When I was little (probably 8 years old), I approached my father and asked: “Why is Kaliningrad called the city of seven bridges?” In response, he told me a most interesting story, putting everything into perspective. It was exciting and very educational. Naturally, I no longer remember this story in its original form, but I will try to tell it as excitingly as possible.

As you know, the city of Königsberg, founded in 1255, consisted of three independent urban settlements. They were located on the islands and banks of the Pregel River (now Pregolya), dividing the city into four parts:

• Altstadt;
• Kneiphof;
• Lomze;
• Forstadt.

To connect the city parts, bridges began to be built in the 14th century. Due to the constant military danger from neighboring Poland and Lithuania, the Königsberg bridges began to have a second function - defensive. In front of each of the bridges, a defensive tower was built with lockable lifting or double-leaf gates made of oak and with wrought iron lining. The piers of some bridges had a pentagonal shape, typical of bastions. Inside these supports there were casemates from which it was possible to fire through embrasures.

All seven bridges in Königsberg were drawbridges. Due to the decline of navigation along the Pregola, the bridges were no longer opened. The only exception was the High Bridge, which is raised periodically to prevent the mechanism and wiring of mast ships.

There was a tradition: a guest of the city, in order to subsequently return to Königsberg, had to throw a coin into Pregel from one of the bridges.

Here's an interesting fact for you, associated with tradition: during the cleaning of the Pregolya riverbed with a dredge in the nineties of the 20th century, numismatist collectors literally fought for the right to stand with a sieve at the “gut” from which the bottom silt was pouring out.

And here is the second fact:"The Problem of the Seven Bridges of Königsberg." The famous philosopher and scientist Immanuel Kant, walking along the bridges of the city of Konigsberg, posed a problem: is it possible to walk across all these bridges and at the same time return to the starting point of the route so as to cross each bridge only once. Many have tried to solve this problem both practically and theoretically. But no one succeeded, nor was it possible to prove that it was impossible even theoretically.

In 1736, this problem interested the scientist Leonhard Euler, an outstanding and famous mathematician and member of the St. Petersburg Academy of Sciences. He wrote about this in a letter to his friend, the scientist, Italian engineer and mathematician Marioni dated March 13, 1736. He found a rule, using which he could easily and simply get an answer to this question that interested everyone. In the case of the city of Königsberg and its bridges, this turned out to be impossible. But he managed to create a theory of graphs (mathematicians will understand), which is still used today.

You too can try to solve this problem. Here is a diagram of the city's bridges:

Let's figure out what these seven bridges are.

Krämerbrücke (Bench Bridge).

It is considered the oldest of the seven bridges. It was built in 1286 with the aim of connecting the cities of Altstadt and Kneiphof, and at its entrance there was a statue of Hans Sagan, the son of a Kneiphof shoemaker. The legend said: during the battle between the troops of the Teutonic Order and Lithuania, Hans caught the falling order banner from the hands of a wounded knight.

The bridge got its name from the fact that the adjacent banks of the Pregel, and he himself, were a place of trade.

It was rebuilt in 1900, and in 1972 it was demolished due to the construction of the Estakadny Bridge.

Grünebrücke (Green Bridge).

The Green Bridge was built in 1322 and connected Kneiphof and Forstadt. It got its name from the paint color that was traditionally used to paint the supports and span of the bridge.

In the 17th century, at the Green Bridge, a messenger distributed letters that had arrived in Königsberg. In anticipation of correspondence, business people of the city gathered here and discussed their daily affairs while waiting for mail. According to legend, it was for this reason that the first building of the Königsberg Trade Exchange was built near the Green Bridge in 1623.

In 1875, a new trading exchange building was built on the other side of the bridge, which still stands today. Now this building is the Palace of Culture of Sailors.

In 1907, the bridge was rebuilt, and in 1972 it suffered the same fate as the Lavochny Bridge: they were replaced by the Estakadny Bridge.

Köttelbrücke (Working Bridge).

The working bridge was built in 1337. Connected Kneiphof and Forstadt. Sometimes its name is translated as "Giblet", which is associated with the slaughterhouse located nearby. From where the offal was transported by swimming along the Pregel through this bridge.

Initially, the bridge was a drawbridge and consisted of three spans. In 1621 it was washed away by a flood and was rebuilt without a lifting mechanism.

During the development of Forstadt in 1886, the Workers' Bridge was rebuilt in stone and metal. The divorce function was returned to him.

The bridge burned down during the Great Patriotic War and was demolished along with the bull supports in the 70s of the twentieth century.

Schmiedebrücke (Forge Bridge).

The Forge Bridge was built in 1397. Connected Altstadt and Kneiphof.

Blacksmiths were traditionally located next to this bridge on the banks of the Pregel, and apparently this is where it got its name.

After construction, the bridge took over part of the load from the Lavochny Bridge, located parallel, slightly downstream. It was originally equipped with two stone piers covered with plank spans, which were badly worn out by 1787 and were replaced. In 1896, the Kuznechny Bridge underwent reconstruction and received decorative supports, steel spans and became a drawbridge. On the Altstadt side, a caretaker's tower was built, which housed an installation for raising bridge spans using the water pressure of the city water supply, and controlled the lifting mechanism.

During the Great Patriotic War it was destroyed and was not restored after the war.

Holzbrücke (Wooden Bridge).

The wooden bridge was built in 1404 and connected Altstadt and Lomse.

On it there was a memorial plaque with excerpts from the “Prussian Chronicle” by Albrecht Luhel David. This ten-volume work told about pagan Prussia and the history of the Teutonic Order.

The wooden bridge was reconstructed in 1904 and still exists in this form.

Hohebrücke (High Bridge).

The high bridge was built in 1520, connecting Lomse and Forstadt. In 1882 it was rebuilt, adding the “Bridge Keeper's House” (a room for distributing the bridge raising mechanisms). This neo-Gothic style building still stands today.

The High Bridge was demolished in 1938.

A few tens of meters from the surviving stone pillars of the old High Bridge, a new High Bridge was erected, which still stands today. It has an adjustable middle part for guiding mast ships.

Honigbrücke (Honey Bridge).

The youngest of the seven bridges, connects Lomse and Kneiphof. There are different versions about the origin of the name:

1. Besenrode, a member of the Kneiph Town Hall, paid for the construction of the bridge with barrels of honey.
2. The same Bezenrode paid for the construction of a trading post on the territory beyond the river with barrels of honey.
3. The name comes from the word “Hon”, which means ridicule or mockery. By building this bridge, the residents of Kneiphof gained direct access to the city of Lomse, bypassing the High Bridge, which belonged to Altstadt. Thus, the Honey Bridge became a mockery of the main Königsberg bridge.

Now it has a pedestrian character and leads to Kant Island to the Cathedral and the sculpture park. It is prohibited for private vehicles to travel there.

The father of graph theory (as well as topology) is Euler (1707-1782), who in 1736 solved a problem that was widely known at that time, called the Königsberg bridges problem. In the city of Koenigsberg there were two islands connected by seven bridges to the banks of the Pregolya River and to each other as shown in Figure 4.

The task was as follows: find a route through all four parts of the land that would start from any of them, end on the same part and pass over each bridge exactly once. It is easy, of course, to try to solve this problem empirically, searching through all the routes, but all attempts will end in failure.

Figure 4 - Problem about the Königsberg bridges.

Euler's exceptional contribution to solving this problem is that he proved the impossibility of such a route.

To prove that the problem had no solution, Euler designated each part of the land with a point (vertex), and each bridge with a line (edge) connecting the corresponding points. The result was a count. The statement about the non-existence of a positive solution to this problem is equivalent to the statement about the impossibility of traversing this graph in a special way.

Figure 5 – Graph.

Elements of the graph. Methods for specifying a graph. Subgraphs.

A structure such as a graph as a synonym (the term “network” is also used) has a wide variety of applications in computer science.

CountGcalled system (V, U) ,

Where V={ v} - many elements called peaks graph;

U=={ u} - .set of elements called ribs graph.

Each edge is defined either by a pair of vertices (v1,v2) or by two opposite pairs (v1,v2) and (v2,v1).

If an edge from U is represented by only one pair (v1,v2) , then it's called oriented edge, leading from v1 to v2. In this case, v1 is called the beginning, and v2 is the end of such an edge.

If an edge U is represented by two pairs (v1,v2) and (v2,v1), then U is called unoriented edge. Every undirected edge between vertices v1 and v2 leads both from v1 Vv2, back and forth. In this case, the vertices v1 and v2 are both the beginnings and the ends of this edge. They say that the rib leads like fromv1 inv2, so and fromv2 inv1.

Any two vertices that are connected by an edge are adjacent.

Based on the number of elements, graphs are divided into final And endless.

A graph whose edges are all undirected is called unoriented count.

If the edges of a graph are defined by ordered pairs of vertices, then such a graph is called oriented.

R
Figure 6 – Directed graph.

There are mixed graphs, consisting of both oriented and unoriented edges.

If two vertices are connected by two or more edges, then these edges are called parallel.

If the beginning and end of an edge coincide, then such an edge is called loop .

A graph without loops and parallel edges is called simple.

If an edge is defined by vertices v1 and v2, then edge incident vertices v1 and v2.

A vertex that is not incident to any edge is called isolated.

A vertex incident to exactly one edge and this edge itself are called end, or hanging.

Edges that are associated with the same pair of vertices are called multiple or parallel.

Two vertices of an undirected graph v1 and v2 are called adjacent, if there is an edge (v1,v2) in the graph.

Two vertices of a directed graph v1 and v2 are called adjacent, if they are different and there is an edge leading from vertex v1 to v2.

Let's look at some concepts for a directed graph.

Figure 7 – Directed graph.

Simple way:

Elementary path:

Elementary outline:

Circuit:

For undirected graphs the concepts “simple path”, “elementary path”, “circuit”, “elementary circuit” replace, respectively, the concepts “chain”, “simple circuit”, “cycle”, “simple cycle”. The graph is called coherent, if for any two vertices there is a path (chain) connecting these vertices.

An undirected connected graph without cycles is called tree.

Undirected disconnected graph without cycles - forest.

Figure 8 – Connected graph.

Figure 9 –Forest.

Figure 10 – Tree.

The foundations of graph theory as a mathematical science were laid in 1736 by Leonhard Euler, considering the problem of the Königsberg bridges. Today this task has become a classic one.

Former Koenigsberg (now Kaliningrad) is located on the Pregel River. Within the city, the river washes two islands. Bridges were built from the shores to the islands. The old bridges have not survived, but a map of the city remains, where they are depicted. The Koenigsbergers offered visitors the following task: to cross all the bridges and return to the starting point, and each bridge had to be visited only once.

The Problem of the Seven Bridges of Königsberg

The Seven Bridges of Königsberg Problem or the Königsberger Bridges Problem (German: Königsberger Brückenproblem) is an ancient mathematical problem that asked how one could walk across all seven bridges of Königsberg without crossing any of them twice. It was first solved in 1736 by the German and Russian mathematician Leonhard Euler.

The following riddle has long been common among the residents of Königsberg: how to cross all the bridges (across the Pregolya River) without passing over any of them twice. Many Königsbergers tried to solve this problem both theoretically and practically during walks. However, no one could prove or disprove the possibility of the existence of such a route.

In 1736, the problem of seven bridges interested the outstanding mathematician, member of the St. Petersburg Academy of Sciences, Leonhard Euler, which he wrote about in a letter to the Italian mathematician and engineer Marioni dated March 13, 1736. In this letter, Euler writes that he was able to find a rule, using which it is easy to determine whether it is possible to walk across all the bridges without passing over any of them twice. The answer was “no”.

Solving the problem according to Leonhard Euler

In a simplified diagram of parts of a city (graph), bridges correspond to lines (arcs of the graph), and parts of the city correspond to points connecting lines (vertices of the graph). During his reasoning, Euler came to the following conclusions:

The number of odd vertices (vertices to which an odd number of edges lead) of the graph must be even. There cannot be a graph that has an odd number of odd vertices.
If all the vertices of the graph are even, then you can draw a graph without lifting your pencil from the paper, and you can start from any vertex of the graph and end it at the same vertex.
A graph with more than two odd vertices cannot be drawn with one stroke.
The graph of Königsberg bridges had four (blue) odd vertices (that is, all of them), therefore it is impossible to walk over all the bridges without passing over one of them twice

Graph theory, created by Euler, has found very wide application in transport and communication systems (for example, to study the systems themselves, create optimal routes for delivering goods, or routing data on the Internet).

Further history of the bridges of Königsberg

In 1905, the Imperial Bridge was built, which was later destroyed by bombing during World War II. There is a legend that this bridge was built on the orders of the Kaiser himself, who was unable to solve the problem of the Koenigsberg bridges and became the victim of a joke played on him by the learned minds present at a social reception (if you add the eighth bridge, then the problem becomes solvable). The Jubilee Bridge was built on the pillars of the Imperial Bridge in 2005. At the moment there are seven bridges in Kaliningrad, and the graph built on the basis of the islands and bridges of Kaliningrad still does not have an Euler path.

Or the Seven Bridges of Königsberg Problem - an ancient mathematical problem that asked how one could walk across all seven bridges of Königsberg without crossing any of them twice. It was first solved in 1736 by the mathematician Leonhard Euler , who proved that it was impossible and thus invented Euler cycles .

The following riddle has long been common among the residents of Königsberg: how to cross all the city bridges (across the Pregolya River) without passing over any of them twice. Many Königsbergers tried to solve this problem both theoretically and practically during walks. However, no one could prove or disprove the possibility of the existence of such a route.

In 1736, the problem of seven bridges interested the outstanding mathematician, member of the St. Petersburg Academy of Sciences, Leonhard Euler, which he wrote about in a letter to the Italian mathematician and engineer Marinoni dated March 13, 1736. In this letter, Euler writes that he was able to find a rule, using which it is easy to determine whether it is possible to walk across all the bridges without passing over any of them twice. In this case, the answer was “no”.

Solving the problem according to Leonhard Euler

In a simplified city diagram (graph), bridges correspond to lines (edges of the graph), and parts of the city correspond to points connecting lines (vertices of the graph). During his reasoning, Euler came to the following conclusions:

• The number of odd vertices (vertices to which an odd number of edges lead) of the graph must be even. There cannot be a graph that has an odd number of odd vertices.

• If all the vertices of the graph are even, then you can draw a graph without lifting your pencil from the paper, and you can start from any vertex of the graph and end it at the same vertex.

• If exactly two vertices of the graph are odd, then you can draw a graph without lifting your pencil from the paper, and you can start from any of the odd vertices and end it at another odd vertex.

• A graph with more than two odd vertices cannot be drawn with one stroke.

• The graph of Königsberg bridges had four odd vertices (that is, all of them) - therefore, it is impossible to walk over all the bridges without passing over one of them twice.

But the most interesting thing is that historians believe that there is a person who solved this problem; he was able to cross all the bridges only once, although theoretically, but there was a solution... And this is how it happened...

Kaiser (Emperor) Wilhelm was famous for his simplicity of thinking, directness and soldierly “narrow-mindedness.” One day, while at a social event, he almost became the victim of a joke that the learned minds present at the reception decided to play on him. They showed the Kaiser a map of the city of Königsberg and asked him to try to solve this famous problem, which, by definition, was simply unsolvable.

To everyone’s surprise, the Kaiser asked for a piece of paper and a pen, and at the same time specified that he would solve this problem in just a minute and a half. The stunned scientists could not believe their ears, but ink and paper were quickly found for him. The Kaiser put the paper on the table, took the pen, and wrote: “I order the construction of the eighth bridge on the island of Lomze.” And that’s it: the problem is solved...

So in the city of Königsberg a new one appeared 8th bridge across the river, which was named so - Kaiser Bridge, which was subsequently destroyed by bombing during World War II.

The Jubilee Bridge was built on the pillars of the Imperial Bridge in 2005. As of 2017, there are eight bridges in Kaliningrad.

____________________

A short popular science film that tells how an abstract mathematical theory that originated 300 years ago unexpectedly found its application in modern science.

In 1735, mathematician Leonhard Euler solved the famous riddle of the seven bridges of Königsberg, marking the beginning of a new field of mathematics - graph theory. Initially, no applied significance was seen in the theory, and it remained “purely mathematical”. However, in the 21st century, graph theory is finding its application in many fields of science. With its help, for example, the problem of DNA decryption is solved.

From the bridges of Königsberg to genome assembly

Municipal autonomous educational institution

"Secondary school No. 6" Perm

History of mathematics

The old, old problem about the bridges of Königsberg

Completed by: Zheleznov Egor,

student of 10th grade

Head: Orlova E. V.,

math teacher

2014, Perm

Introduction…………………………………………………………………………………..3

History of Königsberg bridges ……………………………………………………………4

The problem of the seven bridges of Königsberg ……………………………………………………......8

Drawing figures with one stroke……………………………………….12

Conclusion…………………………………………………………………………………15

References………………………………………………………………………………….16

Appendix 1………………………………………………………………………………18

Appendix 2…………………………………………………………………………………22

Appendix 3………………………………………………………………………………23

Appendix 4 ………………………………………………………………………………26

Maintaining

Koenigsberg is the historical name of Kaliningrad, the center of the westernmost region of Russia, famous for its mild climate, beaches and amber products. Kaliningrad has a rich cultural heritage. The great philosopher I. Kant, the storyteller Ernst Theodor Amadeus Hoffmann, the physicist Franz Neumann and many others, whose names are inscribed in the history of science and creativity, once lived and worked here. One interesting problem is connected with Konigsberg, the so-called Konigsberg bridge problem.

The purpose of our research: study the history of the problem of the Königberg bridges, consider its solution, find out the role of the problem in the development of mathematics.

To achieve the goal, it is necessary to solve the following tasks:

study literature on this topic;

systematize the material;

select problems in the solution of which the method of solving the problem of Köntgsberg bridges is used;

compile a bibliographic list of references.

History of bridges in Königsberg

Originated in city ​​of Königsberg (now) consisted of three formally independent urban settlements and several more “settlements” and “villages”. They were located on islands and river banks(now Pregolya), dividing the city into four main parts:, , And . For communication between city parts already in began to build . Due to the constant military danger from neighboring And , as well as due to civil strife between the Königsberg cities (in- there was even a war between the cities, caused by the fact that Kneiphof went over to the side of Poland, and Altstadt and Löbenicht remained loyal) V Königsberg bridges had defensive qualities. In front of each of the bridges, a defensive tower was built with lockable lifting or double-leaf gates made of oak and with wrought iron lining. And the bridges themselves acquired the character of defensive structures. The piers of some bridges had a pentagonal shape, typical of bastions. Inside these supports there were casemates. From the supports it was possible to fire through the embrasures.

The bridges were the site of processions, religious and festive processions, and during the years of the so-called “First Russian Time” (-), when Königsberg briefly became part of the city during the Seven Years' War, religious processions took place across the bridges. Once such a religious procession was even dedicated to the Orthodox holiday of the Blessing of the Waters of the Pregel River, which aroused genuine interest among the residents of Königsberg.

By the end of the 19th century, 7 main bridges were built in Königsberg (Appendix 1).

The oldest of the seven bridges Shopbridge(Krämerbrücke / Krämer-brücke). It was built in 1286. The name of the bridge speaks for itself. The square adjacent to it was a place of lively trade. It connected the two medieval cities of Altstadt and Kneiphof. It was built immediately in stone. In 1900 it was rebuilt and made adjustable. Trams began to run across the bridge. It was heavily damaged during the war, but was restored until it was dismantled in 1972.

The second oldest wasGreen Bridge (Grüne Brücke/Grune-brücke). It was built in. This bridge connected the island of Kneiphof with the southern shore of the Pregel. It was also made of stone and had three spans. In 1907, the bridge was rebuilt, the middle span became movable and trams began to run along it. During the war, this bridge was badly damaged, it was restored, and in 1972 it was dismantled.The name of the bridge comes from the paint color that was traditionally used to paint the supports and span of the bridge. INat the Green Bridge, a messenger distributed letters that had arrived in Königsberg. Business people of the city gathered here in anticipation of correspondence. Here, while waiting for mail, they discussed their affairs. It is not surprising that it is in the immediate vicinity of the Green Bridge inthe Königsberg trading center was built. IN on the other bank of the Pregel, but also in close proximity to the Green Bridge, a new building of the trading exchange was built, which has survived to this day (now the Palace of Culture of Sailors).In 1972, the Estakadny Bridge was built instead of the Green and Lavochny bridges.

After Lavochny and Zeleny was builtWorking bridge (Koettelbrucke / Kettel or Kittel-brücke), also connecting Kneiphof and Forstadt. Sometimes the name is also translated as Giblet Bridge. Both translation options are not ideal, since the German name comes fromand in Russian it means approximately “worker, auxiliary, intended for transporting garbage”, etc. This bridge was built in . It connected the city of Kneiphof with the suburb of Forstadt. The bridge was half stone, and the spans were wooden decks. In 1621, during a severe flood, the bridge was torn off and carried into the river. The bridge was returned to its place. In 1886 it was replaced by a new, steel, three-span, movable one. Trams also ran along it. The bridge was destroyed duringand was not restored later.

Seven bridges of Königsberg - Wikipedia (ru /wikipedia .ord)

Graph theory – website www .ref .by /refs

Appendix 1

Lavochny Bridge

Green Bridge

Giblet Bridge

Kuznechny Bridge

Wooden bridge

High Bridge

Honey Bridge. Side view of

former drawbridge.

Honey Bridge. Remains of the adjustable mechanism.

Kaiser Bridge

Appendix 2

Leonard Euler

N German and Russian mathematician, mechanic and physicist. Born April 15, 1707 in Basel. He studied at the University of Basel (1720–1724), where his teacher was Johann Bernoulli. In 1722 he received the degree of Master of Arts. In 1727 he moved to St. Petersburg, receiving a position as an associate professor at the newly founded Academy of Sciences and Arts. In 1730 he became a professor of physics, in 1733 - a professor of mathematics. During the 14 years of his first stay in St. Petersburg, Euler published more than 50 works. In 1741–1766 worked at the Berlin Academy of Sciences under the special patronage of Frederick II and wrote many essays, covering essentially all sections of pure and applied mathematics. In 1766, at the invitation of Catherine II, Euler returned to Russia. Soon after arriving in St. Petersburg, he completely lost his sight due to cataracts, but thanks to his excellent memory and ability to carry out mental calculations, he was engaged in scientific research until the end of his life: during this time he published about 400 works, the total number of which exceeds 850. Died Euler in St. Petersburg September 18, 1783

Euler's works testify to the extraordinary versatility of the author. His treatise on celestial mechanics “The Theory of the Motion of Planets and Comets” is widely known. Author of books on hydraulics, shipbuilding, artillery. Euler was best known for his research in pure mathematics.

Appendix 3

Tasks

Z
problem 1(problem about the bridges of Leningrad). In one of the halls of the House of Entertaining Science in St. Petersburg, visitors were shown a diagram of the city’s bridges (Fig.). It was necessary to go around all 17 bridges connecting the islands and banks of the Neva, on which St. Petersburg is located. You need to go around so that each bridge is crossed once.

And cutting off the blocks,

Suddenly emerge from the darkness

St. Petersburg canals,

St. Petersburg bridges!

(N. Agnivtsev)

D prove that the required unicursal bypass of all the bridges of St. Petersburg of that time is possible, but cannot be closed, i.e. endV the point from which it began.

Task 2. There are seven islands on the lake, which are connected to each other as shown in the picture. Which island should a boat take travelers to so that they can cross each bridge and only once? Why can't travelers be transported to Island A? 17

Z luck 3. (In search of treasures) .

In Fig. depicts a plan of a dungeon, in one of the rooms of which the knight’s wealth is hidden. To safely enter this room, you must enter through a certain gate into one of the outer rooms of the dungeon, go through all 29 doors in sequence, turning off the alarm. You cannot go through the same doors twice. Determine the number of the room in which the treasure is hidden and the gate through which you need to enter? 20

Z

problem 4. Pavlik, an avid cyclist, drew part of the plan of the area and the village on the chalkboard (Fig. 8), where he lived last summer. According to Pavlik’s story, not far from the village located along the banks of the Oya River, there is a small deep lake fed by underground springs. Oya originates from it, which at the entrance of the village is divided into two separate rivers, connected by a natural channel so that a green sharpwok(in the figure marked with the letterA) with a beach and sports ground. DalekOBehind the village, both rivulets merge to form a wide river. Pavlik claims that when returning on a bicycle from a sportssite located on the island, home (in the picture the letterD ), he passes once over all eight bridges shown on the plan, without ever interrupting his movement. Our experts in the theory of such puzzles have marked with lettersA, B, C, D sections of the village, separated by a river (sections are nodes of the network, bridges are branches), and established that the unicursal route starting atA (odd node), is possible, but it must certainly end in B - in the second odd node, the other two nodesWITH AndD - even. But Pavlik is telling the truth: his route fromA VD really ran along all eight bridges and was unicursal. What's the matter here? What do you think?

Z problem 5 . The English mathematician L. Carroll (author of the world-famous books “Alice in Wonderland”, “Alice Through the Looking Glass”, etc.) loved to ask his little friends a puzzle to walk around a figure (Fig. 9)with a single stroke of the pen and without passing through any section of the contour twice. Crossing lines was allowed. This problem can be solved simply.

Let's complicate it with an additional requirement: with each transition through a node (considering the intersection points of lines in the figure as nodes), the direction of the traversal must change by 90°. (Starting a traversal from any node, you will have to make 23 turns) 6 .

Problem 6 . (Fly in a jar) A fly climbed into a sugar jar. The jar has the shape of a cube. Can a fly sequentially go around all 12 edges of a cube without going over the same edge twice? Jumping and flying from place to place is not allowed. 22

Z problem 7 . The picture shows a bird. Is it possible to draw it with one stroke?

Z problem 8 . OnFig. 10 shows a sketch of one of Euler’s portraits. The artist reproduced it with one stroke of the pen (only the hair is drawn separately). Where in the figure are the beginning and end of the unicursal contour located? Repeat the movement of the artist's pen (hair and dotted lines in the drawing are not includedVdetour route) 6 .

Fig.10

Z

luck 9. Draw the following shapes with one stroke. (Such figures are called unicursal (from the Latin unus - one, cursus - path)).

Appendix 4

Problem solving

1

.

3 . To solve, you need to build a graph, where the vertices are the room numbers and the edges are the doors.

Odd vertices: 6, 18. Since the number of odd vertices = 2, it is possible to safely enter the room with treasures.

You need to start the journey through the gate IN, and finish in room no. 18 .

5. An example of the required bypass is given in the figure

6 . The edges and vertices of the cube form a graph, all 8 vertices of which have a multiplicity of 3 and, therefore, the traversal required by the condition is impossible.

7. Taking the intersection points of the line as the vertices of the graph, we get 7 vertices, only two of which have an odd degree. Therefore, there is an Euler path in this graph, which means that it (i.e., the bird) can be drawn with one stroke. You need to start the path at one odd vertex and end at the other.

8. You need to start the traversal from the odd node in the corner of the right eye and end at the odd node of the eyebrow above the left eye (the dotted lines are not included in the network). All other nodes in the figure are even.

9 .