Sunday Afternoon Maths VIII Answers & Extensions

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

Two

a+(a+A)^-1 = a + A = 2.

So, a+2^-1 = 2

So, a = ³/₂

By the same method, b = 2-√2, c = -2 and d = 2-2^k

Extension

For which values of k does this converge?

Reverse Bases Again

445 in base 10 is equal to 544 in base 7

Extension

Find another pair of bases A and B so that there exist digits d, e and f such that def in base A is equal to fed in base B?

Adding bases

A=8, B=9 and C=7.

Extension

Find bases A, B and C such that 123_A+456_B=789_C

Complex Squares

Any complex number can be written in the form z=re^iθ.

This gives that z²=r²e^2iθ, which will have positive real and complex parts when 0+2πn < 2θ < ^π/₂+2πn.

This will occur when 0 < θ < ^π/₄ and π < θ < ^5π/₄.

This can be represented on an Argand diagram:

A complex number z falls in these regions when |Re(z)|>|Im(z)| and sign(Re(z))=sign(Im(z))

Extension

For which complex numbers, z, are Re(z³) and Re(z³) both positive?

Rebounds

If mirrors were placed along the walls of the rectangle, the ball would appear to travel in a straight line across a grid of rectangles:

Viewed this way, the ball will still stop once it reaches a corner:

For an n by m (in above example: n = 4, m = 3) rectangle, this will occur once the ball has travelled through lcm(n, m) squares.

Let a =	lcm(n, m)
	n

Let b =	lcm(n, m)
	m

On its way to the corner, the ball will bounce b – 1 times off the top and bottom and a – 1 times off the sides. It can be seen that if a – 1 is even, then the ball will end in one of the corners on the right hand side. The complete results can be seen in the following Carroll diagram:

	a – 1 odd	a – 1 even
b – 1 odd	Top left	Top right
b – 1 even	Bottom left	Bottom right

It can be shown that the ball will never finish in the top left (where it started) as this would require it to travel through the bottom right first. Therefore the following holds:

	a even	a odd
b even		Top right
b odd	Bottom left	Bottom right

Extension

For which sizes of rectangle will the path of the ball make the same pattern?