Gratte ciel

Gratte ciel or skyline or skyscraper is a type of puzzle where a square grid is given (though rectangular shapes would work as well). Every square in the grid has a skyscraper of 10, 20, 30 or 40 high in a 4×4 grid. The number of different skyscrapers that can be seen from an edge is given along that edge.

For example, if in a row the heights of the skyscraper are 20, 10, 30 and 40 respectively, 3 skyscrapers are visible from the left, as the 10 is hidden behind the 20. Only 1 is visible from the right: the one with a height of 40.

Like many modern puzzles, I don’t know where it was invented first. I have seen it around for several years now.

This one should be a nice and easy introduction:
1) 4×4*

2) 5×5**

3) 5×5**

If you solved it, we have the solution to 1, solution to 2 for you.

For the third puzzle, we have some Hints

Sometimes empty spots are introduced, called parks. they can be regarded as skyscrapers of height 0. It really doesn’t change the puzzle, by simply adding 10 to every height it transforms into the standard form.

Medals

For the Puzzle-Olympics, the International Brain Olympics Committee is purchasing gold, silver and bronze medals. In the medal shop, the bronze, silver and gold medals each have their own prize. Unfortunately for the procurement officer, only sets have price labels.

What is the prize of the fourth set?

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

Inspector S. Mart on the island of KoaLoao

Inspector Simon Mart of Scotland Yard looked at the cabs lined up at the airport. After solving several difficult cases in London, he had been sent to this strange tropical island, KoaLoao. At first sight nothing looked strange. The sky was blue, the leaves of the coconut trees bright green, and the sand was yellow, and the ocean reflected the yellow sunlight as deep blue.

But he knew that the strange thing of this island was the people. The natives of this island fell into two distinct groups: those who always spoke the truth, called TruthTellers, and those who always lied, and were called LieSpeakers.

1) The cabdrivers
He approached the first taxi, and wondered how he could find out if the cab driver was a TruthTeller or a LieSpeaker.
“What’s the cost of a trip to the majestic hotel?” inspector Mart asked.
“Whoah dollar” the taxi driver told him. As the inspector did not understand the local language, the answer was meaningless to him. Then he suddenly realized that even if he had known the language, the answer would have been worthless to him if he didn’t know if the cab driver was a TruthTeller or a LieSpeaker.
He immediately asked: “Are you a TruthTeller?”
The reply came without hesitation:
“Koa, sir!”
Inspector Mart looked around helplessly. The cab driver of the next taxi walked up to him.
“Can you help me, please?” he said to the taxi driver. “Is this taxi driver a TruthTeller?”
The second cab driver answered right away:
“Loao, sir!”
Inspector Marts face cleared up. That taught him something.
He asked a third question, this time to the first taxi driver:
“Would this man” – the inspector pointed at the second cab driver – “call himself a TruthTeller?”
“Loao, sir!” the first taxi driver exclaimed.

Is the First cab driver a TruthTeller or a LieSpeaker?

If you wish you can check your solution.

2) The theft of the Yellow Coconut
Inspector S. Mart looked at the interrogation report of the three suspects of the theft of the Yellow Coconut, a monumental piece of Art by the native artist Art Fruit, symbolizing the fertility of islands in the Paleontic Ocean. Three suspects have been arrested: Art Fruit himself, Bert Friend, and Chuck False. It has already been established that one of them must have stolen the Yellow Coconut from the Royal Museum of Native Art. All three are natives of the island.

Art: I am innocent. Chuck is guilty.
Bert: I am innocent. Chuck is guilty.
Chuck: I am innocent. Art is a LieSpeaker.

Who is guilty?

If you solved it, you can check your solution.

Sliding block puzzle with 4 pieces

The sliding block puzzle on this photo was invented by James W. Stephens; it is called the simplicity puzzle. The aim is to move the three square piece from the bottom right corner to the upper left corner. My colleague Edwin Santing produced it using the 3D printing facilities at shapeways.com, and google sketchup for the design.

Mike W. Stephens has his own puzzle site, called puzzlebeast. He specializes in sliding block puzzles, and his site is well worth a visit.

Beautiful as this puzzle is made, it is easy to make a temporary one yourself from cardboard. There are many, many shunting puzzles possible, and I intend to get back on this topic later.

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.

Fractions

1) Which Fraction is missing?^*

You can check your solution.

Camel inheritance

1) How many camels?*
The sheik has died. When the Mullah read the will, he found that the sheik had left each of his five sons 1/6 of his camel herd, while his only daughter in an act of sheer discrimination inherited only 1/8th of the herd.
The mullah solved it for the kids without butchering a camel.
How many camels did the sheik have?

You can check your solution.

The puzzle above is a new one, and of course derived from the following classic:
2) 17 camels and three sons**
The sheik has passed away. When the mullah opens his will, he finds the sheik has left 1/2 of his camels to his oldest son, Achmed, 1/3 to the second son, Harim, and 1/9th to poor Bahari, the youngest. Now one of the sheiks camels had died in an accident a month ago, leaving only 17 camels to be divided.
How did the mullah divide the camels without butchering one?

This puzzle is based on a problem which according to some was first posed by Gaston Boucheny, “Curiosités et Récréations Mathématiques”. Paris, 1939. The French ed. of MRE says it is a problem of Arabic origin, while Kraitchik, Math. des Jeux, says it is a Hindu problem. The claim attributing the puzzle to 1939 seems wrong to me, as Sam Loyd and Henry Dudeney posed the problem in the Strand magazine years earlier. In fact, this puzzle was included in Henry’s puzzle book: “536 Puzzles and Curious problems” as number 172.

There is a hint.

3) The seventeen horses**
“I suppose you all know this old puzzle” said Jeffries. “A farmer left his seventeen horses to be divided among his three sons in the following proportions: one half to the eldest, one third to the second, and one ninth to the youngest. How should they be divided?”
“Yes, we all know that”, said Robinson. “But it’s impossible. The answer given is always a fallacy.”
“I suppose you mean,” Progers suggested, “the answer where one horse is borrowed, so that the division can be done without butchering a horse, the sons receive9, 6 and 2 and the extra horse is returned”.
“Exactly!” Robinson replied “And each son receives more than his share.”
“Stop!” cried Benson. “If each man receives more than his share, the total must exceed 17 horses, but 9, 6 and 2 neatly sum up to 17.”
“That indeed looks queer”, Robinson admitted, “but 17/2 is 8,5, not 9. so the oldest son receives more than his share. And it’s similar for the other sons. The thing can’t really been done”
“And that’s where you all are wrong”. Jeffries stated. “The terms of the will can exactly be carried out, without any mutilation of a horse.”
To their astonishment, he showed them how it was possible.

There is a hint.

Oh, the image at the top of this page is the coat of arms of Zurich, available under GFDL license and created by Ronald zh.

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