The foundations of graph theory as a mathematical science were laid in 1736 by Leonhard Euler, considering the problem of Königsberg bridges. Today, this task has become a classic.
Former Koenigsberg (now Kaliningrad) is located on the Pregel River. Within the city, the river washes two islands. Bridges were thrown from the coast to the islands. The old bridges have not been preserved, but there is a map of the city 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 should have been visited only once.
The problem of the seven bridges of Königsberg
The Problem of the Seven Bridges of Königsberg or the Problem of Königsberg Bridges (German: Königsberger Brückenproblem) is an old mathematical problem that asked how it is possible to pass over all seven bridges of Königsberg without passing through any of them twice. It was first solved in 1736 by the German and Russian mathematician Leonhard Euler.
For a long time, such a riddle has been common among the inhabitants of Königsberg: how to pass through all the bridges (across the Pregolya River) without passing through 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, about which he wrote in a letter dated March 13, 1736, to the Italian mathematician and engineer Marioni. In this letter, Euler writes that he was able to find a rule by which it is easy to determine whether it is possible to pass over all bridges without passing over any of them twice. The answer was "no".
Solving the problem according to Leonhard Euler
On a simplified diagram, parts of the city (graph) correspond to bridges with lines (arcs of the graph), and parts of the city correspond to points of connection of lines (vertices of the graph). In the course of reasoning, Euler came to the following conclusions:
The number of odd vertices (vertices to which an odd number of edges lead) 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 a single stroke.
The graph of Königsberg bridges had four (in blue) odd vertices (i.e. all), therefore it is impossible to pass through all the bridges without passing through any of them twice
The graph theory created by Euler has found very wide application in transport and communication systems (for example, for studying the systems themselves, compiling optimal routes for delivering goods or routing data on the Internet).
Further history of the Königsberg bridges
In 1905, the Imperial Bridge was built, which was subsequently destroyed by bombardment during World War II. There is a legend that this bridge was built by order of the Kaiser himself, who could not solve the problem of Königsberg bridges and became a victim of a joke played with him by the learned minds who were present at the secular 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.
Municipal autonomous educational institution
"Secondary school No. 6", Perm
History of mathematics
The old-old problem about the bridges of Koenigsberg
Completed by: Zheleznov Egor,
10 "a" class student
Head: Orlova E. V.,
mathematic teacher
2014, Perm
Introduction …………………………………………………………………………..3
The history of the bridges of Koenigsberg …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
The problem of the seven bridges of Koenigsberg ………………………………………….......8
Drawing figures with one stroke ……………………………………….12
Conclusion ………………………………………………………………………… 15
References ...…………………………………………………………….16
Annex 1 …………………………………………………………………………………………………………18
Annex 2 ………………………………………………………………………22
Annex 3 ………………………………………………………………………23
Annex 4 ………………………………………………………………………26
Doing
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. An interesting problem is related to Koenigsberg, the so-called problem of the bridges of Koenigsberg.
The purpose of our study: study the history of the emergence of the Königberg bridge problem, consider its solution, and clarify the role of the problem in the development of mathematics.
To achieve the goal, it is necessary to solve the following tasks:
study the literature on the topic;
organize the material;
select tasks in the solution of which the method of solving the problem of Kentgsberg bridges is used;
make a bibliographic list of references.
History of the bridges of Koenigsberg
Arising in city of Königsberg (now) consisted of three formally independent urban settlements and several more “settlements” and “villages”. They were located on the islands and the banks of the river.(now Pregol), 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 , and also 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, while Altstadt and Löbenicht remained loyal) in Königsberg bridges had defensive qualities. In front of each of the bridges, a defensive tower was built with closing lifting or double-leaf gates made of oak and with iron forged upholstery. And the bridges themselves acquired the character of defensive structures. The piers of some bridges had a pentagonal shape typical of bastions. Casemates were located inside these supports. From the supports it was possible to fire through the embrasures.
Bridges were a place of processions, religious and festive processions, and during the years of the so-called "First Russian Time" (-), when Königsberg briefly became part of the Seven Years' War, religious processions passed along the bridges. Once such a procession was even dedicated to the Orthodox feast of the Blessing of the Waters of the Pregel River, which aroused genuine interest among the inhabitants of Königsberg.
By the end of the 19th century, 7 main bridges were built in Königsberg (Appendix 1).
The oldest of the seven bridges Lavochnybridge(Krämerbrücke / Kramer-brücke). It was built in 1286. The very 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 movable. Trams began to run across the bridge. During the war, it was badly damaged, but restored until it was dismantled in 1972.
Was the second oldestGreen bridge (Grüne Brücke / Grüne brücke). It was built in. This bridge connected the island of Kneiphof with the southern bank of the Pregel. It was also stone and three-span. In 1907, the bridge was rebuilt, the middle span became drawable and trams began to run along it. During the war, this bridge was badly damaged, was restored, and in 1972 it was dismantled.The name of the bridge comes from the color of the paint, which was traditionally used to paint the supports and the superstructure of the bridge. ATat the Green Bridge, a messenger handed out letters that had arrived in Königsberg. Business people of the city gathered here in anticipation of correspondence. Here, while waiting for the mail, they discussed their affairs. It is not surprising that it is in the immediate vicinity of the Green Bridge inKönigsberg trading house was built. AT on the other side of the Pregel, but also in the immediate vicinity of 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, instead of the Green and Lavochny bridges, the Trestle Bridge was built.
After Lavochnoye and Green was builtworking bridge (Koettelbrucke / Kettel or Kittelbrücke), also connecting Kneiphof and Vorstadt. Sometimes the name is also translated as Gut Bridge. Both translations are not ideal, since the German name comes fromand in Russian means approximately “working, auxiliary, intended for garbage transportation”, etc. This bridge was built in . It connected the city of Kneiphof with the suburb of Vorstadt. The bridge was half stone, and the spans were wooden decks. In 1621, during a severe flood, the bridge was torn off and swept into the river. The bridge was returned to its place. In 1886 it was replaced with a new, steel, three-span, movable one. Trams also ran along it. The bridge was destroyed duringand did not recover later.
Seven bridges of Koenigsberg - Wikipedia (ru /wikipedia .ord)
Graph theory – site www .ref .by /refs
Appendix 1
shop bridge
green bridge
Gut bridge
Blacksmith bridge
wooden bridge
Honey Bridge. Side view of
former drawbridge.
Honey Bridge. Remains of the draw mechanism.
Kaiser Bridge
Annex 2
Leonhard Euler
H 
German and Russian mathematician, mechanic and physicist. Born April 15, 1707 in Basel. He studied at the University of Basel (in 1720-1724), where his teacher was Johann Bernoulli. In 1722 he received a master's degree in arts. In 1727 he moved to St. Petersburg, receiving a position as an adjunct 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 papers. In 1741–1766 worked at the Berlin Academy of Sciences under the special patronage of Frederick II and wrote many works covering essentially all branches of pure and applied mathematics. In 1766, at the invitation of Catherine II, Euler returned to Russia. Shortly after arriving in St. Petersburg, he completely lost his sight due to cataracts, but thanks to his excellent memory and the ability to perform calculations in his mind, he was engaged in scientific research until the end of his life: during this time he published about 400 works, their total number exceeds 850. Died Euler in Saint Petersburg on 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 is best known for his research in pure mathematics.
Annex 3
Tasks
W 
task 1(problem about the bridges of Leningrad). In one of the halls of the House of Entertaining Science in St. Petersburg, visitors showed a diagram of the city's bridges (Fig.). It was required to bypass all 17 bridges connecting the islands and the banks of the Neva, on which St. Petersburg is located. It is necessary to go around so that each bridge is passed once.
And cutting the quarters
Emerge suddenly from the darkness
St. Petersburg channels,
St. Petersburg bridges!
(N. Agnivtsev)
D 
prove that the required unicursal bypass of all the bridges of St. Petersburg at that time is possible, but cannot be closed, i.e., endin
point from which it started.
Task 2. There are seven islands on the lake, which are interconnected as shown in the figure. Which island should the boat take travelers to so that they can cross each bridge and only once? Why can't travelers be taken to island A? 17
W 
hell 3.
(in search of treasure)
.
On fig. the plan of the dungeon is depicted, in one of the rooms of which the treasures of the knight are hidden. To safely enter this room, you must enter through certain gates into one of the extreme rooms of the dungeon, go through all 29 doors in sequence, turning off the alarm. You can't go through the same doors twice. Determine the number of the room in which treasures are hidden and the gate through which you need to enter? 20
W
hell 4. Pavlik - an avid cyclist - depicted on the blackboard part of the plan of the area and the village (fig. 8), where he lived last summer. According to Pavlik, not far from the village, located on the banks of the Oya River, there is a small deep lake fed by underground springs. Oya originates from it, which, at the entrance, the village is divided into two separate streams, connected by a natural channel so that a green island is formed.wok(in the figure marked with the letterBUT)
with beach and playground. Dalekaboutbehind the village, both streams, merging, form a wide river. Pavlik claims that, returning on a bicycle from a sportssite located on the island, home (in the figure, the letterD
),
he passes once over all eight bridges shown on the plan, never once interrupting the movement. Our connoisseurs of the theory of such puzzles marked with lettersA, B, C,
D
sections of the village, separated by a river (sections are network nodes, bridges are branches), and found that a unicursal route starting atBUT
(odd node), it is possible, but it must certainly end in B - in the second odd node, the remaining two nodesWith
andD
- even. But Pavlik, too, is telling the truth: his route fromBUT
inD
really ran along all eight bridges and was unicursal. What is the matter here? What do you think? W 
hell 5
. The English mathematician L. Carroll (the author of the world-famous books Alice in Wonderland, Alice Through the Looking-Glass, etc.) liked to ask his little friends a puzzle to bypass the figure (Fig. 9)with a single stroke of the pen and without passing twice a single section of the contour. Lines were allowed to cross. Such a task is easily solved.
Let's complicate it with an additional requirement: at each transition through a node (considering the points of intersection of the lines in the figure as nodes), the direction of the bypass must change by 90°. (Starting from any node, you will have to make 23 turns) 6 .
Task 6 . (A fly in a jar) A fly has climbed into a sugar jar. The jar is in the shape of a cube. Will the fly be able to sequentially go around all 12 edges of the cube without passing twice along one edge. Jumping and flying from place to place is not allowed. 22
W 
hell 7
.
The picture shows a bird. Is it possible to draw it with one stroke?
W 
hell 8
.
On theFigure 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 figure are not includedinbypass route) 6
.
Fig.10
W
hell 9. Draw the following figures in one stroke. (Such figures are called unicursal (from the Latin unus - one, cursus - path)).
Appendix 4
Problem solving
1
.
3
. To solve it, you need to build a graph where the vertices are the numbers of the rooms, and the edges are the doors.
Odd Vertices: 6, 18. Since the number of odd vertices = 2, it is safe to enter the treasure room.
You need to start the path through the gate AT 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 multiplicity 3 and, therefore, the bypass required by the condition is impossible.
7. Taking the intersection points of the line as graph vertices, 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 (that is, the bird) can be drawn with one stroke. You need to start the path at one odd vertex, and end at another.
8. You need to start bypassing at the odd node in the corner of the right eye and end at the odd node of the eyebrow above the left eye (dotted lines are not included in the network). All other nodes in the figure are even.
9
.
Or the Seven Bridges of Königsberg Problem, an old mathematical problem that asked how one could cross all seven bridges of Königsberg without crossing any of them twice. It was first solved in 1736 by a mathematician Leonhard Euler , proving that this is impossible, and thus inventing euler cycles .
For a long time, such a riddle has been spread among the inhabitants of Königsberg: how to pass through all the city bridges (across the Pregolya River) without passing through 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, Leonard Euler, about which he wrote 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 by which it is easy to determine whether it is possible to pass over all bridges without passing over any of them twice. In this case, the answer was "no".
Solving the problem according to Leonhard Euler
On a simplified city diagram (graph), bridges correspond to lines (graph edges), and parts of the city correspond to line connection points (graph vertices). In the course of reasoning, Euler came to the following conclusions:
- The number of odd vertices (vertices to which an odd number of edges lead) 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 at any of the odd vertices and end it at another odd vertex.
- A graph with more than two odd vertices cannot be drawn with a single stroke.
- The graph of Königsberg bridges had four odd vertices (that is, all) - therefore, it is impossible to pass through all the bridges without passing through any 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 pass through all the bridges only once, though theoretically, but the solution was .... And this is how it happened...
Kaiser (Emperor) Wilhelm was famous for his simplicity of thinking, directness and soldier's "narrowness". Once, being at a social event, he almost became a victim of a joke that the learned minds present at this reception decided to play with him. They showed the Kaiser a map of the city of Koenigsberg, and asked him to try to solve this famous problem, which, by definition, was simply not solvable.
To everyone's surprise, Kaiser asked for a sheet 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 piece of paper on the table, took up his pen, and wrote: "I order the construction of the eighth bridge on the island of Lomse." And that's it: problem solved...
So in the city of Königsberg a new 8th bridge across the river, which they called - 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. For 2017, there are eight bridges in Kaliningrad.
____________________
A small popular science film that tells how an abstract mathematical theory that originated 300 years ago suddenly found its application in modern science.
In 1735, mathematician Leonhard Euler solved the famous puzzle about the seven bridges of Königsberg, initiating a new area of mathematics - graph theory. Initially, no applied value was seen in the theory, and it remained "purely mathematical". However, in the 21st century, graph theory finds its application in many areas of science. With the help of it, for example, the problem of DNA deciphering is solved.
From Königsberg bridges to genome assembly
The location of the seven bridges, according to legend, was also not chosen by chance, and the number seven has long been considered mystical.
By the way, the tradition of throwing a coin from the bridge in order to return has appeared in Koenigsberg since ancient times.
Once in the old city, I walked along its bridges.
Imperial bridge at the beginning of the 20th century
It is impossible to bypass all the bridges by crossing each one only once. Among the townspeople there was an unsolvable problem - how to pass over all the bridges of Kneiphof without crossing any of them twice.
The problem was solved by Emperor Wilhelm. One day at the ball, the conversation turned to the unsolvable riddle of bridges. The emperor said that he could easily solve this problem and ordered a pen and paper to be brought to him. Wilhelm wrote an order - to build the eighth bridge, which was called the Imperial.
Map of the bridges connecting the islet of Kneiphof with the coast. Seven bridges is a mystical number.
Kneiphof gained fame as the "island of magicians", it was said that bridges in foggy twilight could lead to other worlds. The island is located at the crossroads of these worlds. No wonder they became interested in Hitler's sorcerers.
Only three of the seven bridges have survived to this day. The ghosts of the townspeople of past eras appear here in our days, pass importantly, hurrying about their business. Maybe they are in a hurry from one "parallel world" to another through the island?
Each bridge has its own history and legends.
shop bridge
The oldest bridge in Königsberg, built at the end of the 13th century. Then he connected two settlements - Kneiphof on the island and Altstadt (King's Castle) on the coast. It was originally called St. George's Bridge. The settlements then were not a single city and were even at enmity with each other. The bridge became a no man's land where trade took place. Merchants' tents stood along the bridge, so the people called the bridge Shop. They also sold a strong alcoholic drink called Pregel Stink.
The bridge fell into disrepair over the centuries, was dismantled and rebuilt in 1900 into a drawbridge. During the war, it was badly damaged and was restored by Soviet restorers. Unfortunately, in the seventies, as the “party ordered,” the bridge was demolished, and an overpass passed in its place.
green bridge
Built at the beginning of the 14th century. At first, the bridge was wooden and was called the "Bridge of the Long Street", which ran from the castle to the hospital of St. George. The wooden bridge often burned down and was rebuilt. In the 16th century, the bridge, rebuilt after a fire, was painted green, so it became the "Green Bridge". On this bridge, noble merchants of the city met for negotiations. The bridge was "postal", messengers brought letters here. Dear townspeople came for important mail in person and at the same time met with partners.
In the 17th century, an exchange was built next to the bridge, the current building of which is a rebuilding of the late 19th century.
The bridge was modernized at the beginning of the 20th century. Survived the war, was restored. Unfortunately, it suffered the fate of the Lavochny Bridge, it was destroyed "by order of the party" for the construction of an overpass, which runs right on the site of these two bridges.
Green Bridge at the beginning of the 20th century
Exchange building and Green Bridge at the beginning of the 20th century
Overpass, which passes on the site of the Lavochny and Green Bridge
View from part of the overpass (former green bridge) to the stock exchange
Offal (Working) bridge
Built in the second half of the 14th century, next (50 meters) to the green bridge. The bridge was used to transport goods. In the 17th century, on Easter 1621, a terrible flood occurred in Königsberg, which flooded the island of Kneiphof. According to contemporaries "ships were thrown onto the ramparts of the city, rats swam on floating coffins, and in the cathedral the water was knee-deep". During the flood, the bridge was destroyed, hastily rebuilt. Completely rebuilt at the end of the 19th century. The bridge did not survive the war.
Previously, 50 meters here was the Worker Bridge
Königsberg Cathedral, once there was a bridge nearby
Blacksmith bridge
Built in the second half of the 14th century, it was also wooden at first. It got its name thanks to the forges located nearby. It was rebuilt at the end of the 19th century with an adjustable mechanism. Nearby there was a turret, in which there was a "control point" for the bridge.
The bridge was destroyed during the war.
wooden bridge
Built at the beginning of the 15th century. On the bridge was a memorial plaque with quotations from the Prussian Chronicle. Rebuilt at the beginning of the 20th, preserved to this day. Even the pillars of the bridge have been preserved.
The bridge has survived to this day
high bridge
Built at the beginning of the 16th century. A legend about the most "truthful" Baron Munchausen and his lost boot is associated with him. Once, after sorting through the local noble beer, the baron wandered into the High Bridge area. He could not find his home, so he stopped for the night at a nearby hotel. The room turned out to be so small that the baron, when he lay down, could not fit in his full height. He stretched his legs out the open window. Without taking off his boots, the baron fell asleep. In the morning, Munchausen discovered that one of his boots had fallen into the water of the river.
The famous resourceful Baron Munchausen became a legend of Koenigsberg
The bridge was rebuilt in the early 19th century.
The high bridge is no longer so beautiful today, but it has been preserved
And in this turret there is a mechanism for drawing the bridge
honey bridge
Built in the second half of the 16th century.
Several legends are associated with the name of the bridge. According to one version, the bridge was built by the "honey tycoon" of that era to connect Kneiphof with his honey shop on the banks of the Lomse. To do this, he even gave a bribe to the mayor of Kneiphof with barrels of honey. According to another version, the tycoon bought the whole bridge for honey. There is a version that the builders of the bridge were paid with honey. Residents of the neighboring area - Altstadt, who did not like Kneiphof, nicknamed its inhabitants - honey slimes.
Romantic legends are associated with the bridge: “If you carry your beloved girl three times in your arms across the Honey Bridge, circle her three times on each bank and finish the cycle on the banks of the Kneiphof without dropping her from your hands, then she will love you forever”
Honey bridge today
Imperial Bridge
This bridge was built in 1905 by order of Emperor Wilhelm, who solved the riddle of the "seven bridges" in this way. The bridge was destroyed during the war. In 2005, a new bridge was built on its supports in honor of the city's anniversary, which was named Yubileiny.
This is how the bridge looked at the beginning of the 20th century
New jubilee bridge
View of the Jubilee Bridge
7 bridges of the city of Kaliningrad (Köningsberg) led to the creation by Leonhard Euler of the so-called graph theory.
A graph is a certain number of nodes (vertices) that are connected by edges. Two islands and banks on the Pregel River, where he stood, were connected by 7 bridges. The famous philosopher and scientist I. Kant, walking along the bridges of Koenigsberg, came up with a problem that is known to everyone in the world as the problem of "7 Koenigsberg bridges": is it possible to pass through all these bridges and at the same time return to the starting point of the route in such a way as to pass along each bridge only once?
Many have tried to solve this problem both practically and theoretically. But no one succeeded. Therefore, it is believed that in the 17th century, the inhabitants started a special tradition: walking around the city, go through all the bridges only once. But, of course, no one succeeded.
In 1736, the scientist Leonhard Euler, who was an outstanding and famous mathematician and a member of the St. Petersburg Academy of Sciences, became interested in this problem. He was able to find a rule by which this riddle could be solved. In the course of his judgments, Euler drew the following conclusions: 1. 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. 2. 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. 3. A graph with more than 2 odd vertices cannot be drawn in one stroke.
From this follows the conclusion that it is impossible to cross all seven bridges without crossing any of them twice. Subsequently, this graph theory became the basis for the design of communication and transport systems, became widely used in programming, computer science, physics, chemistry and many other sciences and fields.
It is noteworthy that historians believe that there is a person who solved this problem, that he was able to pass through all the bridges only once, though theoretically ....
And it was like that. Kaiser (i.e. Emperor) Wilhelm was famous for his simplicity of thinking, directness and "narrowness". Once, he almost became a victim of a joke played on him by learned minds - pranksters showed the Kaiser a map of the city of Königsberg and asked him to try to solve this famous problem, which, by definition, was unsolvable. But Kaiser only asked for a sheet and a pen, while specifying that he would solve it in just 1.5 minutes. Scientists were amazed - Wilhelm wrote: "I order the construction of the eighth bridge on the island of Lomse." That's all, the problem is solved... So in Kaliningrad, a new eighth bridge across the river, named after the Kaiser, appeared. And even a child can solve the problem with eight bridges...