Argh. Well, I finally made it into work, sweating poisons through all my pores all the way. I'm not as young as I used to be.

Today's "Friday puzzle" is a nice one. Some simple stuff, and some not-so-simple stuff. Plus some advanced mathematics at the end should anyone feel brave.

# STRAIGHT-LINE AND COMPASS CONSTRUCTIONS.

These are something that entertained the Greeks (well, the nerdy Greeks anyway) back when Aristotle was a lad. Hmm, wasn't he Phoenician? Anyway, those Mediterranean lovers used to while away their siestas with junk like this.

You have: a straight edge (like a ruler, but without any distances marked on it) and a set of compasses at your disposal.

The rules of R+C construction are as follows: you can draw straight lines, and set the compasses to any radius you like (obviously you can't measure the radius, because you've no markings on your ruler); in particular, if you've managed to mark two points on the (infinite) plane that you're drawing on (normally points are defined when arcs and lines intersect) then you can set the compass to exactly the distance between those points.

Obviously, this being a Greek philosophical pastime, it's assumed that you have an infinitessimally thin pencil at your disposal, and absolute precision.

This is harder to describe than it is to demonstrate; I'm sure you all remember doing something like this when you were a kid at school.

Anyway, here are some problems. In each case, you get told what points, circles and lines already have been drawn, and are asked to come up with a scheme (if possible) to construct the required figures. As always, for convenience, you can use the answers to previous questions as 'primitive operations' in solving latter problems.

## Problems:

• 0. Draw an equilateral triangle. Solution
• 1. Given a line segment, find the mid-point of that segment. Solution
• 2. Given a straight line and a point on that line, construct a line at right angles to the original line passing through that point. Solution
• 3. Given two lines that intersect at an angle, draw the line that cuts the angle of intersection precisely in half. Solution
• 4. Given a triangle, draw a circle inside it which touches all three sides. Solution
• 5. Given a triangle, draw a circle outside it that touches all three corners. Solution
• 6. Given a circle, find the construction which will enable you to locate its centre. Solution
• 7. Given a line, and a point not on that line, construct a circle whose centre is that point and which touches (tangentially) the line. Solution

OK, the above problems are good starting places. Now:

Suppose you're shown a line segment and told that it is (exactly) one inch long.

• 8. Show how you can construct a line segment 2 inches long, 3, 4, 5, 6, and so on. Solution
• 9. Show how you can construct a line segment 1/2 inch long, 1/4, 1/8, etc. Solution
• 10. Show how you can construct a line segment sqrt(2) inches long (hint: pythagoras). Solution
• 11. Show how you can Construct a line segment sqrt(n) inches long, for any integer n > 2, as well. Solution
I've also provided an added bonus showing how to construct arbitrary square roots.
• 12. Given two additional line segments, of lengths x inches and y inches (you don't necessarily know x and y; they can be arbitrary lengths), can you construct a line segment of length (x times y)? Solution
• 13. Given a line segment of length z, show how you can construct a line segment of length (1/z). Solution
• 14. Show how you can construct a line of length m/n, for any integers m and n. Solution

And for a final stint (you have to decide if these are possible or not)

• 15. Construct a regular 12-sided polygon. Solution
• 16. In honour of this being the 17th question: construct a regular 17-sided polygon. Solution