games with four double faults.

## The mutilated chessboard

You are given a chessboard where two diagonally opposite corners have been removed and a large bag of dominoes of such size that they exactly cover two adjacent squares on the chessboard.

Is it possible to place 31 dominoes on the chessboard so that all the squares are covered? If yes, how? If no, why not?

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On a normal chessboard there are 32 white and 32 black squares. After removing two diagonally opposite corners there will be 30 white and 32 black or 32 black and 30 white squares.

Each domino covers one white square and one black square. Therefore a combination of dominoes will always cover the same number of black and white squares, so it is not possible to cover all the squares.

#### Extension

Is it possible to do a

knight's tour on the mutilated chessboard?

## Cycling digits

I have in mind a number which when you remove the units digit and place it at the front, gives the same result as multiplying the original number by 2. Am I telling the truth?

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A few examples are:

$$105263157894736842 \times 2 = 210526315789473684$$
$$210526315789473684 \times 2 = 421052631578947368$$
$$421052631578947368 \times 2 = 842105263157894736$$

#### Extension

I have in mind a number which when you remove the units digit and place it at the front, gives the same result as multiplying the original number by 3. Am I telling the truth?

## Grand piano

Jack and Jill are moving into a new flat and their grand piano presents a potential problem. Fortunately, it will just pass round the corridor without being tipped or disassembled.

Given that its area, looking down from above, is the largest possible which can be passed around the corner, what is the ratio of its length to its width?

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When halfway around the corner, the grand piano will look like this:

The problem then is finding the largest area rectangle which fits in the highlighted triangle: an isosceles triangle where the base is twice the height. Let the base be 2 and the height be 1.

If the height of the rectangle is \(h\), then its width is \(2(1-h)\). Therefore its area is \(2h-2h^2\). By differentiation, it can be seen that this is maximum when \(h=\frac{1}{2}\), which means that ratio of the rectangle's length to its width is **2:1**.

#### Extension

If the corner was not a 90° angle, then what is the largest area rectangle which could fit round it?