Categories
Logic

The pearl merchant

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?

Categories
Geometry Logic Problems

Mathematicians suck!

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
Area of a square - inside a circle - inside a square

Categories
Logic

Word elimination problems

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
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.

Categories
Logic

“Added Corners”

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.

The problem is:

"Added Corners"
"Added Corners"

Categories
Logic

“Segments”

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.

Any way – the problems it:

"Segments"
"Segments"

Categories
Logic

Which twin am I talking to?

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?

Twin problem
Twin problem

Categories
Logic

“Prohibited Connection”

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.

Prohibited Connection
Prohibited Connection
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.