Category: Book Notes

Categories
Algrebra Math Problems

Find five rectangles that tile a square

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. 

Find five rectangles with sides of length (1..10) that tile a square
Find five rectangles with sides of length (1..10) that tile a square

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.

Categories
Logic Math

Three people crossing a bridge with a bicycle

Problem #96 from Mathproblems.info (at http://mathproblems.info/group5.html) was one I have seen before as a kid :

“Three people (A, B, and C) need to cross a bridge. A can cross the bridge in 10 minutes, B can cross in 5 minutes, and C can cross in 2 minutes. There is also a bicycle available and any person can cross the bridge in 1 minute with the bicycle. What is the shortest time that all men can get across the bridge? Each man travels at his own constant rate.”

When I saw this problem as a kid there was the caveat that only one person could cross the bridge at any one time. For that case the fastest solution is to have 

(1) A crosses with the bicycle. (2) C crosses. (3) C crosses back with the bicycle. (4) B crosses with the bicycle. (5) C crosses. This makes for a total crossing time of (1+2+1+1+2) or 7 mins. 

The two new twist that the Mathproblems.info guys put on the problem was to remove the restriction that only one person could cross at a time, and allowing the bike to be left anywhere on the bridge. I am still working on this new version of the problem.

Categories
Algrebra Math

Problem from mathematical quickies – #264 – More simultaneous equations

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.

Solve the four equations with four unknowns
Solve the four equations with four unknowns

…. and it keeps going, and going, and going…. I am really missing something here. This will give me the right solution but *ick*. 

Solution continued
Solution continued
Categories
Algrebra Math

Problem from mathematical quickies – #94 – “Simultaneous Equations”

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. 

Solve the system of simultaneous equations
Solve the system of simultaneous equations

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. 

simultaneous_equation_mathematical_quickies_numb94_answer 

Categories
Algrebra Math

Problem from mathematical quickies – #136 – “A Consistent Sum”

 

Solve for K
Solve for K
Categories
Calculus Math

Area of a right triangle

Ok, one of the first equations learned in school is that the area of a right triangle is half the base times the height. Obvious since the base times height gives the area of a rectangle and that is twice the are of the triangle in question. Using calculus though you can say the same thing as:

Area of a right triangle via integration
Area of a right triangle via integration
Categories
Geometry Math

Prove the Pythagorean theorem

The pythagorean theorem states that for a right triangle the sum of the squares of the opposite and adjacent sides is equal to the square of the hypotenuse. I used a graphical proof to prove this theorem.

Graphical proof for the pythagorean theorem
Graphical proof for the pythagorean theorem
Categories
Math

Human skills are apparently perishable

I realized that there are things that are important to me other then just martial arts that require constant training if I do not want to loose skills I worked so hard to acquire.  One of those things was the ability to describe aspects of the world around me mathematically. I keep reaching back for things I know I once could do with little effort and finding they were either much harder than I remembered – or worse yet now beyond my ability. 

 So – I am going to try re-working through the basics and working up my skill level again. Hopefully to where I can get better then I once was.