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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 $\sqrt{2}$?

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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?

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Approximating Pi

By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?

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.