## Posts Tagged 'Recreational Maths'

### Mathematics Everywhere (in Blackrock Station)

Mathematics is everywhere. We are often unaware of it but, when we observe our environment consciously, we can see mathematical structures all around us.

This footbridge is a cornucopia of mathematical forms.

### The Imaginary Power Tower: Part II

This is a continuation of last week’s post: LINK

The complex power tower is defined by an infinite tower’ of exponents:

$\displaystyle Z(z) = {z^{z^{z^{.^{.^{.}}}}}} \,.$

The sequence of successive approximations to this function is

$z_0 = 1 \qquad z_{1} = z \qquad z_{2} = z^{z} \qquad \dots \qquad z_{n+1} = z^{z_n} \qquad \dots$

If the sequence ${\{z_n(z)\}}$ converges it is easy to solve numerically for a given ${z }$.

In Part I we described an attempt to fit a logarithmic spiral to the sequence ${\{z_n(i)\}}$. While the points of the sequence were close to such a curve they did not lie exactly upon it. Therefore, we now examine the asymptotic behaviour of the sequence for large ${n}$.

### The Imaginary Power Tower: Part I

The function defined by an infinite tower’ of exponents,

$\displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}}$

is called the Power Tower function. We consider the sequence of successive approximations to this function:

$\displaystyle y_0 = 1 \qquad y_1 = x \qquad \dots \qquad y_{n+1} = x^{y_n} \,.$

As ${n\rightarrow\infty}$, the sequence ${\{y_n\}}$ converges for ${e^{-e}. This result was first proved by Euler. For an earlier post on the power tower, click here.

We all know the festive carol The Twelve Days of Christmas. Each day, “my true love” receives an increasing number of gifts. On the first day there is one gift, a partridge in a pear tree. On the second, two turtle doves and another partridge, making three. There are six gifts on the third day, ten on the fourth, fifteen on the fifth, and so on.

Here is a Christmas puzzle: what is the total number of gifts over the twelve days? [TM083, or search for “thatsmaths” at irishtimes.com]

### Eccentric Pizza Slices

Suppose six friends visit a pizzeria and have enough cash for just one big pizza. They need to divide it fairly into six equal pieces. That is simple: cut the pizza in the usual way into six equal sectors.

But suppose there is meat in the centre of the pizza and some of the friends are vegetarians. How can we cut the pizza into slices of identical shape and size, some of them not including the central region?

A pizza with various toppings. Image: Pizza Masetti Craiova, Romania (Flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)%5D, via Wikimedia Commons.

### Golden Moments

Suppose a circle is divided by two radii and the two arcs a and b are in the golden ratio:

b / a = ( a + b ) / b = φ ≈ 1.618

Then the smaller angle formed by the radii is called the golden angle. It is equal to about 137.5° or 2.4 radians. We will denote the golden angle by γ. Its exact value, as a fraction of a complete circle, is ( 3 – √5 ) / 2 ≈ 0.382 cycles.

### Triangular Numbers: EYPHKA

The maths teacher was at his wits’ end. To get some respite, he set the class a task:

Add up the first one hundred numbers.

That should keep them busy for a while”, he thought. Almost at once, a boy raised his hand and called out the answer. The boy was Carl Friedrich Gauss, later dubbed the Prince of Mathematicians. Continue reading ‘Triangular Numbers: EYPHKA’