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.
So the series was pretty obvious in retrospect – the summation of integer values from 1 to N has a known series that is so close I should have seen I could start there and work backwards. Ugh.
Yeah, I know I could Google this stuff – but where’s the fun in that?
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?
Tristan and his friend Joe are playing a game of catch. They are fifty feet apart from each other and tossing the ball back and forth at approximately seventy-five feet per second. Two thirds of a second elapse between each of the throws of the ball and the corresponding catch. During that time the ball travels one hundred and sixty seven feet. How is this possible?
Ok, so my first attempt at a lateral thinking problem was a bit of a disaster. Konrad and Jason both were not happy with it and I am trying to figure out a way to re-work it so that the problem is not trying to figure out what the heck I am thinking!
Apparently since I want to publish I need to hold off on posting answers here for copy write reasons. So I will post the answers on a separate site and use comments here as a way to discuss the problems.
So I hate “lateral thinking” problems. Seriously. They usually seem to be more of an exercise in thinking about how other people think – normally an interesting if not enjoyable topic – but somehow sucked dry of the positives of the experience. Never thought much about why I feel that way but I will certainly give it more thought now that I need to write some for my puzzle / math book.
So at some stage I need a tutorial on how to approach these sorts of problems algorithmically – to that end I wrote up this example.
A problem I found online for “lateral thinking” was stated as:
Problem: “How can you throw a ball such that it always comes back to you?”
I am paraphrasing the problem but this restatement has the critical property of the original – at least one answer that is obvious from reading it. Answer: “Throw the ball straight up!”.
These problems always have a basis set of assumptions. For example that the person is in a gravity field or on Earth, that that they can’t throw the ball at escape velocity. I broke down some examples below that have easy graphical representations of the problem. The graphics should help with the book and relaying understanding.
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!