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# Puzzles

## Archive

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## Ninety nine

In a 'ninety nine' shop, all items cost a number of pounds and 99 pence. Susanna spent £65.76. How many items did she buy?
Tags: numbers, money

## Triangle numbers

Let $$T_n$$ be the $$n^\mathrm{th}$$ triangle number. Find $$n$$ such that: $$T_n+T_{n+1}+T_{n+2}+T_{n+3}=T_{n+4}+T_{n+5}$$

## Ellipses

A piece of string 10cm long is tied to two pins 6cm apart. The string is used to draw an ellipse. The pins are then moved 2cm further apart and a second ellipse is drawn. Which ellipse has the larger area?

## Folding tube maps

Back in 2012, I posted instructions for folding a tetrahedron from tube maps. When tube maps are used, the sides of the tetrahedron are not quite equal. What ratio would the rectangular maps need to be in to give a regular tetrahedron?

## x to the power of x again

Let $$y=x^{x^{x^{x^{...}}}}$$ [$$x$$ to the power of ($$x$$ to the power of ($$x$$ to the power of ($$x$$ to the power of ...))) with an infinite number of $$x$$s]. What is $$\frac{dy}{dx}$$?

## Odd sums

What is $$\frac{1+3}{5+7}$$?
What is $$\frac{1+3+5}{7+9+11}$$?
What is $$\frac{1+3+5+7}{9+11+13+15}$$?
What is $$\frac{1+3+5+7+9}{11+13+15+17+19}$$?
What is $$\frac{\mathrm{sum\ of\ the\ first\ }n\mathrm{\ odd\ numbers}}{\mathrm{sum\ of\ the\ next\ }n\mathrm{\ odd\ numbers}}$$?

## Two lines

Let A and B be two straight lines such that the gradient of A is the y-intercept of B and the y-intercept of A is the gradient of B (the gradient and y-intercept of A are not the same). What are the co-ordinates of the point where the lines meet?

## x to the power of x

If $$x^{x^{x^{x^{...}}}}$$ [$$x$$ to the power of ($$x$$ to the power of ($$x$$ to the power of ($$x$$ to the power of ...))) with an infinite number of $$x$$s] is equal to 2, what is the value of $$x$$?