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.
Good example on graphically representing a story problem
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?”
Irregular CIrcuit
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?
Finally I have some time to work through daily problems again. Worked this one last week and forgot to write it up.
When I saw this problem I just saw the trigonometry solution and it blinded me to the symmetry. I guess math is good at showing how what we already know can blind us to the better paths.
So why do mathematicians suck? Well that seemed politer for Konrad than saying he sucks – I worked on this problem for a while and just did not see it. He saw the answer before I was finished describing the problem. So I guess I am the one who sucks, – apologies to mathematicians everywhere.
Area of a square - inside a circle - inside a square
I am a moron. Yup. I sat down and worked a 4×4 problem this morning only to prove it was impossible to solve. It helps if you copy the problem down correctly. Normally I am fairly anal about this type of error but I was writing out a detailed description of how to solve the problem and it spilled onto a second page. When copying over the current part of the problem… whoops. So I worked the next problem in the book, this simpler 3×3 of the same type, and will only post one today. After all I need to get to work from the morning coffee break!
Ok, so while I like the reasoning this type of problem encourages – I don’t like it when the longer ones assume a thorough knowledge of English. Mostly, I don’t like relying on the English vocabulary because English has a loosely defined vocabulary that borrows words from all sorts of other languages. Working one of these problems I found several alternate answers with words I considered English but which were borrowed from other languages. I think I will also avoid them as students who do not have English as their first language would be discouraged from solving the problems.
The problem says you are looking for a 3 letter word. You are given four words. The first, HAS, has zero letters in the same location as the word you are looking for, the second and third words, CON and CAB, each have one match and finally the last word SON has two matches. What is the word you are looking for?
Word elimination problems
I think I will work the problems I find like this as they are – but then change the letters to combinations of symbols and colors when making new problems along these lines.
This is another problem that is really simple to introduce but I could quickly make much harder providing problems that reward the reader / student for being clever – punishing brute force attacks.
Right off the bat I am thinking this problem going from a square to a cube – or other three dimensional shapes – so the user has to be doing both reasoning and mapping to spatial objects.
This type of problem might be good to reward the reader / student for clever and not brute force attacks. This problem is simple enough to brute force through it but add in a few extra digits and more rules would stop that. So I can use one like this as an example then grow the string to make the problems harder. I also think this problem would be better with crossed patterns – sort of like numeric scrabble. I think I have seen math puzzle books in the airports with patterns like this. Probably something like what they are doing.
Ok, this is another problem that was kind of lame as worded but could be turned into a useful example
Problem as stated: Suppose that 3% of births give rise to twins. What percentage of the population is a twin 3%, less than 3%, or more that 3%.
So the problem is obviously trying to get the reader to realize the birth of twins increases the population by two people as opposed to “normal” births increasing the population by a single person. Then intuit what that means to the percentage of twins in the population.
No reason that the reader should not be able work out the exact percentage given the problem as stated and the assumption that triplets, quintuplets, and all other multiple births are a negligibly small percentage of the population.
Stated this way I can use this problem to introduce readers to the idea of taking two equations dependent on some third and unknown variable that cancels out later.
Ok, super simple problem, but it would be good as an example for introducing people to truth tables at the start of a section with a bunch of the “liar” problems.
Problem #1 page 6 of hard-to-solve brainteasers
Problem: Peter and Paul are twins. One of them, we don’t know which, always lies. The other always tells the truth. I ask one of them “Is Paul the one that lies?” and he answers “Yes”. Was I speaking to Peter or Paul?
Bought a copy of “Hart-to-solve brainteasers over the weekend. So far that title has been a disappointment – most of the problems assume that the reader has not had any formal training in problem solving – so the hard parts of finding a solution seem to be reinventing the tools needed to tackle the problems. Especially, annoying since the answers section just list answers without telling the reader how to solve the problems. Defiantly something I am going to have to keep in mind if I go through with writing a puzzle book.
This was problem #19 on page 12 of that book:
Using numbers 1,2,3,4,5, and 6 put each of the numbers in a circle with the constraint that no circles connected by a line can contain adjacent numbers. For example the circle with a three in it cannot be connected to either a circle with a two or a four.
