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27.03.2017


Case-lesson «Labyrinths: how to find exits»
Case-lesson «Labyrinths: how to find the exits»

Category: Mathematics and computers

The level (grade): 9 +

Subject: Computers and programming

Objective: Uncover the secrets of creation and technology of passing the mazes based on exact sciences and legitimacies.

What information is waiting for me here?

  • What is a maze and how to create it?
  • What mazes are the most difficult?
  • What algorithms lie in the basis of mazes creation?
  • Are there any cribs on how to pass a maze?
  • What fundamental and practical knowledge will be useful for me?
  • Are there any "mazes" in nature and how do they work?
7 scans of subjects, phenomena and practices:
Introduction

A modern maze is more like a computer game maze, rather than the labyrinth of Minotaur. But they both have common "ancestors": the first labyrinths-amulets, labyrinths-temples.

 

However, all mazes obey certain algorithms of creating and passing them. So, what are they? How to create and pass mazes? Let's create a "labyrinth cheat sheet" together.

History
When did the first mazes appear? What did they look like? For what were they used?

The first images of a labyrinth were found in the upper Paleolithic 38.000 BC.
The age of a clay tablet with a pattern of seven concentric lines, which was found on the Greek island Pylos, as well as Syrian pottery decorated with the image of classical labyrinth, is about 3.5 thousand years.

 

These days, just like thousands of years ago, Indian tribes Tohono, O’Otham and Pima from American Arizona, weave baskets using dried stalks, roots and leaves of plants growing in the desert. They decorate it with patterned labyrinth, which is called "the house of Iito" – in honor of their progenitor, whose spirit rests on the peak of the mountain Baboquivari. It is believed that these patterns are amulets.

 

Egyptian labyrinth was described by an ancient historian Herodotus. It was built in the year 2300 BC. The building was surrounded by a high wall. There were fifteen hundred ground and underground rooms. The maze occupied a total area of 70 thousand square meters.

 

This colossal temple complex was used as a tomb for pharaohs and crocodiles. In Egypt crocodiles and cats were considered sacred animals. And although there is plenty of evidence that the labyrinth was the center from which kings had been ruling the country, mainly it served religious purposes.
The most famous labyrinth is the labyrinth that was, according to a legend, built on Crete by the Athenian sculptor Daedalus. The labyrinth in Knossos (Crete) is also ancient, although it was built later than the Egyptian labyrinth.

 

The myth says that in this labyrinth the king Minos hid a secret of his cheating wife Pasiphaë. She gave birth to a monster Minotaur (what means “the bull of Minos’”) – half man and half bull. Daedalus built the Maze with so intricate passages that no one could find a way out of it.
According to the myth, Athens was defeated in the war with Crete. Every 9 years the Athenians had to send 7 girls and 7 boys to Crete to be sacrificed to Minotaur. These young people were sent into the labyrinth, and, as it was expected, they were eaten by Minotaur. But one day a young man named Theseus volunteered to go to the Labyrinth to fight the monster. Minos’s daughter, Ariadne, fell in love with the hero. She gave him a sharp sword and a ball of thread. She tied one end of the thread at the entrance. Unwinding the ball, Theseus went into the palace to fight Minotaur. Ariadne’s sword helped him to defeat the monster, and thread – to find the way out of the Maze.
The ancient Greek myth tells about it. However, in those times, when Greeks only began to write this history, the Labyrinth had long ceased to exist.
Subsequently, the mazes were spread. They ceased to perform religious functions and became a part of décor and design. For example, the labyrinth in the tomb of Etruscan king Porsena was built in the Italian city Clusone.
Intricate spiral shapes made of stones on beaches (probably for religious purposes) are a frequent phenomenon in Northern Europe (Denmark, Sweden, Estonia, and Russia). There are over 600 stone labyrinths in Scandinavia on the Baltic Sea coast. There is an assumption that many of them were built by local fishermen who believed that it provides good catch and safe return.

 

There are also figures on floors of medieval cathedrals (such as Chartres cathedral). They are called "the Jerusalem road" because repenting sinners had to crawl on their knees there:

 

One of the first English mazes with hedges (now it is demolished) was planted in the XII century during the reign of King Henry II in the garden near his Palace in Woodstock. It was called "the Boudoir of Rosamund". People of that time claimed that Henry created a decoration for the garden with some purpose. Supposedly in the center of the labyrinth the king built a mansion for his favorite Rosamund Clifford.

 

In modern England mazes of hedges are popular and widespread. For example, the year 1991 was declared the Year of the Maze in the UK. There are so many labyrinths and they are so liked by people that during the year almost every weekend the British citizens and tourists can wander the alleys of Hampton Court, Leeds, Hever and other “puzzles” completely for free. There is no lack of volunteers to conquer the next maze.

 

In 2011 a labyrinth was opened in York. The tracks of this maze form a portrait (or even two portraits) of a young British wizard Harry Potter. Interestingly, two images are slightly different from each other. It makes the maze the world's largest picture in the style of "find 10 differences".
Another type of modern mazes – mirror mazes:

Try to get out of it!

Task:


What maze would you like to visit? Try to come up with your "concept of maze". What materials would you like to use? What form should it have? Present your maze to your classmates.

Now try to go through several mazes to learn how to create your own one.
Technology

To overcome this maze you should use hover boards – flying skateboards. There are two types of hover boards, each of which has some advantages and disadvantages.

Here is your "transport":
1. The hover board called "the brick":

ARCA Space Corporation is the space corporation that has created ArcaBoard. It is equipped with an electric engine with a power of 272 horsepower or 203 thousand watt of installed capacity. Therefore, it can be considered the most powerful and lightweight personal transportation that has ever been created by human!
ArcaBoard was built using composite materials. It has a size of 145x76x15 cm and a weight of 82 kg. Technology used for its creation allows a person (weighing up to 110 kg) to fly at the height up to 30 cm at 20 km/h speed.
The drawback is that it cannot fly above the water surface.
2. Lexus has also created its own hover board — the physical implementation of flying boards from fantastic games and movies. To levitate above the ground this hover board requires special conditions — the metal surface must be laid under the surface of a skate park (the board contains powerful magnets). There is only one skate park like this. It was built specifically to test the new product in Barcelona (and in our "virtual maze").

The hover board has superconductors that are enclosed in special cryochambers with a temperature below -197C. They interact with the cloth and allow moving. Since the technical details have not been disclosed yet we can assume that the speed is 50 km/h.

 

 

 

Good luck!

Biology

Take a close look at these mazes:

 

Actually this is the Golgi apparatus and mitochondria! What are these two "labyrinths" of cell structure famous for?
The main function of mitochondria is derivation of energy-rich substrates (fatty acids, pyruvate, and carbon skeleton of amino acids) from the cytoplasm and their oxidative splitting with the formation of CO2 and H2O, connected with ATP synthesis.

Substances needed by a cell, for example, digestive enzymes, are “packaged” into membrane vesicles and spread in the cytoplasm. In the Golgi apparatus there are also accumulated substances (the cell synthesizes them for the needs of the whole organism; they are excreted from the cell to the outside).

So, what does happen during this? The formation of DNA and RNA chains! Since DNA is in the nucleus, while protein synthesis occurs in the cytoplasm, there is a mediator that transmits information from DNA to ribosomes. mRNA is the intermediary:
In protein synthesis the following stages are identified that are running in different parts of the cell:
1. The first stage. The synthesis of mRNA occurs in the nucleus. The information contained in the DNA is rewritten to RNA. This process is called transcription (from Latin “transcriptio” — rewriting).

2. The second stage. Amino acids connect with molecules of t-RNA that consist of three nucleotides — anticodons with the help of which triplet codon is identified.
3. The third stage is the process of direct synthesis of polypeptide bonds and it is called translation. It takes place in ribosomes.
4. The fourth stage is the formation of secondary and tertiary protein structure, i.e. the formation of the final structure of protein.
The processes:

The synthesis is based on the principle of complementarity. Pairs of amino acids are "marked" in advance:

 

Therefore, to pass the biological part of the maze you have to do this task:

 

Computer Science/Mathematics
You have passed a few mazes. Now try to create your own one!

So, the first variant is the virtual maze. It is always based on a certain algorithm, no matter what programming language you will use later.
Many mazes are in the shape of a spiral. Depending on selected spiral, you’ll need to choose formula for calculation:

 

 

If as an example you choose the maze like the maze from the "Technology" section task, it will be necessary to use Eller’s algorithm. What is this algorithm?
Eller's algorithm of generation of mazes:

www.neocomputer.org/projects/eller.html

The Eller's algorithm allows creating mazes with the only path between two points. The algorithm is very fast. It uses computer memory more efficiently than other popular algorithms. It requires the memory to be proportional to the number of written lines. It allows creating large mazes with limited memory.
Before you start you should know that in such labyrinths the leftmost cell has a border on the left and the rightmost – on the right.

Just keep in mind that in terms of mathematics you work with sets.
The best is to consider the algorithm on this example: let's create a simple maze. Let's start adding new rows, moving from top to bottom, left to right. Each cell in the row will belong to a set. For convenience and clarity we’ll put the numbers of sets and write these numbers in the cells. Each cell can have a border on the right and/or bottom. Assume that the borders are present to the left of the first and to the right of the last cells in the row. Here it is the example of the maze that we are creating:

 

The algorithm for generating the first row is the basis, on variational repetition of which the creation of the maze will be based.
The text form of the algorithm is:
1. Create the first row. None of the cells will be a part of any set.
2. Assign unique sets to the cells outside of any set.
3. Create the right borders moving from left to right:
3.1. Decide randomly whether to add a border or not
3.1.1. If the current cell and the cell on the right belong to one set, create a border between them (to prevent looping)
3.1.2. If you decide not to add a border, combine two sets that consist of current cell and the cell on the right.
4. Create bottom borders, moving from left to right:
4.1. Decide randomly whether to add a border or not. Make sure that every set has at least one cell without the bottom border (to prevent isolation of areas)
4.1.1. If the cell is the only one in the set, do not create the bottom border
4.1.2. If the cell is the only one in the set without the bottom border, do not create the bottom border
5. Decide whether to continue adding rows or finish the maze
5.1. If you decide to add another row, then:
5.1.1. Print the current row
5.1.2. Remove all the right borders
5.1.3. Remove cells with the bottom borders from their sets
5.1.4. Remove all bottom borders
5.1.5. Continue with the step 2
5.2. If you decide to finish the maze, then:
5.2.1. Add the bottom border to the each cell
5.2.2. Moving from left to right:
5.2.2.1. If the current cell and the cell on the right are the members of different sets:
5.2.2.1.1. Remove the right border
5.2.2.1.2. Combine sets of the current cell and the cell on the right
5.2.2.1.3. Print the final row

 

Now let's add the next rows:

 

When the maze is created with a sufficient number of rows, it should be completed correctly: the last row is different from the usual in a way that each cell has a bottom border. Thus, each cell must belong to the same set.

 

How else can you create mazes? Here is an example:

You can create a maze using this way:

Alternatively, this:

Here you can see the maze and the scripts for it:

Relaxation/Practicalskills
If you prefer real labyrinths, you can create them using these tips:

A maze is one of the common elements of landscape design. How can we create it?
First, a project is created. Second, you should think about using the topography of the area, and what plants will be used.
It is important to consider the preferences of plants (light, watering and care) and the permissibility of the neighborhood. For example:

 

After that, the project is “carried” to nature and laid out on the ground. Plants are planted. For example, the maze of hedges or flowers:

 

Also you can create a rock maze:

 

You can create mazes for pets, for example hamsters:

There are both good mood for you and your pet and mental pabulum!
The choice of the maze depends on patience and desire. Moreover, if you do not want to create it, you can simply enjoy passing already created mazes.

What tricks are there for successful completion of mazes?

One of the simplest rules for passing a maze is the rule of "one hand": when moving through the maze you have to touch the right or left wall. The ancient Greeks had probably known this algorithm. In this case, you have to go a long way, coming to deadlocks, but eventually the goal will be achieved.
You can act using the following algorithm:

 

If the maze contains freestanding walls, it is not always possible to pass through all the corridors and deadlocks using the rule of "one hand". Mazes with freestanding walls and closed routes are called multiply connected.
Universal algorithm for passing any labyrinths was described in a book of French mathematician E. Lucas "Recreations matematiques" published in 1882. Moreover, it is still working! It is interesting that when Lucas was describing the algorithm he indicated the superiority of another French mathematician M. Tremeau. Thus, the algorithm is known as the algorithm of Lucas-Tremeau.
Tremeau had proposed such rules:
• after coming out of any location of the maze, it is necessary to paint a mark on the wall (a cross) and move in random direction to the impasse or crossroads;
• in the first case you have to go back to put the second cross. It indicates that the path has been passed twice – back and forth. Then you have to choose direction that has not been passed, or has been passed once;
• in the second – you have to go in arbitrary direction, noting every intersection on entering and exit;
• if there is one cross on the intersection, you should choose another way, if not – you should choose the way that has been already passed, noting it with the second cross.
Now you are armed with the tips and algorithms, and you will have success in passing all mazes!

Task:


Using the gained knowledge, try to create your own virtual or real maze (or at least "paper based").

Art

Mazes inspire not only architects or programmers, but also artists. For example, “The Maze” picture belongs to the great Salvador Dali:

 

Swiss artist Fabian Oefner likes the theme of labyrinths. By diluting paints with water, dripping gouache, watercolor or colored ink into a container filled with a clear liquid, you can get amazing pictures that show colorful explosions, or abstract or stylized figures. However, the work of Fabian Oefner is more complicated and unusual. He pours a watercolor paint into water, but then adds a ferrofluid into this substance and uses a magnet. Then in the water channels are formed randomly, like passages of a labyrinth, where bright colors are flowing.

 

Question:


What other “maze” paintings or sculptures do you know?

Quest
A quest and a labyrinth are "relatives". Because passing a maze is also a kind of quest. A quest, in fact, is a cascade of tasks, which can be done sequentially or in some other optimum order. And mazes can have either one or multiple passing options. Here is an example of the quest: http://case.edufuture.biz/m-keys-quest.html.This is the case-lesson "Quest". You can try to pass it!
Spiral mazes are the most popular type of mazes. The majority of historical mazes in Northern Europe are the same type.

In addition, thematic quests of “maze” type are often based on them. “Maze quest” is quite a popular topic in computer games, including online ones. Even special "cheat sheets" for playing can be found in the Internet:

Lesson summary:

 

Title

The content

1

The results of the case-lesson can be complemented by findings of students

 

2

What three websites helped find important information?

http://edufuture.biz/

http://www.chem.msu.su/rus/teaching/kolman/212.htm

http://shkolo.ru/sintez-belkov-v-kletke/

 

3

To help student and coach:

http://edufuture.biz/

http://akamar.narod.ru/Labirint/labi1.htm

4

Where can you get information for the case?

http://edufuture.biz/

http://www.chem.msu.su/rus/teaching/kolman/212.htm

http://shkolo.ru/sintez-belkov-v-kletke/

http://news.discovery.com/tech/gear-and-gadgets/real-hoverboard-makes-its-flying-debut-151228.htm?utm_source=facebook.com&utm_medium=social&utm_campaign=DNewsSocial

https://meduza.io/shapito/2015/08/04/parit-nad-zemley-i-vodoy-kak-rabotaet-hoverbord-lexus-motor

http://akamar.narod.ru/Labirint/labi1.htm

http://myrobot.ru/articles/logo_mazesolving.php

 

5

Location of the lesson:

The case-lesson takes place in the classroom. It is possible to conduct in the museum or the library.

6

Competition:

Teams of boys and girls.The score was:....

Tasks:

1. Create a small maze (5 by 5 cells) using the proposed algorithm. Is the "one hand rule" suitable for your maze?
2. Pass the proposed labyrinth. Who copes with the task quicker is the winner.

 

7

Homework:

Prepare a presentation about one of the mazes (artifact, archaeological, mythological, artwork, unusual task, a maze from computer game etc.) How was it created? What algorithm can be applied for passing (if there is an algorithm).)?

8

Duration:

90 min (double lesson)

9

The possibility to conduct the lesson with a student-understudy 

It is possible.

10

The acquired knowledge and developed competence:

• Knowledge about labyrinths, their types and ways of creation.
• The ability to find, organize and analyze the information quickly.
• The ability to compare the obtained information, to establish the causal connection on the example of labyrinths and algorithms of their creation and passage.

11

Tags:

The labyrinth, Knossos, Minotaur, Ariadne, Theseus, the Golgi complex, mitochondria, DNA, RNA, adenine, thymine, cytosine, valine, algorithm, set, Eller's algorithm, the algorithm of the Lukas-Tremeau.

12

Authors:

Hrabowskaya Larisa Leonidovna

13

Participated in the case upgrading:

 

The End

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