### Climbing Powers

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

### Little and Large

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

### Ruler

The interval 0 - 1 is marked into halves, quarters, eighths ... etc. Vertical lines are drawn at these points, heights depending on positions. What happens as this process goes on indefinitely?

# How Does Your Function Grow?

##### Age 16 to 18 Challenge Level:

Four enthusiastic mathematicians are asked to think of a function involving the number 100. The challenge is to think of the function which is biggest for big values of n
• Archimedes chooses a logarithm function $$A(n) = \log(100n)$$
• Bernoulli decides to take 100th powers $$B(n) = n^{100}$$
• Copernicus takes powers of 100 $$C(n) = 100^n$$
• and, finally, de Moivre, who likes to be different, chooses the factorial function which he claims will be quite big enough without any reference to 100 at all $$D(n) = n\times (n-1)\times (n-2)\times \dots \times 2\times 1$$

Which function is biggest for large values of n? Can you determine a value beyond which you know this function will be biggest?

[Extension: To find the exact switch-over value will be difficult and will require the clever use of a spreadsheet or computer.]

What could you say if the 100s were replaced by a million? billions? Create a convincing argument to prove your results to the mathematicians.