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# Puzzles

## Archive

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## Placing Plates

Two players take turns placing identical plates on a square table. The player who is first to be unable to place a plate loses. Which player wins?

## 20 December

Earlier this year, I wrote a blog post about different ways to prove Pythagoras' theorem. Today's puzzle uses Pythagoras' theorem.
Start with a line of length 2. Draw a line of length 17 perpendicular to it. Connect the ends to make a right-angled triangle. The length of the hypotenuse of this triangle will be a non-integer.
Draw a line of length 17 perpendicular to the hypotenuse and make another right-angled triangle. Again the new hypotenuse will have a non-integer length. Repeat this until you get a hypotenuse of integer length. What is the length of this hypotenuse?

## 17 December

The number of degrees in one internal angle of a regular polygon with 360 sides.

## 3 December

What is the volume of the smallest cube inside which a rectangular-based pyramid of volume 266 will fit?

## 2 December

What is the maximum number of lines that can be formed by the intersection of 30 planes?

## Cross Diagonal Cover Problem

Draw with an $$m\times n$$ rectangle, split into unit squares. Starting in the top left corner, move at 45° across the rectangle. When you reach the side, bounce off. Continue until you reach another corner of the rectangle:
How many squares will be coloured in when the process ends?

## Two Semicircles

The diagram shows two semicircles.
$$CD$$ is a chord of the larger circle and is parallel to $$AB$$. The length of $$CD$$ is 8m. What is the area of the shaded region (in terms of $$\pi$$)?

## 21 December

This year, I posted instructions for making a dodecahedron and a stellated rhombicuboctahedron.
To get today's number, multiply the number of modules needed to make a dodecahedron by half the number of tube maps used to make a stellated rhombicuboctahedron.
© Matthew Scroggs 2017