Cubes

Cubes are wonderful things. With six side surfaces, eight vertices and 12 edges, they are highly symmetrical. There are 11 ways to flatten a cube into a plane by cutting the edges. Here are 6 of the 11 ways:

Can you tell which cube is different? You can ignore the orientation of the letters – they are merely for identification. The symbols have been added for those readers who are colourblind.

If you are puzzled, we have a solution for you.

Did you know?
The subiculum plays a role in spatial navigation, mnemonic (symbol) processing. You probably already understood that this puzzle challenges the 3D representation facilities of your brain.

Playing with numbers – quickies

The next two problems are variants of easy problems from the Dutch 2008 Mathematics Olympiad.
1) 2016^*
What is the smallest integer number that, when multiplying its digits, gives 2016?

2) Odd numbers ^*
Multiply all numbers between 1913 and 2012 (including 1913 and 2012). What is the last digit of the product?

You can check your solutions at 16 and 26 respectively.

Books

1) books^*
The professors assistant entered the office room of the professor and noticed that the professor had not only labeled the plants as shown in a previous post, but also his books. Only a new book on his desk was not yet labeled. Can you help him out?

You can look up hint here.

As you may have noticed I like this type of puzzle. Though I am no brain expert, i think you need to active different parts of your brain, and though I’m no expert in the field here’s a list of those parts of the brain which i think will be utilized while solving these puzzles:

• According to research by J. R. Binder and others, the Angular gyrus was used heavily when processing abstract keywords. Finding the correct abstract concepts play an important role in these puzzles.
• Prefrontal Cortex: used for: planning, reasoning, and judgment. Once you have an idea which properties of the objects play a role in the codes, you will need deductive reasoning to check that. Deductive reasoning activates the left frontal lobes, as researched in a meta study in 2011 by Jérôme Prado, Angad Chadha, and James R. Booth.
• The Occipital Lobes are used by visual activities, and as these puzzles are highly visual the solver needs to use this part of his/her brains.
• The inferior frontal gyrus and middle temporal gyrus are utilized according to research by Jing Wang, Julie A. Conder, David N. Blitzer, Svetlana V. Shinkareva.
• Corpus Callosum: This allows information to move between the left and right hemispheres of the brain and is thus a very important integrative structure.

Next weekend we hope to be away a few days, so the puzzle may be later than usual.

Pattern code post-its

1) Post-its^*
You know those super busy people at the office? Their desk is full of yellow post-its. Every office seem to have at least one such person. One fellow worker had a system to code their priority. Can you find his system?

You can look up a Hint

Coffee with milk, please

Tanya Khovanova publishes an irregular but excellent blog about math problems. Of Russian descent, she often uses Russian sources, which are otherwise not very accessible in the Western world. The next problem comes from her blog, and has the Moscow 2011 mathematics olympiad as origin:

1) Coffee with milk, please^***
1) Coffee and milk^**
In a certain family everyone likes their coffee with milk. At breakfast everyone had a full cup of coffee. Given that Alex consumed a quarter of all consumed milk and one sixth of all coffee, how many people are there in the family?

The above problem would go into the class of problems for which you have n equations and n+1 unknowns. Here’s a classic in this category:

2) A farmer went to the market^*
A farmer buys 100 animals for 100 dollars but lost his receipt. Cows are $10 each, pigs are $3 each and chicks are $.50 each. How many of each did he buy?
This puzzle is a ‘classic’, but I don’t know its source. If you do, I’d welcome this information!

When you solved both, you will notice that the solving methods of the two puzzles are totally different.

You can find hints at 126, 116 respectively.

patterns – ballpoints

ballpoints
Which code goes to the question mark?

You can look up hint here.

7 nails and a block of wood

How can you balance the six nails on the nail in the wood?
No other objects are allowed, and the nails should balance on the nail, not rest on the wood.

I would like to thank my colleague Theo Sweres for showing this puzzle to me. There is a slight manual dexterity required, but not much. Though this puzzle has been exclusively on logic and math puzzles so far, I’ll add some physical puzzles for variety.

You can check your solutions at solution 222

Pattern – flowers

When the new assistant of Professor F. Lower walked into the lab, he noticed that the professor had labeled 5 of the plants. The professor seemed to have walked out of the lab without labeling the sixth plant. Can you help out the assistant?

You can look up a Hint

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?
You can check your solution.

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?
You can check your solution.

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?”

You can check your solution.

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.

You can check your solution.

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.