Cambridge Pieces

1.  Prove that the shoe may be represented by an equation of the fifth degree.  Find the equation to a man blacking a shoe:  (1) in rectangular co-ordinates; (2) in polar co-ordinates.

2.  A had 500 shoes to black every day, but being unwell for two days he had to hire a substitute, and paid him a third of the wages per shoe which he himself received.  Had A been ill two days longer there would have been the devil to pay; as it was he actually paid the sum of the geometrical series found by taking the first n letters of the substitute’s name.  How much did A pay the substitute? (Answer, 13s. 6d.)

3.  Prove that the scraping-knife should never be a secant, and the brush always a tangent to a shoe.

4.  Can you distinguish between meum and tuum?  Prove that their values vary inversely as the propinquity of the owners.

5.  How often should a shoe-black ask his master for beer notes?  Interpret a negative result.