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Indices and Surds
Climbing Powers
Age 16 to 18
Challenge Level 
$2\wedge 3\wedge 4$ 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 $\sqrt{2}$?
Irrational Arithmagons
Age 16 to 18
Challenge Level
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?