# Boolean logic

1) The first Sudoku toilet paper*
Inspector Simon Mart of Scotland Yard was looking at the interrogation statements of 3 well known criminals. It had already been established that one of them had stolen the Very First Role of Sudoku Toilet Paper, which of course is an object of immense historical value.
It also had already been established that of the four suspects, exactly one spoke the truth.

Inspector Simon Mart looked at their statements:
Albert: I am innocent
Bill: Charles stole it
Charles: I am innocent
Who stole the toilet paper?

There is a rather recent class of puzzles which have to do with statements which are either true or not true. In the branch of mathematics which is called Logic, these statements are called propositions. Though in everyday life we use the term logic rather loosely, in Mathematics it is a rather tricky field with sub-fields such as Arestotelian Logic, logical positivism, fuzzy logic, hypothetical syllogism, Propositional calculus, Predicate logic, Mathematical logic, Intuitionistic logic and many others. It also has practical applications, such as in computer science, and in Argumentation theory.

2) The blue towel*
Everyone of course knows that the blue towel really is yellow, but it is always called the blue towel because king Henry the 87th, of true blue blood, had washed his face with it in the 13th century. A small blue streak of blood, said to have been originated when the king cut his finger, testifies to it.
Inspector S. Mart of Wales Yard looked at the report of the interrogation. He knew the two suspects: Dirty Dave and Big Barry. None of them was able to utter two consecutive sentences without lying at least once. The police officer, who had questioned them after the theft of the “blue towel” from hotel “the palace”, had written a short summary:
Big Barry’s statement, alas, had been in a downtown accent which the police officer had been unable to understand and his notes were completely unintelligible. Dirty Dave’s statements were very short and clear:
Dirty Dave: I am innocent. Big Barry did it.
As other investigations revealed that one of the two must have been the thief, who did Inspector Mart keep under arrest?
Don’t peek at the solution, just use it to check your own solution.

3) The white waste-paper basket *
Inspector S. Mart of Scotland Yard interrogates two suspects of a theft of the white waste-paper basket from the local museum. This famous waste-paper basket is so old it dates back to the previous century.
Mr Brown declares: Both Green and I are guilty.
Mr Green: Brown stole it.
Given the premise that one of them is lying and the other one speaks the truth, who should he arrest?
Don’t peek at the solution, just check your solution.

4) The blue eye paper envelope **
Chief police inspector S. Mart interrogates the 3 suspects of the robbery of the famous Blue Eye paper envelope. All three suspects are well known criminals, and he knows that none of them can utter two consecutive sentences without lying at least once.
Mr Black: “I am innocent, inspector. It was White who stole the envelope.”
Mr Green: “Black is innocent, inspector. Black is lying when he says White is guilty.”
Mr White: “Black is innocent. Green is innocent.”
Who did Smart arrest?
Don’t peek at the solution, just check your solution.

5) The stolen washing-glove **
In his next case, Inspector S. Mart of Wales Yard interrogated the infamous villains Awful All, Boney Bill and Cold Charley about the theft of a hotel washing-glove.
All: Bill lies and Charley stole it.
Bill: I am innocent
Charley: All lies or Bill did it
If only one of them speaks the truth, whom should the inspector arrest?

5) The stolen chocolate**
In the famous royal family of the isle of Kids a chocolate has been stolen. The suspects are no less than the five princesses! Inspector S. Mart is immediately called upon when the queen discovers that a chocolate is missing from the chocolate box: princesses are supposed to be absolutely honest!
Anna: Cindy is guilty;
Belinda: I am innocent;
Cindy: Diana is guilty;
Diana: Charles lies if he says I am guilty;
Elizabeth: Anna tells the truth and Cindy lies;
Assuming that only 1 of them lies and all the others speak the truth, who stole the chocolate?
If you find this one a bit hard, you can look up a hint.
And assuming that only 1 spoke the truth, who would be guilty?

You may have noticed that the first puzzle in this post had three people, and the culprit could be deduced if either one of the lied or one of them told the truth.
This puzzle has 5 people and the same conditions. If you omit Elizabeth, you still have a puzzle with 4 people and the same condition. Can you construct a puzzle with 6 people and the same condition?
And a more intriguing question: can this sequenced be epanded to any number of people?

This area of puzzles has been investigated by the logician Raymond Smullyan, in lovely books as Alice in Puzzleland, This book has no title, and other books. According to the English language wikipedia, he has about invented this type of puzzle. I also found puzzles of this type in J.A.H. Hunters “Mathematical brainteasers”, copyrighted 1965, preceding Smullyans books by over 10 years, which seems to make the statement on wikipedia doubtful.

The looking glass at the top of this article was drawn by an unknown artist at commons.wikipedia.

# jars and pearls

1) 6 jars with pearls**
Sultan Oil-well decided that his beautiful daughter had reached the age of marriage, and of course numerous princes of the neighbouring states were interested in her hand.
Instead of choosing the handsomest or richest, he decided to choose the most intelligent candidate.

He put 6 jars in front of the assembled princes, and told them that each jar contained a number of pearls. Jar 2 contained 1 pearl more than jar 1; jar 3 had 1 pearl more than jar 2, and so on.
Then he ordered his daughter to take 1 pearl from jar 1 and put it in jar 2. Next she took 2 pearls from jar 2 and put it in jar 3, and so on, competing a complete circle by moving pearls from jar 6 into jar 1.
“Gentlemen” the sultan told the princes “jar 1 now has twice as many pearls as it had at start. How many pearls were in each jar at start?”

2) The boxes*
In one of his books (‘Test your wits’) Eric Doubleday presents the following, simplified version:
The daughter of the sultan had 4 boxes in front of her: each one contained one more than the previous one. The last one had twice as many as the first one.
What is the total number of pearls?

3) Men in a circle with shillings*
This one goes back to Lewis Carroll. It is one of his “pillow problems”, problems thought out during sleepless hours.

Some men sat in a circle, so that each had 2 neighbours. Each had a certain number of shillings. The first had 1 more than the second, who had 1 more than the third, and so on. The first gave 1 to the second, who gave 2 to the third, and so on, each giving 1 more than he received, as long as possible. There were then 2 neighbours, one of them had 4 times as much as the other.
How many men were there? And how many had the poorest at start?

Feel free to take the entire night to solve this one. Lewis Carroll solved them by head, and I’m sure that with some exercise you can too.

Incidentally, This blog is now slightly over 1 year old. The general speed has been 1 post in 2 weeks. I’m trying to move up speed to once a week, and I seem to have sufficient puzzles in store to keep up this pace during the month of May. I may have to slow down somewhere in the future again, but we’ll see that when we get there. As many posts contain more than 1 puzzle, the general pace of puzzles has been over 1 puzzle a week.

# Logic puzzles – about old jacks and new coats

Here are some new puzzles:
1) Old jacks
(1) Leather jacks always become very old
(2) Old jacks turn brown
Which conclusion is valid?
a) All brown jacks are old
b) No old jack is blue
c) Some leather jacks are brown
d) None of the above

2) Dirty Pigs
(1) No dirty pig is fat;
(2) No meager pig is pink;
Which conclusion is valid?
a) All fat pigs are clean;
b) All dirty pigs are meager;
c) All pink pigs are fat;
d) All pink pigs are clean;
e) All of the above;

3) New coats
(1) No new coat of mine is not made of plastic;
(2) All plastic coats are closed with zips;
Which conclusions are valid?
a) All plastic coats are mine;
b) All zipped coats are mine;
c) All coats closed with zips are old;
d) All my new coats are closed with zips;

These puzzles are based on a mathematical concept called Sets, but you can solve them without this knowledge. Sets are now taught at all high schools in the Netherlands, and I suppose also in other countries. Some Googling on Venn diagrams and puzzles revealed little or no sites. This surprises me as it usually regarded as a good educational practice if students are also taught to apply their knowledge to practical problems, although this may depend on the preferred learning style of the student.

Puzzles of this type do have a history. One not so well know puzzle master from Victorian England is Charles Lutwidge Dodgson (1832 – 1898), better known as Lewis Carroll, author of children tales as Alice in Wonderland and Through the looking glass.

In his daily life he taught mathematics, and the story goes that Queen Victoria loved Alice in Wonderland so much that she wrote the author and asked for a copy of his next book. Charles felt honoured, and duly send her a copy of his next book – a treatise on a mathematical subject.

Charles was also an avid puzzle designer. In his books “Symbolic logic” and “The game of Logic”, later republished by Dover press, he devised an elaborate mechanism to solve set problems like those above. Modern mathematicians would use Venn diagrams to solve the same class of problems. Puzzles like these are somtimes called Soriteses.

Here are some examples of the type of puzzles you can find in his book:
4) Wasps
(1) All wasps are unfriendly
(2) All unfriendly creatures are unwelcome
What conclusion can be found?

The puzzles can be made slightly more complex by adding mote statements, by more types of objects that play a role, or by adding negatives. Here are some of his puzzles that comprise 3 statements:
5) Sane
(1) Everyone who is sane can do Logic
(2) No lunatics are fit to serve on a jury
(3) None of your sons can do Logic
What conclusion can be found?

6) Flowers
(1) Coloured flowers are always scented
(2) I dislike flowers that are not grown in the open air
(3) No flowers grown in the open air are colourless.
What conclusion can be found?

7) Showy talkers
(1) Showy talkers think too much of themselves;
(2) No really well informed people are bad company;
(3) People who think too much of themselves are not good company.
What conclusion can be found?
Lewis Carroll was thinking in terms of modern day sets, and made no secret of it. At the end of each puzzle, he added a note about what the Universe of discourse in that puzzle was, wand what sets needed to be considered. For example, in puzzle 4 this was:
Universe = persons; a=good company; b=really well-informed; c=show talkers; d=thinking too much of ones self;

It is of course possible to create puzzles in this vein with more statements. Here is one from Charles L. Dodgsons book with four statements:
8) Birds in the aviary
(1) No birds, except ostriches, are 9 feet high;
(2) There are no birds in this aviary that belong to any one but me;
(3) No ostrich lives on mince pies;
(4) I have no birds less than 9 feet high;
What conclusion can be found?

Dodgson constructed puzzles much more complex; his longest is made up of 10 statements:
9) Animals in the house
(1) The only animals in this house are cats;
(2) Every animal is suitable for a cat, that loves to gaze at the moon;
(3) When I detest an animal, I avoid it;
(4) No animals are carnivorous, unless they prowl at night;
(5) No cat fails to kill mice;
(6) No animals ever take to me, except what are in this house;
(7) Kangaroos are not suitable for pets;
(8) None but carnivora kill mice;
(9) I detest animals that do not take to me;
10) Animals, that prowl at night, always love to gaze at the moon.
What conclusion can be found?