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.
Problem 179 from http://mathproblems.info/group9.html
A drag racer accelerates at a uniform rate from its starting point. It travels the last one fourth of the distance from the starting point to the finish line in 3 seconds. How long did it take to travel the entire distance from starting point to finish line?
This is basic “interview algebra” – it is *not* a differential equations problem. I suspect the whole trick to this problem is making you *think* you are supposed to be setting up and solving a differential equation – so basically if you are getting asked a problem like this in an interview the interviewer is trying to see if you make things hard for yourself under pressure. Useful information to know for both parties.
Grass grows in a field at some rate r, where r is the units of grass grown per day. It is known that if 10 sheep are turned out in the field, the grass will be gone in 20 days. On the other hand, if 15 sheep are turned out in the field, the grass will be gone in 10 days. If 25 sheep are turned out in the field, when will the grass be gone? ( Another Microsoft question from http://vijay.techi.googlepages.com/puzzles).
So I ran across pretty much this exact same problem on pg 36 of Mathematical Recreations from 1907 and they attributed the problem to Isaac Newton.
An explorer wishes to cross a barren desert that requires 6 days to cross, but one man can only carry enough food for 4 days. What is the fewest number of other men required to help carry enough food for him to cross, the constraint is that each man should survive? (One of the Microsoft questions from http://vijay.techi.googlepages.com/puzzles).
