of Konigsberg
The Seven Bridges
... and how a new field in mathematics was born!
The Challenge
The Pregel river flows through the city of Konigsberg (pronounced KER-nigs-burg). The river divided the city into four parts - two islands and two mainlands lying on either side of the river. Seven bridges were built to connect these different parts of the city. The people of Konigsberg would go walking, trying to find a route that crosses all the bridges exactly once and return to the starting point.
Many tried, for many years...
Would you like to give it a try?
Start from anywhere. GO through all the bridges only once and return to the starting point.
(Use the printed card and the erasable marker provided in your kit)
How many routes were you able to find?
None
One
Two
Three
Good job! This is an honest answer.
Check again! Did you retrace your path anywhere?
No way! Are you an alien from the planet Guess-a-Lot?
Did the lie-detector beep now?
Don't worry that you couldn't find a suitable route.. Because such a route does not exist, for this configuration.
And it was conclusively proved by the Swiss polymath Euler (pronounced OY-LUH)
A proof that establishes that no solution exists, is called a Negative Proof, or Proof of Impossibility
Leonhard Euler
Let's see how Euler approached the problem...
He represented this map...
... like this.
It is called a GRAPH
WHAT IS A GRAPH?
Graphs are drawings that are used to show connections or relationships. But 'connection' or 'relationship' can mean different things, even for the same information. For example, let's say we are analysing the air connectivity across 3 cities (Delhi, Mumbai, Chennai), serviced by 3 airlines (FlyFast, AirJet, and SkyGo). FlyFast operates between Delhi and MumbaiAirjet operates between Delhi and ChennaiSkyGo operates between Chennai and Bengaluru
How to build a graph for this information?
There are different ways to do it. What becomes dots and what becomes lines will depend on the question we are trying to answer.
Same data. Different questions --> different graphs!
But this graph below is more suitable to analyse the question: "Which airlines operates the most number of connecting flights?"
We can see that AirJet operates connecting flights in two cities, whereas FlyFast and SkyJet can do so in only one city each.
FlyFast
SkyGo
Airjet
Delhi
Chennai
The graph below would be helpful in analysing the question: "How well are the cities connected by direct flights?"
We can see that Delhi and Chennai are more connected (with two links each), than Mumbai or Bengaluru (which have only one link each).
SkyGo
Delhi
Mumbai
Chennai
Bengaluru
FlyFast
Airjet
Thus Euler invented a whole new flexible method for representing relationships.
This gave birth to GRAPH THEORY, a new field in mathematics!
Click on these portals to get a peek into the world of graph theory. Understanding these terms will help you conquer challenges that are coming up next...
What is an Euler Path? What is an Euler Circuit?
What are Odd nodes? What are Even nodes?
What are Graphs? What are Nodes and Links?
Check out this river that divides a land mass into two (A and B). And there are two bridges across the river.
Figure 1. Here is a graph that represents the picture on the left. The nodes are the land masses. The links are the bridges.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Note down your answers in the printed table given in your kit.
Figure 2. Here is a different graph that represents a totally different scenario.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Note down your answers, FOR THIS AND ALL THE FOLLOWING QUESTIONS, in the printed table given in your kit.
Figure 3. Here is a different graph that represents a totally different scenario.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Figure 4. Here is a different graph that represents a totally different scenario.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Figure 5. Here is a different graph that represents a totally different scenario.
10
How many odd nodes does this graph have?
10
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Figure 6. Here is a different graph that represents a totally different scenario.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Figure 7. Here is a different graph that represents a totally different scenario.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Figure 8. Here is a different graph that represents a totally different scenario.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Figure 9. Here is a different graph that represents a totally different scenario.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
WHAT PATTERNS DO YOU OBSERVE FROM THE TABLE?
What do you notice about the number of odd nodes?
What type of numbers are these?
CLUE #1
Wow, ALL their nodes are...
Their number of odd nodes is EXACTLY...
When do graphs have an Euler path, but not an Euler circuit?
CLUE #2
What is common for graphs which have an Euler circuit?
CLUE #3
TAKEAWAY 1
The number of odd nodes cannot be an odd number !
WHY?? Every link has two ends. One end connects to one node, and the other end connects to another node. So: 1 link → will have 2 endpoints 2 links → 4 endpoints 3 links → 6 endpoints and so on. No matter how many links there are, the total number of endpoints will always be even. It is these endpoints that get distributed among the nodes, to make them odd or even nodes. The way they are distributed (for eg., 6 endpoints) can be a sum of: - only even numbers (like 2+4) OR - a mix of even and odd numbers (like 2+3+1). But the number of odd numbers in this mix will always be an even number, since their sum should result in an even number. So the number of odd nodes can never be an odd number.
All these are EVEN numbers
TAKEAWAY 2
If the graph has exactly two odd nodes, it has an Euler path, but not an Euler circuit .
WHY?? There can be two special places in the route: the START and the END. At the START, we leave the node but we never entered it. At the END, we enter the node but we do not leave it. So these two nodes each can have one unpaired link, which makes them odd nodes. But in all other nodes, their links should get used in pairs: one link to enter and one link to leave. So they should all be even nodes. That means, if there are exactly two odd nodes, there can be an Euler path that starts at one odd node and ends at the other odd node. Since the start and end are different nodes, we cannot return to the starting point. So the graph has an Euler path, but not an Euler circuit.
TAKEAWAY 3
If the graph has more than two odd nodes, it can have neither an Euler path nor an Euler circuit .
WHY?? In the previous slide, we saw that only the START and END of the walk can have one unpaired link, making them odd nodes. That means AT MOST two nodes can be odd. If there are more than two odd nodes, we would need more than one start or end point for the walk, which is impossible. So neither an Euler path nor an Euler circuit can exist in that case.
TAKEAWAY 4
If all the nodes of a graph are even, it will have an Euler path as well as an Euler circuit .
WHY?? If every node in the graph has an even number of links, all those links can be grouped in pairs. This means that whenever we reach a node during our walk, there will always be another unused link to leave through, so we will never get stuck. So we can keep walking and eventually use every link exactly once, which means an Euler path exists. And since there are no odd nodes, there is no special place where the route must start or end. This means we can start at any node and eventually return to the same node, forming an Euler circuit.
There are NO odd nodes
The Konigsberg bridge puzzle was basically an effort to find an Euler circuit.
Konigsberg today
The original seven bridges of Königsberg no longer exist in their 18th-century form. The city itself is now called Kaliningrad, and is located in Russia. Two bridges got destroyed by bombing during World War II. Two were removed for building of new highways. One was damaged, but rebuilt in 2009. Only two of the original bridges survive in the old form. With new additions, there are 5 bridges today, and their configuration allows an Euler path!
Bristol Bridges Walk
Like historical Königsberg, the city of Bristol in England occupies two river banks and two river islands. However, the configuration of the 45 major bridges in Bristol is such that an Euler Circuit exists.
Here are a few bonus puzzles for you...
Since this figure has two odd nodes, the Euler paths must always start at one of these two nodes and end at the other.There are 44 unique such Euler paths possible for the House of Saint Nicholas!! How many did you find? (But Euler circuits are not possible, because all nodes are not even).
Since this figure has more than two odd-nodes, (all four nodes are odd), it has neither an Euler path nor an Euler circuit.
Use this side of the card to provide more information about a topic. Focus on one concept. Make learning and communication more efficient.
You can start at ANY node and trace an Euler circuit back to it! Try it out.
Use this side of the card to provide more information about a topic. Focus on one concept. Make learning and communication more efficient.
Use this side of the card to provide more information about a topic. Focus on one concept. Make learning and communication more efficient.
House of Saint Nicholas
Modified version 1
Modified version 2
How many unique Euler paths exist for this image? In other words, in how many ways can you draw the House of Saint Nicholas in one-stroke without lifting your pen?
Title
Title
Title
What is your prediction for one-stroke drawings of this figure?
What happens now? Did you notice that there are no odd nodes?
Write a brief description here
Write a brief description here
Write a brief description here
... and another.
Starting from anywhere (even from outside the rooms), try and navigate through all 16 doors of this five-room house, without lifting your finger. You should pass through each door exactly once, without retracing your path or crossing your line anywhere.
Now try doing the same with the two rooms below:
What changed??
... and another.
Starting from anywhere (even from outside the rooms), try and navigate through all 16 doors of this five-room house, without lifting your finger. You should pass through each door exactly once, without retracing your path or crossing your line anywhere.
Upload a pic of your Anthotype
Where are graphs used in real life ?
Graphs can be used to represent any scenario where there are connections or relationships or networks. Here are some examples:
Social media - users are connected by being 'friends' or by other actions like commenting or subscribing.
The internet - webpages are connected by links. It is the largest network created by humans.
Bus and train routes - stops/stations are connected by roads/tracks.
Electronic and electrical circuits - components and appliances are connected by wires
Graphs can be used to make models of other things too - like the spread of diseases, structure of languages, molecular structures, etc. Converting these into graphs makes it easier to analyse them and discover useful patterns. For example, apps like Google maps use graph theory concepts to find out the shortest route between two points!
Share your feedback...
Exit
The nodes which have an even number of nodes connected to them are called Even Nodes.
Even Nodes
The nodes which have an odd number of links connected to them are called Odd Nodes.
Odd Nodes
An Euler circuit is an Euler path that starts and ends at the same node.
Euler Circuit
A path that traces every link exactly once is called an Euler Path (though nodes can be revisited many times). It is like drawing a one-stroke diagram without going back on the same line or lifting your pen.
Euler Path
A graph represents how objects are 'related'. The above graph of Konigsberg shows which pairs of land masses are connected by bridges.
Graph
Nodes & Links
Nodes (a.k.a vertices) represent the objects that are being analysed. Links (a.k.a edges) mean that the two nodes at their ends are related.
The Seven-bridge problem
Mathu Shalini
Created on March 2, 2026
Start designing with a free template
Discover more than 1500 professional designs like these:
View
Sorting Cards
View
Interactive Scoreboard
View
Semicircle Mind Map
View
Visual Thinking Checklist
View
Choice Board Flipcards
View
Team Retrospective
View
Fill in the Blanks
Explore all templates
Transcript
of Konigsberg
The Seven Bridges
... and how a new field in mathematics was born!
The Challenge
The Pregel river flows through the city of Konigsberg (pronounced KER-nigs-burg). The river divided the city into four parts - two islands and two mainlands lying on either side of the river. Seven bridges were built to connect these different parts of the city. The people of Konigsberg would go walking, trying to find a route that crosses all the bridges exactly once and return to the starting point.
Many tried, for many years...
Would you like to give it a try?
Start from anywhere. GO through all the bridges only once and return to the starting point.
(Use the printed card and the erasable marker provided in your kit)
How many routes were you able to find?
None
One
Two
Three
Good job! This is an honest answer.
Check again! Did you retrace your path anywhere?
No way! Are you an alien from the planet Guess-a-Lot?
Did the lie-detector beep now?
Don't worry that you couldn't find a suitable route.. Because such a route does not exist, for this configuration.
And it was conclusively proved by the Swiss polymath Euler (pronounced OY-LUH)
A proof that establishes that no solution exists, is called a Negative Proof, or Proof of Impossibility
Leonhard Euler
Let's see how Euler approached the problem...
He represented this map...
... like this.
It is called a GRAPH
WHAT IS A GRAPH?
Graphs are drawings that are used to show connections or relationships. But 'connection' or 'relationship' can mean different things, even for the same information. For example, let's say we are analysing the air connectivity across 3 cities (Delhi, Mumbai, Chennai), serviced by 3 airlines (FlyFast, AirJet, and SkyGo). FlyFast operates between Delhi and MumbaiAirjet operates between Delhi and ChennaiSkyGo operates between Chennai and Bengaluru
How to build a graph for this information?
There are different ways to do it. What becomes dots and what becomes lines will depend on the question we are trying to answer.
Same data. Different questions --> different graphs!
But this graph below is more suitable to analyse the question: "Which airlines operates the most number of connecting flights?"
We can see that AirJet operates connecting flights in two cities, whereas FlyFast and SkyJet can do so in only one city each.
FlyFast
SkyGo
Airjet
Delhi
Chennai
The graph below would be helpful in analysing the question: "How well are the cities connected by direct flights?"
We can see that Delhi and Chennai are more connected (with two links each), than Mumbai or Bengaluru (which have only one link each).
SkyGo
Delhi
Mumbai
Chennai
Bengaluru
FlyFast
Airjet
Thus Euler invented a whole new flexible method for representing relationships.
This gave birth to GRAPH THEORY, a new field in mathematics!
Click on these portals to get a peek into the world of graph theory. Understanding these terms will help you conquer challenges that are coming up next...
What is an Euler Path? What is an Euler Circuit?
What are Odd nodes? What are Even nodes?
What are Graphs? What are Nodes and Links?
Check out this river that divides a land mass into two (A and B). And there are two bridges across the river.
Figure 1. Here is a graph that represents the picture on the left. The nodes are the land masses. The links are the bridges.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Note down your answers in the printed table given in your kit.
Figure 2. Here is a different graph that represents a totally different scenario.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Note down your answers, FOR THIS AND ALL THE FOLLOWING QUESTIONS, in the printed table given in your kit.
Figure 3. Here is a different graph that represents a totally different scenario.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Figure 4. Here is a different graph that represents a totally different scenario.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Figure 5. Here is a different graph that represents a totally different scenario.
10
How many odd nodes does this graph have?
10
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Figure 6. Here is a different graph that represents a totally different scenario.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Figure 7. Here is a different graph that represents a totally different scenario.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Figure 8. Here is a different graph that represents a totally different scenario.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
Figure 9. Here is a different graph that represents a totally different scenario.
How many odd nodes does this graph have?
How many even nodes does this graph have?
NO
YES
Does this graph have an Euler path?
NO
YES
Does this graph have an Euler circuit?
WHAT PATTERNS DO YOU OBSERVE FROM THE TABLE?
What do you notice about the number of odd nodes?
What type of numbers are these?
CLUE #1
Wow, ALL their nodes are...
Their number of odd nodes is EXACTLY...
When do graphs have an Euler path, but not an Euler circuit?
CLUE #2
What is common for graphs which have an Euler circuit?
CLUE #3
TAKEAWAY 1
The number of odd nodes cannot be an odd number !
WHY?? Every link has two ends. One end connects to one node, and the other end connects to another node. So: 1 link → will have 2 endpoints 2 links → 4 endpoints 3 links → 6 endpoints and so on. No matter how many links there are, the total number of endpoints will always be even. It is these endpoints that get distributed among the nodes, to make them odd or even nodes. The way they are distributed (for eg., 6 endpoints) can be a sum of: - only even numbers (like 2+4) OR - a mix of even and odd numbers (like 2+3+1). But the number of odd numbers in this mix will always be an even number, since their sum should result in an even number. So the number of odd nodes can never be an odd number.
All these are EVEN numbers
TAKEAWAY 2
If the graph has exactly two odd nodes, it has an Euler path, but not an Euler circuit .
WHY?? There can be two special places in the route: the START and the END. At the START, we leave the node but we never entered it. At the END, we enter the node but we do not leave it. So these two nodes each can have one unpaired link, which makes them odd nodes. But in all other nodes, their links should get used in pairs: one link to enter and one link to leave. So they should all be even nodes. That means, if there are exactly two odd nodes, there can be an Euler path that starts at one odd node and ends at the other odd node. Since the start and end are different nodes, we cannot return to the starting point. So the graph has an Euler path, but not an Euler circuit.
TAKEAWAY 3
If the graph has more than two odd nodes, it can have neither an Euler path nor an Euler circuit .
WHY?? In the previous slide, we saw that only the START and END of the walk can have one unpaired link, making them odd nodes. That means AT MOST two nodes can be odd. If there are more than two odd nodes, we would need more than one start or end point for the walk, which is impossible. So neither an Euler path nor an Euler circuit can exist in that case.
TAKEAWAY 4
If all the nodes of a graph are even, it will have an Euler path as well as an Euler circuit .
WHY?? If every node in the graph has an even number of links, all those links can be grouped in pairs. This means that whenever we reach a node during our walk, there will always be another unused link to leave through, so we will never get stuck. So we can keep walking and eventually use every link exactly once, which means an Euler path exists. And since there are no odd nodes, there is no special place where the route must start or end. This means we can start at any node and eventually return to the same node, forming an Euler circuit.
There are NO odd nodes
The Konigsberg bridge puzzle was basically an effort to find an Euler circuit.
Konigsberg today
The original seven bridges of Königsberg no longer exist in their 18th-century form. The city itself is now called Kaliningrad, and is located in Russia. Two bridges got destroyed by bombing during World War II. Two were removed for building of new highways. One was damaged, but rebuilt in 2009. Only two of the original bridges survive in the old form. With new additions, there are 5 bridges today, and their configuration allows an Euler path!
Bristol Bridges Walk
Like historical Königsberg, the city of Bristol in England occupies two river banks and two river islands. However, the configuration of the 45 major bridges in Bristol is such that an Euler Circuit exists.
Here are a few bonus puzzles for you...
Since this figure has two odd nodes, the Euler paths must always start at one of these two nodes and end at the other.There are 44 unique such Euler paths possible for the House of Saint Nicholas!! How many did you find? (But Euler circuits are not possible, because all nodes are not even).
Since this figure has more than two odd-nodes, (all four nodes are odd), it has neither an Euler path nor an Euler circuit.
Use this side of the card to provide more information about a topic. Focus on one concept. Make learning and communication more efficient.
You can start at ANY node and trace an Euler circuit back to it! Try it out.
Use this side of the card to provide more information about a topic. Focus on one concept. Make learning and communication more efficient.
Use this side of the card to provide more information about a topic. Focus on one concept. Make learning and communication more efficient.
House of Saint Nicholas
Modified version 1
Modified version 2
How many unique Euler paths exist for this image? In other words, in how many ways can you draw the House of Saint Nicholas in one-stroke without lifting your pen?
Title
Title
Title
What is your prediction for one-stroke drawings of this figure?
What happens now? Did you notice that there are no odd nodes?
Write a brief description here
Write a brief description here
Write a brief description here
... and another.
Starting from anywhere (even from outside the rooms), try and navigate through all 16 doors of this five-room house, without lifting your finger. You should pass through each door exactly once, without retracing your path or crossing your line anywhere.
Now try doing the same with the two rooms below:
What changed??
... and another.
Starting from anywhere (even from outside the rooms), try and navigate through all 16 doors of this five-room house, without lifting your finger. You should pass through each door exactly once, without retracing your path or crossing your line anywhere.
Upload a pic of your Anthotype
Where are graphs used in real life ?
Graphs can be used to represent any scenario where there are connections or relationships or networks. Here are some examples:
Social media - users are connected by being 'friends' or by other actions like commenting or subscribing.
The internet - webpages are connected by links. It is the largest network created by humans.
Bus and train routes - stops/stations are connected by roads/tracks.
Electronic and electrical circuits - components and appliances are connected by wires
Graphs can be used to make models of other things too - like the spread of diseases, structure of languages, molecular structures, etc. Converting these into graphs makes it easier to analyse them and discover useful patterns. For example, apps like Google maps use graph theory concepts to find out the shortest route between two points!
Share your feedback...
Exit
The nodes which have an even number of nodes connected to them are called Even Nodes.
Even Nodes
The nodes which have an odd number of links connected to them are called Odd Nodes.
Odd Nodes
An Euler circuit is an Euler path that starts and ends at the same node.
Euler Circuit
A path that traces every link exactly once is called an Euler Path (though nodes can be revisited many times). It is like drawing a one-stroke diagram without going back on the same line or lifting your pen.
Euler Path
A graph represents how objects are 'related'. The above graph of Konigsberg shows which pairs of land masses are connected by bridges.
Graph
Nodes & Links
Nodes (a.k.a vertices) represent the objects that are being analysed. Links (a.k.a edges) mean that the two nodes at their ends are related.