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Standard problems on backtracking

• The Knight's tour problem
• Rat in a Maze
• N Queen Problem
• Subset Sum Problem using Backtracking
• M-Coloring Problem
• Algorithm to Solve Sudoku | Sudoku Solver
• Magnet Puzzle
• Remove Invalid Parentheses
• A backtracking approach to generate n bit Gray Codes
• Permutations of given String

Easy Problems on Backtracking

• Print all subsets of a given Set or Array
• Check if a given string is sum-string
• Count all possible Paths between two Vertices
• Find all distinct subsets of a given set using BitMasking Approach
• Find if there is a path of more than k length from a source
• Print all paths from a given source to a destination
• Print all possible strings that can be made by placing spaces

Medium prblems on Backtracking

• 8 queen problem
• Combinational Sum
• Warnsdorff's algorithm for Knight’s tour problem
• Find paths from corner cell to middle cell in maze
• Find Maximum number possible by doing at-most K swaps
• Rat in a Maze with multiple steps or jump allowed
• N Queen in O(n) space

Hard problems on Backtracking

• Power Set in Lexicographic order
• Word Break Problem using Backtracking
• Partition of a set into K subsets with equal sum
• Longest Possible Route in a Matrix with Hurdles
• Find shortest safe route in a path with landmines
• Printing all solutions in N-Queen Problem
• Print all longest common sub-sequences in lexicographical order
• Top 20 Backtracking Algorithm Interview Questions

The Knight’s tour problem

Backtracking is an algorithmic paradigm that tries different solutions until finds a solution that “works”. Problems that are typically solved using the backtracking technique have the following property in common. These problems can only be solved by trying every possible configuration and each configuration is tried only once. A Naive solution for these problems is to try all configurations and output a configuration that follows given problem constraints. Backtracking works incrementally and is an optimization over the Naive solution where all possible configurations are generated and tried. For example, consider the following Knight’s Tour problem. 

Problem Statement: Given a N*N board with the Knight placed on the first block of an empty board. Moving according to the rules of chess knight must visit each square exactly once. Print the order of each cell in which they are visited.

The path followed by Knight to cover all the cells Following is a chessboard with 8 x 8 cells. Numbers in cells indicate the move number of Knight. 

Let us first discuss the Naive algorithm for this problem and then the Backtracking algorithm.

Naive Algorithm for Knight’s tour   The Naive Algorithm is to generate all tours one by one and check if the generated tour satisfies the constraints. 

Backtracking works in an incremental way to attack problems. Typically, we start from an empty solution vector and one by one add items (Meaning of item varies from problem to problem. In the context of Knight’s tour problem, an item is a Knight’s move). When we add an item, we check if adding the current item violates the problem constraint, if it does then we remove the item and try other alternatives. If none of the alternatives works out then we go to the previous stage and remove the item added in the previous stage. If we reach the initial stage back then we say that no solution exists. If adding an item doesn’t violate constraints then we recursively add items one by one. If the solution vector becomes complete then we print the solution.

Backtracking Algorithm for Knight’s tour  

Following is the Backtracking algorithm for Knight’s tour problem. 

Following are implementations for Knight’s tour problem. It prints one of the possible solutions in 2D matrix form. Basically, the output is a 2D 8*8 matrix with numbers from 0 to 63 and these numbers show steps made by Knight.   

Time Complexity :  There are N 2 Cells and for each, we have a maximum of 8 possible moves to choose from, so the worst running time is O(8 N^2 ).

Auxiliary Space: O(N 2 )

Important Note: No order of the xMove, yMove is wrong, but they will affect the running time of the algorithm drastically. For example, think of the case where the 8th choice of the move is the correct one, and before that our code ran 7 different wrong paths. It’s always a good idea a have a heuristic than to try backtracking randomly. Like, in this case, we know the next step would probably be in the south or east direction, then checking the paths which lead their first is a better strategy.

Note that Backtracking is not the best solution for the Knight’s tour problem. See the below article for other better solutions. The purpose of this post is to explain Backtracking with an example.  Warnsdorff’s algorithm for Knight’s tour problem

References:  http://see.stanford.edu/materials/icspacs106b/H19-RecBacktrackExamples.pdf   http://www.cis.upenn.edu/~matuszek/cit594-2009/Lectures/35-backtracking.ppt   http://mathworld.wolfram.com/KnightsTour.html   http://en.wikipedia.org/wiki/Knight%27s_tour    

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The Knight's Tour Problem (using Backtracking Algorithm)

• Jun 30, 2023
• 6 Minute Read
• Why Trust Us We uphold a strict editorial policy that emphasizes factual accuracy, relevance, and impartiality. Our content is crafted by top technical writers with deep knowledge in the fields of computer science and data science, ensuring each piece is meticulously reviewed by a team of seasoned editors to guarantee compliance with the highest standards in educational content creation and publishing.
• By Vedanti Kshirsagar

Ever wondered how a computer playing as a chess player improves its algorithm to defeat you in the game? In this article, we will learn about the Knight's Tour Problem, the various ways to solve it along with their time and space complexities. So, let's get started!

What is Knight's Tour Problem?

Just for a reminder, the knight is a piece in chess that usually looks like a horse and moves in an L-shaped pattern. This means it will first move two squares in one direction and then one square in a perpendicular direction.

The Knight's Tour problem is about finding a sequence of moves for the knight on a chessboard such that it visits every square on the board exactly one time. It is a type of Hamiltonian path problem in graph theory, where the squares represent the vertices and the knight's moves represent the edges of the graph.

This problem has fascinated many mathematicians for centuries, but the solutions they found were very difficult. The simple solution will be to find every conceivable configuration and selecting the best one is one technique to solve this problem. But that will take a load of time.

One popular solution to solve the Knight's tour problem is Warnsdorff's rule, which involves choosing the next move that leads to a square with the fewest available moves. There is also a backtracking approach. 

 But first moving to all that, let's take a quick understanding of the Hamiltonian path problem.

Hamiltonian Path Problem

The Hamiltonian path problem is a well-known problem in graph theory that asks whether a given graph contains a path that visits every vertex exactly once.

A path that visits every vertex exactly once is called a Hamiltonian path, and a graph that contains a Hamiltonian path is called a Hamiltonian graph. 

Let's take an example of the Hamiltonian path problem. Suppose we have a graph with five vertices and the following edges:

This graph can be represented as:

Knight's Tour Backtracking Algorithm

The backtracking algorithm works by exploring all possible moves for the knight, starting from a given square, and backtracking to try different moves if it reaches a dead end.

Here's the basic outline of the backtracking algorithm to solve the Knight's tour problem:

• Choose a starting square for the knight on the chessboard.
• Mark the starting square as visited.
• For each valid move from the current square, make the move and recursively repeat the process for the new square.
• If all squares on the chessboard have been visited, we have found a solution. Otherwise, undo the last move and try a different move.
• If all moves have been tried from the current square and we have not found a solution, backtrack to the previous square and try a different move from there.
• If we have backtracked to the starting square and tried all possible moves without finding a solution, there is no solution to the problem.

We have given the full C++ program for Backtracking Algorithm to solve Knight's Tour Problem below:

Check the image below before we explain the code:

In this implementation, we first define a function validMoves  which takes the current row and column of the knight as input and returns a vector of pairs representing all the valid moves that the knight can make from that position.

The solve function is a recursive function that takes the current row and column of the knight, as well as the current move number, as input. We mark the current square as visited by setting board[row][column] it to the current move number, and then we recursively try all possible moves from the current position.

If we reach the last move, then we found a solution and return true. If no valid move is found from the current position, we backtrack by setting the current square to 0 and returning false.

In the main function, we start the solve function at a specified starting position and then output the number of moves it took to find a solution and print the final chessboard with the solution.

Time & Space Complexity for Backtracking

The backtracking algorithm used to solve the Knight's Tour problem has an exponential time complexity. The number of possible paths for the knight grows very quickly as the size of the chessboard increases, which means that the time taken to explore all possible paths grows exponentially.

The exact time complexity of the Knight's Tour Backtracking algorithm is O(8^(n^2)), where n is the size of the chessboard. This is because each move has a maximum of 8 possible directions, and we have to explore all possible moves until we find a solution.

The space complexity of the backtracking algorithm is O(n^2), where n is the size of the chessboard. So, we can say that the backtracking algorithm is efficient for smaller chessboards.

Warnsdorff's Rule

Warndorff's rule is a heuristic greedy algorithm used to solve this problem. It tries to move the knight to the square with the fewest valid moves, hoping that this will lead to a solution.

Here's an overview of how Warndorff's rule algorithm works:

• Start with a random square on the chessboard.
• From the current square, consider all possible moves and count the number of valid moves from each adjacent square.
• Move to the square with the lowest number of valid moves. In case of a tie, move to the square with the lowest number of valid moves from its adjacent squares.
• Repeat steps 2 and 3 until all squares on the chessboard have been visited.

Here is the pseudocode for Warndorff's rule algorithm:

The time complexity of Warndorff's rule algorithm is O(n^2 log n), where n is the size of the chessboard. This is because we have to visit each square once, and for each square, we have to compute the number of valid moves from each adjacent square. The space complexity of the algorithm is O(n^2) since we need to store the chessboard and the current position of the knight.

Overall, The Knight's Tour problem is related to chess, and solving it can help chess players improve their strategy and become better at the game. In the real world, you can also use it to design efficient algorithms for finding the shortest path between two points in a network.

Now we know The Knight's Tour Problem and its solutions using Backtracking and Warnsdorff's Rule. It has several applications in various fields such as Game theory, Network Routing etc, making it an important problem to study in computer science and mathematics.

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Knight's tour problem

What is knight's tour problem.

In the knight's tour problem, we are given with an empty chess board of size NxN, and a knight. In chess, the knight is a piece that looks exactly like a horse. Assume, it can start from any location on the board. Now, our task is to check whether the knight can visit all of the squares on the board or not. When it can visit all of the squares, then print the number of jumps needed to reach that location from the starting point.

There can be two ways in which a knight can finish its tour. In the first way, the knight moves one step and returns back to the starting position forming a loop which is called a closed tour . In the second way i.e. the open tour , it finishes anywhere in the board.

For a person who is not familiar with chess, note that the knight moves in a special manner. It can move either two squares horizontally and one square vertically or two squares vertically and one square horizontally in each direction. So, the complete movement looks like English letter 'L' .

Suppose the size of given chess board is 8 and the knight is at the top-left position on the board. The next possible moves are shown below −

Each cell in the above chess board holds a number, that indicates where to start and in how many moves the knight will reach a cell. The final values of the cell will be represented by the below matrix −

Remember, this problem can have multiple solutions, the above matrix is one possible solution.

One way to solve the knight's tour problem is by generating all the tours one by one and then checking if they satisfy the specified constraint or not. However, it is time consuming and not an efficient way.

Backtracking Approach to Solve Knight's tour problem

The other way to solve this problem is to use backtracking. It is a technique that tries different possibilities until a solution is found or all options are tried. It involves choosing a move, making it, and then recursively trying to solve the rest of the problem. If the current move leads to a dead end, we backtrack and undo the move, then try another one.

To solve the knight's tour problem using the backtracking approach, follow the steps given below −

Start from any cell on the board and mark it as visited by the knight.

Move the knight to a valid unvisited cell and mark it visited. From any cell, a knight can take a maximum of 8 moves.

If the current cell is not valid or not taking to the solution, then backtrack and try other possible moves that may lead to a solution.

Repeat this process until the moves of knight are equal to 8 x 8 = 64.

In the following example, we will practically demonstrate how to solve the knight's tour problem.

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Knight's Tour Visualization

Tip: An n * n chessboard has a closed knight's tour iff n ≥ 6 is even.

Note: The pieces of chess are placed inside a square, while the pieces of Chinese chess are placed on the intersections of the lines.

Board size: (Board size should be an even number).

Time of a stroke (ms):

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Standard problems on backtracking

• The Knight's tour problem
• Rat in a Maze
• N Queen Problem
• Subset Sum Problem using Backtracking
• M-Coloring Problem
• Algorithm to Solve Sudoku | Sudoku Solver
• Magnet Puzzle
• Remove Invalid Parentheses
• A backtracking approach to generate n bit Gray Codes
• Permutations of given String

Easy Problems on Backtracking

• Print all subsets of a given Set or Array
• Check if a given string is sum-string
• Count all possible Paths between two Vertices
• Find all distinct subsets of a given set using BitMasking Approach
• Find if there is a path of more than k length from a source
• Print all paths from a given source to a destination
• Print all possible strings that can be made by placing spaces

Medium prblems on Backtracking

• 8 queen problem
• Combinational Sum
• Warnsdorff's algorithm for Knight’s tour problem
• Find paths from corner cell to middle cell in maze
• Find Maximum number possible by doing at-most K swaps
• Rat in a Maze with multiple steps or jump allowed
• N Queen in O(n) space

Hard problems on Backtracking

• Power Set in Lexicographic order
• Word Break Problem using Backtracking
• Partition of a set into K subsets with equal sum
• Longest Possible Route in a Matrix with Hurdles
• Find shortest safe route in a path with landmines
• Printing all solutions in N-Queen Problem
• Print all longest common sub-sequences in lexicographical order
• Top 20 Backtracking Algorithm Interview Questions
• DSA to Development Course

The Knight’s tour problem

Backtracking is an algorithmic paradigm that tries different solutions until finds a solution that “works”. Problems that are typically solved using the backtracking technique have the following property in common. These problems can only be solved by trying every possible configuration and each configuration is tried only once. A Naive solution for these problems is to try all configurations and output a configuration that follows given problem constraints. Backtracking works incrementally and is an optimization over the Naive solution where all possible configurations are generated and tried. For example, consider the following Knight’s Tour problem. 

Problem Statement: Given a N*N board with the Knight placed on the first block of an empty board. Moving according to the rules of chess knight must visit each square exactly once. Print the order of each cell in which they are visited.

The path followed by Knight to cover all the cells Following is a chessboard with 8 x 8 cells. Numbers in cells indicate the move number of Knight. 

Let us first discuss the Naive algorithm for this problem and then the Backtracking algorithm.

Naive Algorithm for Knight’s tour   The Naive Algorithm is to generate all tours one by one and check if the generated tour satisfies the constraints. 

Backtracking works in an incremental way to attack problems. Typically, we start from an empty solution vector and one by one add items (Meaning of item varies from problem to problem. In the context of Knight’s tour problem, an item is a Knight’s move). When we add an item, we check if adding the current item violates the problem constraint, if it does then we remove the item and try other alternatives. If none of the alternatives works out then we go to the previous stage and remove the item added in the previous stage. If we reach the initial stage back then we say that no solution exists. If adding an item doesn’t violate constraints then we recursively add items one by one. If the solution vector becomes complete then we print the solution.

Backtracking Algorithm for Knight’s tour  

Following is the Backtracking algorithm for Knight’s tour problem. 

Following are implementations for Knight’s tour problem. It prints one of the possible solutions in 2D matrix form. Basically, the output is a 2D 8*8 matrix with numbers from 0 to 63 and these numbers show steps made by Knight.   

Time Complexity :  There are N 2 Cells and for each, we have a maximum of 8 possible moves to choose from, so the worst running time is O(8 N^2 ).

Auxiliary Space: O(N 2 )

Important Note: No order of the xMove, yMove is wrong, but they will affect the running time of the algorithm drastically. For example, think of the case where the 8th choice of the move is the correct one, and before that our code ran 7 different wrong paths. It’s always a good idea a have a heuristic than to try backtracking randomly. Like, in this case, we know the next step would probably be in the south or east direction, then checking the paths which lead their first is a better strategy.

Note that Backtracking is not the best solution for the Knight’s tour problem. See the below article for other better solutions. The purpose of this post is to explain Backtracking with an example.  Warnsdorff’s algorithm for Knight’s tour problem

References:  http://see.stanford.edu/materials/icspacs106b/H19-RecBacktrackExamples.pdf   http://www.cis.upenn.edu/~matuszek/cit594-2009/Lectures/35-backtracking.ppt   http://mathworld.wolfram.com/KnightsTour.html   http://en.wikipedia.org/wiki/Knight%27s_tour   Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.  

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Here are 83 public repositories matching this topic..., mohamedmahmoudhassan / backtracking-visualizer.

Visualization of some standard problems solved by Backtracking

• Updated May 30, 2024

sgalal / knights-tour-visualization

An online Knight's tour visualizer using divide and conquer algorithm

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NiloofarShahbaz / knight-tour-neural-network

Implementation Of Knight Tour Problem Using Neural Networks

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NikSWE / warnsdorff-algorithm-visualizer

warnsdorff's algorithm to solve the knight's tour problem.

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mrugendrakul / Knight-s-tour

Gives 3D visualization on knight problem using backtracking

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kamoloff / N-Queen-Puzzle-Knights-Tour

Solution for both N Queens Puzzle and Knight's Tour (with GUI)

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LucasAlv3s / Computer-Intelligence

• Updated Nov 24, 2021
• Jupyter Notebook

bellmcp / Knights-Tour-Solver

A Java implementation of the Knight's Tour algorithm.

• Updated Aug 26, 2019

ognjenvucko / knights-tour

Knight's Tour puzzle 4x8 solver in JavaScript

• Updated Mar 11, 2017

Carleslc / Puzzles

Different puzzles to think and enjoy programming.

• Updated Feb 14, 2022

AliRn76 / Knights-Tour

Knights Tour Problem

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veysiertekin / knights-tour

A complete solution with heuristic & non-heuristic ways to knights-tour problem in chess

• Updated Aug 9, 2017

Rocksus / Knight-Tour-Genetic-Algorithm-With-Repair

A Knight Tour Genetic Algorithm simulation with p5js javascript library.

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NiloofarShahbaz / knight-tour-warnsdroff

Implementation Of Knight Tour Problem Using Warnsdroff Rule

• Updated Jul 20, 2019

akshaybahadur21 / KnightsTour

Knight's tour algorithm for humans ♞

• Updated Jun 4, 2021

gkhays / knights-tour

Visualization of the Knight's Tour move sequence

• Updated Feb 5, 2018

0x0584 / Knight-Problem

♞ Knight's tour problem

• Updated Sep 25, 2016

nachiketbhuta / knight-tour

Graphical Representation of Knight Tour using Warnsdorff's Algorithm

• Updated Nov 3, 2019

jfjlaros / hamilton

Hamiltonian path and cycle finder.

• Updated Apr 8, 2022

cyrillkuettel / knights-tour

Two different methods are being utilized to solve this famous puzzle. They are backtracking and genetic algorithms. The second approach is still in development. Backtracking method seems to be faster, but GA delivers more consistent results for a diverse palette of starting positions. All in all, a very interesting project to me personally.

• Updated Aug 20, 2022

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FIFA World Cup 26 format FIFA World Cup 26 format

The FIFA Council also unanimously made the decision to approve the proposed amendment to the World Cup 2026 group stage format from 16 groups of three, to 12 groups of four.

The top two teams from each group and the eight best third-place teams from the groups will now progress to the round of 32. Hopefuls will now have to play eight games from group stage through to the final, rather than seven.

FIFA World Cup 26 host countries FIFA World Cup 26 host countries

For the first time in history, the organisation of the biggest football tournament on the planet will be divided among three countries. In total, 16 cities will host matches. Most of them (11) are located in the USA: Seattle, San Francisco, Los Angeles, Dallas, Houston, Kansas City, Philadelphia, Atlanta, Miami, Boston and New York. The Mexican headquarters will be in Monterrey, Guadalajara and Mexico City. Finally, the Canadian venues are Vancouver and Toronto. Read more about the host nations and cities for the 2026 World Cup .

FIFA World Cup 26 participating teams FIFA World Cup 26 participating teams

Also for the first time, the 2026 World Cup will see the participation of 48 teams, 16 more than the last seven editions of the tournament since 1998. The division of places by continental confederations has already been defined:

AFC (Asia): Eight direct spots + one inter-confederation play-off place CAF (Africa): Nine direct spots + one inter-confederation play-off place Concacaf (North and Central America, plus the Caribbean): Six direct spots + two inter-confederation play-off places CONMEBOL (South America): Six direct spots + one inter-confederation play-off place OFC (Oceania) : One direct spot + one inter-confederation play-off place UEFA (Europe) : 16 direct spots

The three host countries will automatically qualify for the tournament, thus occupying three of the Concacaf slots.

FIFA World Cup 26 qualifiers FIFA World Cup 26 qualifiers

Each continental confederation has the autonomy to define the competition model and the period of their qualifying tournaments. Hosts Canada, Mexico and USA are already confirmed as qualifiers for FIFA World Cup 26 .

Find out more about the FIFA World Cup 26 qualifiers Find out more about the FIFA World Cup 26 qualifiers

Fifa world cup 26 stadiums fifa world cup 26 stadiums.

A total of 16 stadiums have been chosen to host the games, the most since Korea/Japan 2002, with some truly spectacular venues set to showcase the best in the beautiful game.

FIFA World Cup 26 Tickets FIFA World Cup 26 Tickets

To receive information on how to apply for FIFA World Cup 26™ tickets, you can register your interest here . We will keep you up to date with the latest news and information regarding FIFA World Cup 26™ tickets straight to your mailbox.

FIFA World Cup 26 Official Brand FIFA World Cup 26 Official Brand

The FIFA World Cup™ Trophy, the most widely recognised and prized sporting asset globally, was unveiled at the forefront of the FIFA World Cup 26™ Official Brand on Wednesday, 17 May 2023.

Romania coach Edward Iordănescu on his coaching journey, Romania's EURO 2024 dream and Ianis Hagi – interview

Thursday, June 6, 2024

Article summary

"We brought honour to Romania, but we still have a lot to do," says Edward Iordănescu as he prepares to follow in his father's footsteps by leading the national team at a EURO.

Article top media content

Article body.

Edward Iordănescu's father Anghel was one of the stars of the Steaua Bucureşti side that won the European Cup in 1986. He led the club to another European Cup final as coach in 1989 before guiding the national team to the quarter-finals of the 1994 FIFA World Cup.

Edward had quite an act to live up to, but since taking on the Romania job in January 2022 he has carved out his own niche, leading his side unbeaten through qualifying to book their place at EURO 2024 – the national team's first major finals since EURO 2016 (when Anghel was in charge).

On what competing at the finals means for Romania

For the country, the fans and the Romanian people – [it's] a celebration. And together with the Romanian people, we had some unique moments when we qualified. This excellent campaign was received with great enthusiasm by the Romanian people. We felt a lot of joy and warmth. We felt that Romanians were supporting the national team once again.

On becoming a coach

I really wanted to succeed as a player at a high level. Unfortunately, things didn't go as planned, and all my energy and ambition, plus everything that I learnt at home, from my parents, from my father – all that gave me the drive to get into coaching. There was a whole process behind it.

It was a lot of hard work, a lot of effort and sacrifice, and it was a natural process because I started as an assistant coach, and then continued as a head coach in Liga III, in Liga II and then Liga I. We won trophies, we won the championship, we played in European competitions. It's the best moment of my career right now: leading Romania to a major tournament after [an eight-year absence].

On Ianis Hagi

The team really missed Ianis because, sadly, he picked up a bad injury that kept him out of action for more than a year and, consequently, from the national team. He is a player with special qualities. He is a special player but, for me, all the boys, all the players are special. I trust them completely.

Ianis is a cog in our complex mechanism. The national team players play with a sense of responsibility and are full of enthusiasm. As I always say, each player has to play their part in getting results for the national team. This is what Ianis is doing: he is fully focused, ambitious and with that combination, we will manage to produce results for Romania.

On EURO 2024 ambitions

I believe that if we continue on this good path that we are on now, we can go to the European Championship and create some beautiful moments for Romania and the Romanian fans. That's all I hope for. We have to take it step by step, enjoy every moment, every day, every hour, every minute spent in the tournament, and treat it with the utmost responsibility and give it all during the campaign.

It's been a long time since Romania's last tournament proper and the fans went through many tough moments. Our hard work was based on trust, on [maintaining] consistency; every player understood that they had to give 200% – not 100%, but 200%. This has led us to believe that we can progress together as a group and as individuals, and then as a team. We brought honour to Romania, but we still have a lot to do.

More on Romania

Romania at EURO 2024: Fixtures, stats, coach, tickets

UEFA EURO 2024 Group E: Belgium, Slovakia, Romania, Ukraine

Romania: All their EURO records and stats

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