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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 $\sqrt{2}$?

Dalmatians

Stage: 4 and 5 Challenge Level: Challenge Level:1

Investigate the sequences obtained by starting with any positive 2 digit number $(10a+b)$ and repeatedly using the rule

$10a + b \to 10b -a$

to get the next number in the sequence.
 

NOTES AND BACKGROUND

You can take any number and write it in the form $10a+b$ , that is as a multiple of ten plus a number $b$ between 0 and 9, for example:

$$57 = 10 \times 5 + 7\quad\quad -6 = 10 \times (-1) + 4 \quad\quad 123 = 10\times 12 + 3$$

This iterative procedure is an example of a dynamical system which can be studied in more detail at university; you may read an introduction to this fascinating subject in Whole Number Dynamics 1 . Dynamical systems using decimals can have many strange and interesting properties; they form the foundation of the subject of chaos, which you can read about on the Plus website .