The Konigsberg Bridge Problem Reading Answers

Aug 13, 2024

The Konigsberg bridge problem Reading Answers is an academic reading answers topic. The Konigsberg bridge problem Reading Answers has a total of 13 IELTS questions in total. In the questions, you have to choose the correct paragraph from the options. In the next section, you have to choose the correct option.

Candidates should read the IELTS Reading passage thoroughly to recognise synonyms, identify keywords, and answer the questions below. IELTS Reading practice papers, which feature topics such as the Konigsberg bridge problem Reading Answers. Candidates can use IELTS reading practice questions and answers to enhance their performance in the reading section.

Section 1

THE KONIGSBERG BRIDGE PROBLEM

A. During the Middle Ages, the city of Konigsberg was an important trading centre and the capital of the German province of East Prussia. Today, however, Konigsberg is less famous for its trading history, than for the role it played in the creation of a new field of mathematics: Graph Theory.

B. Konigsberg once lay where the Russian city of Kaliningrad is now, on both banks of the river Pregel (present-day Pregolya). Between the two sides of the city were two large islands, thus dividing the city into four districts-all of which were connected through seven bridges. At some point in history, the citizens of Konigsberg came up with a game: was it possible, they said, for someone to travel around the city in a complete circuit using all seven bridges, but crossing each one only once?

C. While for most the question was just a silly riddle, for one prominent mathematician of the time, Carl Gottlieb Ehler, it soon became an obsession. No one could find a way to solve the riddle-but, at the same time, Ehler theorised that one could not be certain there was no solution without solid mathematical proof.

D. Unable to find a solution to his conundrum alone, Ehler turned to Swiss mathematician Leonhard Euler for help. Euler was at first dismissive of the Konigsberg Bridge problem, describing it as "banal" and having "little relationship to mathematics". Nevertheless, he too soon found himself tormented by the question. Relenting, he agreed to help his colleague to find a solution. By 1741, Euler had published a paper ('Solutio problematis ad geometriam situs pertinentis') which not only gave a definitively negative answer to the famous Konigsberg Bridge question but also provided a formula that could be used to solve any similar question in the future.

E. So how did Euler's solution work exactly? As he explained in his paper, one possible way to confirm once and for all whether it was possible to cross all bridges once and end up back where you started would've been to just write down all the possible paths by giving codenames both to the bridges (a, b, c, d, e, f, g) and the landmasses they connected (A, B, C, D). Such a tactic, however, would not only be time-consuming and impractical, it would also be impossible for larger-scale problems than the Konigsberg one.

F. To simplify things, then, Euler drew a diagram ('network', or 'graph') which represented the four landmasses ('nodes', or 'vertices') and seven bridges ('edges"). Each node could have an even or odd degree, i.e. an even or odd number of edges connected to it. For a graph to function the way the Konigsberg Bridge Problem wanted it to function, it would need to have either zero or precisely two nodes with an odd degree; any other variation and it would be impossible for anyone to travel through the graph and traverse each edge only once. In the case of Konigsberg, all four nodes had an odd degree, as one landmass had five bridges touching it and the remaining ones had three it was impossible to travel all seven bridges once in a single round trip. Ehler finally had his answer so it was that Graph Theory, or Geometry of Position as Gottfried Wilhelm Leibniz had earlier called it, was born.

G. As Teo Paoletti of the College of New Jersey explains, Graph Theory took off almost immediately once Euler's paper was published, with mathematicians like Augustin Cauchy, Arthur Caley and George Polya working towards discovering "just about everything that is known about large but ordered graphs". It is thanks to Graph Theory, for example, that we can now tell with certainty which order of moves a knight on a chess board should follow to land on each square once and return to his original square. Even today, Graph Theory is a source of numerous conundrums like the Konigsberg one, many of which have yet to receive answers.

H. Unfortunately, the city of Konigsberg was not fated to enjoy its fame for very long. Konigsberg was one of the many German cities targeted by the RAF during the Second World War, receiving its first attack on the night of August 26th 1944 and it's second on the 30th. and leaving behind them nothing but ruins. Nowadays, very little remains of the original city in Kaliningrad besides a few restored historical buildings such as the Dom or Sackheim Gate. Still, Konigsberg remains preserved in the minds of numerous mathematicians who are well aware that without it, Graph Theory might never have existed.

Questions 28-33

Reading Passage 3 has eight paragraphs labelled A.

Which paragraph contains the following information?

NB: You may use any letter more than once.

28. An Imperfect Approach to the Konigsberg Bridge Problem

Answer: E

Supporting statement: “......Such a tactic, however, would not only be time-consuming and impractical, it would also be impossible for larger-scale problems than the Konigsberg one........”

Keywords: impractical, impossible

Keyword Location: para E, Line 3-4

Explanation: This statement shows that the initial method of solving the Konigsberg Bridge Problem by listing all possible paths was recognized as imperfect due to its impracticality and impossibility for larger problems.

29. A similar problem to the Konigsberg Bridge Problem

Answer: G

Supporting statement: “......It is thanks to Graph Theory, for example, that we can now tell with certainty which order of moves a knight on a chess board should follow to land on each square once and return to his original square.......”

Keywords: similar, knight's

Keyword Location: para G, lines 3-4

Explanation: The knight's tour problem is presented as a similar problem to the Konigsberg Bridge Problem, which is solved using Graph Theory, just like the original problem.

30. The original name for Graph Theory

Answer: F

Supporting statement: “......Graph Theory, or Geometry of Position as Gottfried Wilhelm Leibniz had earlier called it, was born........”

Keywords: name, Geometry

Keyword Location: para F, lines 7-8

Explanation: The statement reveals that Graph Theory was originally called "Geometry of Position," providing the historical name for the mathematical field.

31. The paper which answered the Konigsberg Bridge Problem

Answer: B

Supporting statement: “......By 1741, Euler had published a paper ('Solutio problematic ad geometrical situs pertinent) which not only gave a definitively negative answer to the famous Konigsberg.......”

Keywords: Euler, paper

Keyword Location: para D, lines 4-6

Explanation: This statement identifies Euler’s paper, which answered the Konigsberg Bridge Problem and laid the foundation for solving similar problems, confirming the connection between the paper and the solution.

32. The mathematical term for Konigsberg's districts

Answer: F

Supporting statement: “.......Euler drew a diagram ('network', or 'graph') which represented the four landmasses ('nodes', or 'vertices') and seven bridges ('edges')......”

Keywords: nodes, vertices

Keyword Location: para F, lines 2-3

Explanation: The statement explains that Euler used the terms "nodes" or "vertices" to represent the districts of Konigsberg in his diagram, providing the mathematical terminology.

33. The name of a heritage site in Konigsberg

Answer: H

Supporting statement: “.......Nowadays, very little remains of the original city in Kaliningrad besides a few restored historical buildings such as the Dom or Sackheim Gate......”

Keywords: site, historical

Keyword Location: para H, lines 5-6

Explanation: This statement mentions the Dom and Sackheim Gate as restored historical buildings, which are heritage sites in the former city of Konigsberg.

Questions 34-38

Write NO MORE THAN TWO WORDS for each.

34. Konigsberg functioned as a … during Medieval Times.

Answer: TRADING CENTER / CENTRE

Supporting statement: “......During the Middle Ages, the city of Konigsberg was an important trading centre and the capital of the German province of East Prussia........”

Keywords: Ages, trading

Keyword Location: para A, line 1-2

Explanation: This statement clarifies that Konigsberg was known as a trading centre during the Medieval Times, highlighting its economic significance.

35. Due to the geography of the area, Konigsberg consisted of ......

Answer: FOUR DISTRICTS

Supporting statement: “.......Between the two sides of the city were two large islands, thus dividing the city into four districts—all of which were connected through seven bridges......”

Keywords: geography, four

Keyword Location: para B, lines 3-5

Explanation: The paragraph describes the geographical layout of Konigsberg, where the river and islands divide the city into four districts, which are relevant to the structure of the bridge problem.

36. Carl Gottlieb Ehler believed ……. was needed to confirm that the Konigsberg

The bridge problem was unsolvable.

Answer: MATHEMATICAL PROOF

Supporting statement: “.......Ehler theorised that one could not be certain there was no solution without solid mathematical proof......”

Keywords: Ehler, mathematical

Keyword Location: para C, lines 4-5

Explanation: This paragraph highlights Ehler's belief that mathematical proof was essential to confirm the impossibility of solving the Konigsberg Bridge Problem, indicating his logical approach to the riddle.

37. Leonhard Euler was originally unwilling to assist Ehler because he considered the Konigsberg problem ………

Answer: BANAL

Supporting statement: “.......Euler was at first dismissive of the Konigsberg Bridge problem, describing it as 'banal' and having 'little relationship to mathematics'......”

Keywords: Euler, banal

Keyword Location: para D, lines 2-4

Explanation: The paragraph explains that Euler initially found the problem trivial and unworthy of serious mathematical consideration. Hence the use of the word "banal." is correct.

38. To solve the problem, Euler created a simplified map of Konigsberg which he

called a ………....

Answer: NETWORK/ GRAPH

Supporting statement: “.......To simplify things, then, Euler drew a diagram ('network', or 'graph') which represented the four landmasses ('nodes', or 'vertices') and seven bridges ('edges').......”

Keywords: Simplified map, network/graph

Keyword Location: para F, lines 2-4

Explanation: The paragraph explains how Euler created a simplified representation of the city using a "network" or "graph," which became crucial in solving the problem, providing the term he used.

Questions 39-40

Choose the correct letter.

39. Euler proved it was impossible to cross each of Konigsberg's bridges only once in a round trip because

A. there was an odd number of bridges.

B. there were too many bridges connected to each district.

C. all the bridges had an odd degree.

D. all the landmasses had an odd degree.

Answer: D

Supporting statement: “......In the case of Konigsberg, all four nodes had an odd degree, as one landmass had five bridges touching it and the remaining ones had three—so it was impossible to travel all seven bridges once in a single round trip.......”

Keywords: degree, four

Keyword Location: para F, lines 6-7

Explanation: The paragraph explains that all the landmasses in Konigsberg had an odd degree, meaning it was mathematically impossible to cross each bridge only once in a complete circuit, leading to the conclusion that the task was impossible.

40. During World War II,

A. the city of Konigsberg was destroyed by the Germans.

B. the city of Konigsberg was renamed

C. a large number of Konigsberg citizens were killed by Russian soldiers.

D. a few damaged historical buildings were renovated.

Answer: C

Supporting statement: “.......Nowadays, very little remains of the original city in Kaliningrad besides a few restored historical buildings such as the Dom or Sackheim Gate......”

Keywords: War II, restored

Keyword Location: para H, lines 5-6

Explanation: The paragraph mentions that, following the destruction during World War II, only a few historical buildings were restored, indicating that some heritage sites were renovated after the war.

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