The Crazy Bugs - 07/02/2013

[showmyiq]

The spring is coming and so the bugs are ready for action! Please inspect the diagram below. The bugs are situated on corners of a square 10 inches on a side. Believe it or not the bugs on corners A and C are male, and the bugs on corners B and D are female. Simultaneously all the bugs are starting to move in a pattern – male to female. So the A crawls directly toward B, B toward C, C toward D and D toward A. They crawl at the same constant rate and someone will correctly guess that they will describe logarithmic spirals which meet at the center of the square.
How far does each bug travel before they meet? The problem can be solved without Calculus!

Let the abbles hit you!

Let-the-apples-hit-you.gif (12.22 KiB) Viewed 320 times

[showmyiq]

A solution via facebook was proposed and I want to share it here:

Димитър Марков It is just a GUESS based on the fact that each insect will move towards its chosen one. And since the chosen one will not stand still the chaser will move not only ahead but a little bit to the "right" in each moment. It seems to me reasonable the trajectory to be a circle with center the middle of the side AB (for instance) and radius 5. To the center each insect will travel a quarter of the length of the circumference. I'm trying to proove this. Got apples?

I replied with few hints I want to share here, so the race to be fair!

Your calculations are absolutely right, but having such a scheme as you described means that the bug A path will be generated by two directions only – up and right (I mentioned A, but the case for the other three bugs is the same, just using two other directions).

Are you sure that will be the case? Are you sure the curve will behave like that? I can give you a hint – the curve will not behave like that. The curve will use all the four directions. No matter what curves we are going to draw, using only 2 or 3 directions to generate their path – will just approximate the solution to a specific point when the rest of the directions should participate. If you start from point A of the square moving towards the center of the square – using for example only Up and Right directions would be a wrong approach. Furthermore the private case you have stated (a quarter from a circle) is just only one such possible drawing from the infinite set of possible cases.

My last hint will be that the answer is greater than 2.5 * pi.
The answer is an integer number and as such pi is not involved.
Excellent thought so far, just you need to dig a little bit deeper!
Thanks for your answer!
Let’s apple ourselves!

[ljsense]

I think the answer is 10.

As the bugs walk, the distance between them will shrink, but they will continue to form a square. The square's sides (the distance between the bugs) continues to shrink as the square rotates. Eventually the square's sides equal 0 and the bugs touch, therefore each bug traveled 10 units?

[showmyiq]

Your conclusion is absolutely right!
I will attach later today a graphical presentation of the bugs movement.

Thanks for your answer, after several minutes I will update your solution to the site and attach your nickname as the one who solved it.

In fact I found your solution really brilliant!

Puzzle is closed.

The puzzle is archived here: http://www.showmyiq.com/2013/February/0 ... uzzle.html

[showmyiq]

The graphical presentation of the bugs is shown on the diagram below.

Bugs

Bugs Puzzle.gif (3.26 KiB) Viewed 123 times

Do you wander how you can draw such graphics by yourself?
Join our new project “The missing manual for PostScript” and feel free to ask if you have questions. The project is still in development and after 3 or 4 chapters (which are still in development) we are going to write exactly the same figure representing the crazy bugs puzzle.


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