Here are over 30 maths and mechanics problems and my attempts at answers. The more recent ones are at the bottom.

Some questions are my own invention, some were collected by maths lecturers at the University of Manchester and used within an undergraduate problem solving course.  A few are from maths olympiad competitions, as organised by the Mathematical Association of America.  Others are from English A and S Levels maths examination papers dating from the1950s, when questions were quite hard!

In writing my own solutions I have NOT looked at any published answers -- that would be cheating!   It is quite possible, therefore, that my solutions are distinctly inferior or even downright wrong.

 Q21 : A rounding-up process with integers which tends to PI. Choose an integer N, e.g 10. Round 10 up to the next multiple of 10-1=9. You get 2*9=18. Now round 18 up to the next multiple of 9-1=8. You get 3*8=24. Round 24 up to 4*(8-1)=28, then to 5*6=30. Now 6-1=5, and 5 divides 30, so leave it and go on to divide by 5-1=4. The next multiple of 4 is 32. Then round to 3*11=33 and finally ro 17*2=34. Call this F for final. The contention is that as N get large, N^2/F tend to pi.   (Answer) Q22 : A revolving mass on an elastic chord. This is a question in classical dynamics. A mass is revolving at speed on a smooth horizontal table. It is attached to the centre of the circle of motion by a spring or elastic chord. The mass is suddenly given an extra displacement radially outwards then released, causing it to oscillate radially. Describe the subsequent motion. (Answer)
 Q23 : Prove that every non-zero coefficient of the Taylor series of (1-x-x^2) exp(x)about 0 is a rational number whose numerator in lowest terms is either 1 or a prime. A Putnam competition question from 2014. (Answer) Q24 : Define the exponential function and derive its relation to the logarithm from first principles. This is revisiting very old ground; a piece of analysis using series. I use it to demonstrate the binomial theorem with general exponent (not just integer). (Answer) Q25 : A new continued fraction for e = exp(1). The problem is to prove that it does in fact converge to e and not to some other number close to e = 2.71828.... (Answer) Q26 : Curve fitting and musical phrase shapes. Finding mathematical formulae to model the shapes of typical musical phrases..... (Answer)

Q27 : Find a formula for the sum of n consecutive integers j each raised to the power k: Sum j^k. This is an investigation of a classic problem that was solved 300 years ago by Bernoulii, Euler and Maclaurin. I compare numerical integration schemes with the Euler-Maclaurin formula. (Answer)

 Q28 : Why does a mandarin orange (and any other object with a flat bottom) oscillate appear to speed up to a shudder when tilted and released? Have you noticed this amusing motion at the breakfast table? The object is slowing down but, paradoxically, the frequency of rocking side to side gets faster at a rate which depends on how flat the base is. I have modelled the dynamics, studying some simpler cases first, so there are full solutions of several cases including a cylindrical tube with an eccentric weight inside it. There are several comparisons with experiment and generally good agreement. (Answer)

Q29 : Prove that the infinite product over k of 1 + (-1)^(k+1)/(2k-1) equals sqrt 2. (Answer)

Q30 : Write 2/3 as the sum of two or more fractions each of which has 1 as its numerator - so called `Egyptian fractions'. Generalise. (Answer)

Q31 : Find the largest 2D `sofa' object which can be manoeuvred along a corridor and round a sharp 90 degree corner. (Answer, with videos animations 1 and 2)

Q32 : An arbitrary closed plane figure (e.g. polygon, ellipse, etc.) is overlayed by a lattice of points and the number N of points inside the figure is counted. N times the cell size of the lattice is used to measure the area of the figure. What is the likely error? (Answer)

Q33 : A ladder leans against a wall obstructed by a large box. How high can the ladder reach up the wall? This leads to the astroid curve and its intriguing properties. (Answer)

Q34 : Demonstrate how to fit an ellipse inside an arbitrary convex pentagon so that all five sides are tangents to the ellipse. (Answer)