Why are puzzle kings so damn pushy?

Once, there was a king who had a perfectly round castle. He had two turrets on this circular castle and they were connected by a single bridge. This King, for no good reason other than to make things difficult for his poor, hapless architects and builders, requested a third turret, anywhere on the circumference of his castle. He also required another bridge connecting that

turret to the other two. A few months later, the kings freudian preoccupations seized ahold of him again and he ordered an even bigger, more

erect turret (making a total of four) to be built anywhere on the circumference of his circular castle and he also demanded that bridges be built to connect that turret to the other three.

To cut a long story short, the king requested another four turrets built (bringing the total amount of turrets to eight). He then asked for bridges to be built so as to connect each turret to every other turret. This meant building a hell of a lot of bridges but to make it a little easier on his builders the king decreed that whenever two bridges crossed eachother they could intersect.

Now, each time a bridge is built, the circular castle is divided up into smaller sections, at least it is from an ariel view.

If we imagine this perfectly circular castle as it would appear from the sky

(It's an open top castle and assume that it has a radius of 200 meters and that the turrets are equally spaced around the castle, how long is each and every bridge?

Quoted text is spoiler space. 'Dave' writes:

If the bridges are straight, then there are:

8 bridges of length 200*sqrt(2-sqrt(2)) [~ 153.07] m 8 bridges of length 200*sqrt(2) [~ 282.84] m 8 bridges of length 200*sqrt(2+sqrt(2)) [~ 369.55] m 4 bridges of length 400 m

by repeated applications of the Pythagorean theorem. If not, then insuf- ficient information was given for the question to be answered. If 'how long is each and every bridge?' was meant to imply that all bridgs are the same length, then it's impossible.

My answer was that each bridge is 200 * SQRT(2).

My confusion came about due to the fact that bridges could intersect - this implied to me that not all turrets needed to be connected to all destinations and further that 'how long is each and every bridge?' implied all bridges must be the same length.

When I submitted my answer to Dave I didn't have space to explain it - I was sending it via a mobile (cell) phone whilst away on Business.

I got my answer as follows:

I began by labelling the bridges as follows (clockwise): A1, B1, A2, B2, A3, B3, A4, B4.

Then joining 'A1, A2, A3, and A4' to form a square and 'B1, B2, B3 and B4' to form another square. Each bridge was also the same length. Travel was now possible from any turret to any other (although your journey might mean you had to travel between intermediate turrets).

Using Pythagoras it was simply:

(A1 to A2)^2 = 200^2 + 200^2 = 2*200^2

(A1 to A2) = 200 * SQRT(2)

I think the wording of the question was lacking - but nonetheless it was a fun puzzle. Good job, Dave!

all intersecting at the center. Furthermore, none of the sets of 8 bridges needs to be completed. It suffices to built 7 sides of the octagon, 5 bridges of the intersecting squares (3 of one square, 2 opposite sides of the other), or 4 bridges of the 8-pointed star (2 parallel pairs each at right angles to the other). (Of course, this does not change the length of the bridges.)