The reason you never want to argue with a mathematician is simple – the homework. I awoke to find this waiting for me on the couch where I do my morning reading.
Talking about metrics with my roommate quickly revealed a hole in how I thought about metrics, which was simply put wrong. I hadn’t realized that I tend to project higher dimensional problems into Euclidian spaces – and that included metrics. Still homework waiting for me in the morning was cold, funny, but ice cold.
This problem kind of bugs me. Here is the problem and my first step:
So I like my first step – but then my way of solving {a(a-1)(a-2)(a-3) = 120} is wonky. I want to expand it and use the quadratic equation – which is not slick enough to be what they are after. I am not seeing synthetic division, or any usual suspects – so I think I am missing something obvious.
Rick, someone I work with, looked at the problem and immediately said you could tell that the solution needed to be divisible by 5, which happens to be a valid solution here. However I don’t believe that is always the case though.
My way to get the solution shows a solution of {-2,5}. Work shown here.
This is another example of a substitution problem. Normally I would have just worked it out long hand, expanding thing and solving. That’s a lot of work and frankly kind of a foolish way to attack the problem looking at it now. Since I started looking for substitutions first – the problem collapses to simple in one substitution. Unfortunately its still fairly un-elegant – so I don’t think this is the solution they book is looking for.
The problem is stated as: “Solve: (6x+28)^1/3 – (6x-28)^1/3 = 2”
I kind of hate problems where the solution is to find the most elegant answer possible. This problem is an excellent example of why. My solution (here), solves the problem as stated – but I took a grinder solution just wearing the problem away. My gut says that there is no way that this is the solution they want – however I do solve it with multiple applications of the quadratic equation, which the problem framing hinted at, so I am posting it for now.
Knowing that this is not the solution they wanted, I want to go look at the back of the book to confirm that is the case, and see what they were after. The problem with looking though is that I ruin the problem. So, I pretty much never want to look – hence my dislike of problems I where I can not prove I have the right solution. They all seem to be asking me to prove that someone smarter than I am could not solve the problem in a more elegant manner, but since the person in question is by definition smarter than I am – well – it seems a might ridiculous.
Here we have the problem:
“Solve this equation using nothing higher tan quadratic equations:
X = Sqrt( (X-(1/X)) + Sqrt(1-(1/X)) )”
So for some reason I never started looking for substitutions when solving equations. It was certainly something I learned to do when analyzing circuits, but when I see a math problem I never started looking for substitutions that might simplify the problem. Until recently.
Looking at, I think this problem, it “just clicked” – and I started looking for substitutions. Then used them to nock out answers for the next half dozen puzzle problems I tackled. Weird, since I am not doing anything I did not know before, but I just started looking at problems differently.
This one is fairly straight forward:
Solve: (x-a)/b + (x-b)/a = b/(x-a) + a/(x-b)
It becomes way easier to solve after a simple substitution.
My Solution.
Testing out another problem for my book. This is really just my trying to re-word a centuries old classic. So let me know if the wording is acceptable here:
On his twentieth year working for a pearl merchant Isaac was offered a reward. He was presented with three vases. He was told that each vase contains an equal number of pearls however the vases contain different types of pearls. One vase is labeled as containing white pearls, one vase is labeled as containing black pearls, and the remaining vase is labeled as contain an equal mixture of black and white pearls. The vases are opaque and have long throats the drawer has no way of seeing what the vases contain or the color of the pearl they have chosen until the pearl is removed from the vase.
The merchant tells Isaac that each vase is labeled inaccurately. The merchant suggests that they take turns drawing pearls from the vases in an effort to correctly re-label them.
As a reward for faithful service he tells Isaac that if he can identify how to correctly label each vase first – he can keep any pearls he has drawn. However if the merchant can correctly re-label the jars first Isaac has to return any pearls he has drawn.
The merchant offers Isaac the chance to draw first, should he accept?
So I already noted this correction in the previous post – but this is what they find a repeated / missing element in a collection of values from 1…N-1 should have looked like.
Just writing this problem down is making me realize how much I have forgotten. Time to crack open the books. I mean I even forgot my notation for mapping into a set with conditions!
This is another one from hard to solve brain teasers. It is just another algebra story problem that I have no idea why the included in the book. You just write down what they tell you as equations and end up with two linear equations with two unknowns – so it is fairly straightforward to solve . I do like the way they present the information here as it acts as a good example problem for introducing the idea of graphing or drawing a problem in order help set up equations to solve it.
It would still be a fairly simple problem to solve, but making the problem 3 equations in two unknowns, and using the extra information to nail one of several solutions might make it more fun. So one like this should go in the book.
So this was problem 5 from hard to solve brain teasers. I am working through the book and this problem is representative of about 60-70% of the problems which are just annoying. This would not be a hard problem in a 6th or 7th grade algebra course. I am fairly sure the book is targeted at adults – so are these problems hard because you are supposed to not remember junior high? Ugh.
The problem:
“Two cars start from point A at the same time and drive around a circuit more than one mile in length. While they are driving laps around the circuit, each car must maintain a steady speed. SInce one car is faster than the other, one car will pass the other at certain points. The first pass occurs 150 yards from point A.
At what distance from A will one car pass the other again?”
The thing I don’t like about this problem is it really is just a story problem where you just get the answer reading it. How? Because the problem lacks enough information to have any other way to set it up. You either get that starting at 0 and ending at A away from 0 means the second pass will be 2A away from the origin as long as the track is longer than 4A in length.
This could be turned into an interesting problem if the speed around the track was not constant in such a way that for the Nth lap A would be in the lead, for the Mth B would be in the lead. Then knowing who wins the race would require knowing the track length. If that was not given but was instead easily calculable the way to solve the problem would not jump out so much. I will have to try and write up a version like that for the book.
Update: Interesting – the problem directly before this one in the book is the same problem class I was proposing to modify this too. I wonder if the author was thinking that when clustering these problems of if that was just on my mind since I just solved that problem two hours ago or so. Huh.
So after that last problem I got to thinking that the area between a circle and its bounding or contained square would be potentially useful to know. Really. It just seemed cool. Only problem is that every use for it I could think of I could think of other ways to get there. Grrrr. Is this really cool – or is it the mathematical equivalent of carrying around a small dog in a shoulder bag?
