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Monday, 2 June 2014

Sunday Afternoon Maths XV Answers & Extensions

This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

Grand Piano

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?

Cycling Digits

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?

The Mutilated Chessboard

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?

Tennis

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.