## Posts Tagged 'Recreational Maths'

### Bloom’s attempt to Square the Circle

The quadrature of the circle is one of the great problems posed by the ancient Greeks. This “squaring of the circle” was also an issue of particular interest to Leopold Bloom, the central character in James Joyce’s novel Ulysses, whom we celebrate today, Bloomsday, 16 June 2016 [see TM093, or search for “thatsmaths” at irishtimes.com].

Joyce’s Tower, Sandycove, Co Dublin.

The challenge is to construct a square with area equal to that of a given circle using only the methods of classical geometry. Thus, only a ruler and compass may be used in the construction and the process must terminate in a finite number of steps.

### 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.