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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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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.