The function defined by an `infinite tower’ of exponents,
is called the Power Tower function. We consider the sequence of successive approximations to this function:
As , the sequence converges for . This result was first proved by Euler. For an earlier post on the power tower, click here.
Continue reading ‘The Imaginary Power Tower: Part I’
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]
Continue reading ‘How many Christmas Gifts?’
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?
Have a think about this before reading on. There is more than one solution.
Continue reading ‘Eccentric Pizza Slices’
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.
Continue reading ‘Golden Moments’
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’
The powerful and versatile computational software program called Mathematica is widely used in science, engineering and mathematics. There is a related system called Wolfram Alpha, a computational knowledge engine, that can do Mathematica calculations and that runs on an iPad.
Yogi Bear Curve. The Mathematica command to generate this is given below.
Continue reading ‘Cartoon Curves’
Have you ever tried to build a high stack of coins? In theory it’s fine: as long as the centre of mass of the coins above each level remains over the next coin, the stack should stand. But as the height grows, it becomes increasingly trickier to avoid collapse.
Ten chocolate gold grain biscuits, with a hangover of about one diameter.
In theory it is possible to achieve an arbitrarily large hangover — most students find this out for themselves! In practice, at more than about one coin diameter it starts to become difficult to maintain balance.
Continue reading ‘Biscuits, Books, Coins and Cards: Massive Hangovers’