This is more along the lines of proving what I know. Kind of struck me that this is a visual representation of something you see in martial arts all the time. You can use techniques you know to force your way through to a valid but un-elegant solution or you can do it elegantly. Both are technically correct but it is pretty obvious which way you should be solving the problem. *ugh* I think I am not up to reviewing though freshman year of high school. Who would have thought I forgot so much.
This problem is representative of a class of problems that I can solve – but only thought brute force. In some cases – like this one – I can winnow my guessing spaces though the use of logic and algebra – but it is still a brute force attack. What I don’t see is how to make a non-brute force algorithmic attack on this problem.
This problem was phrased with five rectangles tiled to cover a square – with a much larger number of squares I could not easily do it by hand and if I was to write a program to do it that attack would also be brute force.
Ugly as it is – here is how I solved this problem.
I just looked at the Mathproblems.info site where I saw this problem and they show three more solutions. One more with side length of 13 and two more with side length of 11. I was less thorough with my search of the 11 options and stopped on 13 after I found the first solution but the techniques seem to be the same for all solutions. Still brute force-ish though. There must be a better way to formalize solving this type of problem.
Ok, so I just ground through this solution long hand – which sucks because I am fairly sure there was some slick trick I was supposed to have seen here. All the problems in this book are searching for an elegant solution and this one is anything but. I should ask Konrad what I missed when he gets back.
…. and it keeps going, and going, and going…. I am really missing something here. This will give me the right solution but *ick*.
The problem is to solve an equation of 5 variables with 5 equations. Making it into a matrix the left side looks like an identity matrix. The first time I solved this I did it long hand – which took forever – then I took another look and saw what they person framing the problem was thinking. Which is cool since this trick should make it easier to attack matrix reduction for some annoying cases.
Sum all of the rows to create the new vector S. Then since all the columns of S have the same value it can be normalized so all the values of S are unity. Then by subtracting S from each of the rows (a..e) we create a new matrix M. Multiply M by negative 1 to keep the values at unit as opposed to negative unity and the solution is done. Most important it is done without the page of long hand reduction.
