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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 $\sqrt{2}$?

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

# Weekly Challenge 11: Unit Interval

##### Stage: 5 Short Challenge Level:

Although this solution is quite short to write down, you do need to keep a clear head to find it! As you read this proof, think carefully about each step to be sure that you follow it.

Suppose that $0< x< 1$ and $0< y< 1$ for real numbers $x, y$.

Since $0< y< 1$ we must also have $0< 1-y< 1$. Similarly, $0< 1-x< 1$.

Since the product of two real numbers between $0$ and $1$ must also be between $0$ and $1$ we have

$$0< (1-x)(1-y)=1-x-y+xy< 1$$

Looking at the left hand side of this inequality we have

$$0< 1-x-y+xy$$
Rearranging gives the desired result.

Note: In this proof we assume standard properties of real numbers, such as "the product of two real numbers between $0$ and $1$ must also be between $0$ and $1$". You might wish to read this proof carefully and try to note where assumptions such as these have been made.