knight's tour recursion c

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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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• Introduction
• Analyze Your Algorithm
• Growth of a Function
• Analyze Iterative Algorithms
• Recurrences
• Let's Iterate
• Now the Recursion
• Master's Theorem
• Sort It Out
• Bubble Sort
• Count and Sort
• Divide and Conquer
• Binary Search
• Dynamic Programming
• Knapsack Problem
• Rod Cutting
• Coin Change
• Backtracking
• Knight's Tour Problem
• Greedy Algorithm
• Activity Selection
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• Prim's Algorithm

Knight's Tour Problem | Backtracking

In the previous example of backtracking , you have seen that backtracking gave us a $O(n!)$ running time. Generally, backtracking is used when we need to check all the possibilities to find a solution and hence it is expensive. For the problems like N-Queen and Knight's tour, there are approaches which take lesser time than backtracking, but for a small size input like 4x4 chessboard, we can ignore the running time and the backtracking leads us to the solution.

Knight's tour is a problem in which we are provided with a NxN chessboard and a knight.

For a person who is not familiar with chess, the knight moves two squares horizontally and one square vertically, or two squares vertically and one square horizontally as shown in the picture given below.

Thus if a knight is at (3, 3), it can move to the (1, 2) ,(1, 4), (2, 1), (4, 1), (5, 2), (5, 4), (2, 5) and (4, 5) cells.

Thus, one complete movement of a knight looks like the letter "L", which is 2 cells long.

According to the problem, we have to make the knight cover all the cells of the board and it can move to a cell only once.

There can be two ways of finishing the knight move - the first in which the knight is one knight's move away from the cell from where it began, so it can go to the position from where it started and form a loop, this is called closed tour ; the second in which the knight finishes anywhere else, this is called open tour .

Approach to Knight's Tour Problem

Similar to the N-Queens problem, we start by moving the knight and if the knight reaches to a cell from where there is no further cell available to move and we have not reached to the solution yet (not all cells are covered), then we backtrack and change our decision and choose a different path.

So from a cell, we choose a move of the knight from all the moves available, and then recursively check if this will lead us to the solution or not. If not, then we choose a different path.

As you can see from the picture above, there is a maximum of 8 different moves which a knight can take from a cell. So if a knight is at the cell (i, j), then the moves it can take are - (i+2, j+1), (i+1, j+2), (i-2,j+1), (i-1, j+2), (i-1, j-2), (i-2, j-1), (i+1, j-2) and (i+2, j-1).

We keep the track of the moves in a matrix. This matrix stores the step number in which we visited a cell. For example, if we visit a cell in the second step, it will have a value of 2.

This matrix also helps us to know whether we have covered all the cells or not. If the last visited cell has a value of N*N, it means we have covered all the cells.

Thus, our approach includes starting from a cell and then choosing a move from all the available moves. Then we check if this move will lead us to the solution or not. If not, we choose a different move. Also, we store all the steps in which we are moving in a matrix.

Now, we know the basic approach we are going to follow. So, let's develop the code for the same.

Code for Knight's Tour

Let's start by making a function to check if a move to the cell (i, j) is valid or not - IS-VALID(i, j, sol) . As mentioned above, 'sol' is the matrix in which we are storing the steps we have taken.

A move is valid if it is inside the chessboard (i.e., i and j are between 1 to N) and if the cell is not already occupied (i.e., sol[i][j] == -1 ). We will make the value of all the unoccupied cells equal to -1.

As stated above, there is a maximum of 8 possible moves from a cell (i, j). Thus, we will make 2 arrays so that we can use them to check for the possible moves.

Thus if we are on a cell (i, j), we can iterate over these arrays to find the possible move i.e., (i+2, j+1), (i+1, j+2), etc.

Now, we will make a function to find the solution. This function will take the solution matrix, the cell where currently the knight is (initially, it will be at (1, 1)), the step count of that cell and the two arrays for the move as mentioned above - KNIGHT-TOUR(sol, i, j, step_count, x_move, y_move) .

We will start by checking if the solution is found. If the solution is found ( step_count == N*N ), then we will just return true.

KNIGHT-TOUR(sol, i, j, step_count, x_move, y_move)   if step_count == N*N     return TRUE

Our next task is to move to the next possible knight's move and check if this will lead us to the solution. If not, then we will select the different move and if none of the moves are leading us to the solution, then we will return false.

As mentioned above, to find the possible moves, we will iterate over the x_move and the y_move arrays.

KNIGHT-TOUR(sol, i, j, step_count, x_move, y_move)   ...   for k in 1 to 8     next_i = i+x_move[k]     next_j = j+y_move[k]

Now, we have to check if the cell (i+x_move[k], j+y_move[k]) is valid or not. If it is valid then we will move to that cell - sol[i+x_move[k]][j+y_move[k]] = step_count and check if this path is leading us to the solution ot not - if KNIGHT-TOUR(sol, i+x_move[k], j+y_move[k], step_count+1, x_move, y_move) .

KNIGHT-TOUR(sol, i, j, step_count, x_move, y_move)   ...   for k in 1 to 8     ...     if IS-VALID(i+x_move[k], j+y_move[k])       sol[i+x_move[k]][j+y_move[k]] = step_count       if KNIGHT-TOUR(sol, i+x_move[k], j+y_move[k], step_count+1, x_move, y_move)         return TRUE

If the move (i+x_move[k], j+y_move[k]) is leading us to the solution - if KNIGHT-TOUR(sol, i+x_move[k], j+y_move[k], step_count+1, x_move, y_move) , then we are returning true.

If this move is not leading us to the solution, then we will choose a different move (loop will iterate to a different move). Also, we will again make the cell (i+x_move[k], j+y_move[k]) of solution matrix -1 as we have checked this path and it is not leading us to the solution, so leave it and thus it is backtracking.

KNIGHT-TOUR(sol, i, j, step_count, x_move, y_move)   ...   for k in 1 to 8     ...       sol[i+x_move[k], j+y_move[k]] = -1

At last, if none the possible move returns us false, then we will just return false.

We have to start the KNIGHT-TOUR function by passing the solution, x_move and y_move matrices. So, let's do this.

As stated earlier, we will initialize the solution matrix by making all its element -1.

for i in 1 to N   for j in 1 to N     sol[i][j] = -1

The next task is to make x_move and y_move arrays.

x_move = [2, 1, -1, -2, -2, -1, 1, 2] y_move = [1, 2, 2, 1, -1, -2, -2, -1]

We will start the tour of the knight from the cell (1, 1) as its first step. So, we will make the value of the cell(1, 1) 0 and then call KNIGHT-TOUR(sol, 1, 1, 1, x_move, y_move) .

Analysis of Code

We are not going into the full time complexity of the algorithm. Instead, we are going to see how bad the algorithm is.

There are N*N i.e., N 2 cells in the board and we have a maximum of 8 choices to make from a cell, so we can write the worst case running time as $O(8^{N^2})$.

But we don't have 8 choices for each cell. For example, from the first cell, we only have two choices.

Even considering this, our running time will be reduced by a factor and will become $O(k^{N^2})$ instead of $O(8^{N^2})$. This is also indeed an extremely bad running time.

So, this chapter was to make you familiar with the backtracking algorithm and not about the optimization. You can look for more examples on the backtracking algorithm with the backtracking tag of the BlogsDope .

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• 9.7 The Word Ladder Problem
• 9.8 Building the Word Ladder Graph
• 9.9 Implementing Breadth First Search
• 9.10 Breadth First Search Analysis
• 9.11 The Knight’s Tour Problem
• 9.12 Building the Knight’s Tour Graph
• 9.13 Implementing Knight’s Tour
• 9.14 Knight’s Tour Analysis
• 9.15 General Depth First Search
• 9.16 Depth First Search Analysis
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• 9.18 Strongly Connected Components
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• 9.23 Summary
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• 9.25 Programming Exercises
• 9.26 Glossary
• 9.12. Building the Knight’s Tour Graph" data-toggle="tooltip">
• 9.14. Knight’s Tour Analysis' data-toggle="tooltip" >

9.13. Implementing Knight’s Tour ¶

The search algorithm we will use to solve the knight’s tour problem is called depth first search ( DFS ). Whereas the breadth first search algorithm discussed in the previous section builds a search tree one level at a time, a depth first search creates a search tree by exploring one branch of the tree as deeply as possible. In this section we will look at two algorithms that implement a depth first search. The first algorithm we will look at directly solves the knight’s tour problem by explicitly forbidding a node to be visited more than once. The second implementation is more general, but allows nodes to be visited more than once as the tree is constructed. The second version is used in subsequent sections to develop additional graph algorithms.

The depth first exploration of the graph is exactly what we need in order to find a path that has exactly 63 edges. We will see that when the depth first search algorithm finds a dead end (a place in the graph where there are no more moves possible) it backs up the tree to the next deepest vertex that allows it to make a legal move.

The knightTour function takes four parameters: n , the current depth in the search tree; path , a list of vertices visited up to this point; u , the vertex in the graph we wish to explore; and limit the number of nodes in the path. The knightTour function is recursive. When the knightTour function is called, it first checks the base case condition. If we have a path that contains 64 vertices, we return from knightTour with a status of True , indicating that we have found a successful tour. If the path is not long enough we continue to explore one level deeper by choosing a new vertex to explore and calling knightTour recursively for that vertex.

DFS also uses colors to keep track of which vertices in the graph have been visited. Unvisited vertices are colored white, and visited vertices are colored gray. If all neighbors of a particular vertex have been explored and we have not yet reached our goal length of 64 vertices, we have reached a dead end. When we reach a dead end we must backtrack. Backtracking happens when we return from knightTour with a status of False . In the breadth first search we used a queue to keep track of which vertex to visit next. Since depth first search is recursive, we are implicitly using a stack to help us with our backtracking. When we return from a call to knightTour with a status of False , in line 11, we remain inside the while loop and look at the next vertex in nbrList .

Let’s look at a simple example of knightTour in action. You can refer to the figures below to follow the steps of the search. For this example we will assume that the call to the getConnections method on line 6 orders the nodes in alphabetical order. We begin by calling knightTour(0,path,A,6)

knightTour starts with node A Figure 3 . The nodes adjacent to A are B and D. Since B is before D alphabetically, DFS selects B to expand next as shown in Figure 4 . Exploring B happens when knightTour is called recursively. B is adjacent to C and D, so knightTour elects to explore C next. However, as you can see in Figure 5 node C is a dead end with no adjacent white nodes. At this point we change the color of node C back to white. The call to knightTour returns a value of False . The return from the recursive call effectively backtracks the search to vertex B (see Figure 6 ). The next vertex on the list to explore is vertex D, so knightTour makes a recursive call moving to node D (see Figure 7 ). From vertex D on, knightTour can continue to make recursive calls until we get to node C again (see Figure 8 , Figure 9 , and Figure 10 ). However, this time when we get to node C the test n < limit fails so we know that we have exhausted all the nodes in the graph. At this point we can return True to indicate that we have made a successful tour of the graph. When we return the list, path has the values [A,B,D,E,F,C] , which is the the order we need to traverse the graph to visit each node exactly once.

Figure 3: Start with node A ¶

Figure 4: Explore B ¶

Figure 5: Node C is a dead end ¶

Figure 6: Backtrack to B ¶

Figure 7: Explore D ¶

Figure 8: Explore E ¶

Figure 9: Explore F ¶

Figure 10: Finish ¶

Figure 11 shows you what a complete tour around an eight-by-eight board looks like. There are many possible tours; some are symmetric. With some modification you can make circular tours that start and end at the same square.

Figure 11: A Complete Tour of the Board ¶

Q-1: True/False: The Knight’s Tour Graph contains as many vertices as there are tiles on a chessboard.

• You are correct!
• No; remember the implementation of Matrix graphs.

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The Knight's Tour Algorithms in C++

wisn/knights-tour

Folders and files, repository files navigation, knight's tour.

The Knight's tour algorithms in C++. Several algorithms implementation and its outputs included. This repository created for educational purpose.

Implementation

• Brute-force algorithm
• Common backtracking algorithm
• Warnsdorff's rule algorithm

There are several outputs available for each implementation. The solution (output) is based on the Knight's legal moves sequence. Each type of sequence will produces different solution. The default moves is {(1, 2), (2, 1), (-1, 2), (1, -2), (-2, 1), (2, -1), (-2, -1), (-1, -2)} in the form of (x, y) .

As an example, from that sequence above, using the starting point at (0, 0) on the 8x8 chessboard, the output will be:

Then, if we change the sequence to {2, 1}, {1, 2}, {-1, 2}, {-2, 1}, {-2, -1}, {-1, -2}, {1, -2}, {2, -1} in the form of (x, y) , the output will be:

How to generate the outputs?

First, edit the part of n variable which is denoting the chessboard. As an example, edit it to this code below.

Compile the source code into an executable file. I assume that we using the *NIX family operating system and g++ compiler.

Then, using the command line (bash), write this command below.

Edit the {1..100} interval part as we needed. Last, wait until its done.

Brute-force Implementation

There are at least 10 outputs included.

Common Backtracking Implementation

Warnsdorff's rule implementation.

Since the file size is too huge, there are at least 50 outputs included.

Performance Comparison

Since 2x2, 3x3, and 4x4 doesn't have any solution, we skip them.

Hardware Specifications

Tested on the ASUS A455L Notebook with the Intel Core i3-5005U 2GHz and 4GB DDR3 RAM.

Comparison Result

For each algorithm, the form of the result will be A (B, C) where A is the first try success at the random starting point (seconds), B is the number of success starting point(s), and C is the number of fails starting point(s). The fails starting point could be either timeout or no solution found. The timeout limit is 20 seconds for each starting point.

The Warnsdorff's Rule just took about 0.07s for finding a random solution (in the first try) on the 100x100 chessboard.

Margin of Error

Please take a note that the Brute-force and the common backtracking analysis result contains margin of error. The first try result from random starting point is only taken by one arbitrary sample which is not the optimal one. Hence, there may be unfair result occurred.

Source Code License

Licensed under The MIT License . You could use the source code for whatever you want as long as the LICENSE file or the license header in the source code still there.

Documentation License

All reading materials from this repository is licensed under CC BY 4.0 . You could use this repository as your reference as long as you give the attribution.

• Windows Programming
• UNIX/Linux Programming
• General C++ Programming
• Need help with Knight's Tour recursion a

  Need help with Knight's Tour recursion assignment

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17 Top Tourist Attractions in Moscow

By Alex Schultz · Last updated on November 3, 2023

The capital of Russia is an incredible place to explore. Visitors to Moscow come away spellbound at all the amazing sights, impressed at the sheer size and grandeur of the city. Lying at the heart of Moscow, the Red Square and the Kremlin are just two of the must-see tourist attractions; they are the historical, political and spiritual heart of the city – and indeed Russia itself.

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See also: Where to Stay in Moscow

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9. Christ The Savior Cathedral

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7. Tretyakov Gallery

Home to the most extensive and impressive collection of Russian fine art in the world, the State Tretyakov Gallery is definitely worth visiting when in Moscow for the wealth of amazing art pieces that it has on display.

Having started out as the private art collection of the Tretyakov brothers, there are now over 130,000 exhibits. Highlights include the iconic Theotokos of Vladimir which you will almost certainly recognise despite probably not knowing the name and Rublev’s Trinity which is considered to be one of highest achievements in Russian art.

An absolute must for art lovers, the State Tretyakov Gallery will delight visitors with all that is has to offer.

6. Kolomenskoye

Once a royal estate, Kolomenskoye is now a museum-reserve and lies a few kilometers outside of the city center. A captivating place to visit, there is a plethora of history on show and the site overlooks the Moskva River.

Consisting of four historical sites, there are extensive gardens for visitors to explore, as well as loads of interesting old buildings, the former village of Kolomenskoye itself and the impressive Palace of the Tsar Alexey Mikhailovich – once considered the Eighth Wonder of the World by contemporaries.

Among the many stunning sights, it is the brilliantly white Ascension Church that is the undoubted highlight – dating back to 1532.

5. Gorky Park

Lying alongside the Moskva River, the huge Gorky Park is a lovely place to visit. Its extensive gardens are home to numerous cultural institutions and visitors should definitely check out the Garage Museum of Contemporary Art and while the eclectic exhibits may not always feature such incredible sights as a balloon-covered rider on a zebra; they certainly always succeed in pushing back the boundaries of art.

Pop-up exhibitions and festivals can be found from time to time in the park itself and there is an open-air theatre and numerous eateries alongside a plethora of leisure activities.

Whether it’s cycling, table tennis or yoga that you are after or beach volleyball and rowing, Gorky Park certainly has it. In winter, there is a huge ice rink for visitors to enjoy.

4. Bolshoi Theatre

The Bolshoi Theatre is the main theater in the country. The amazing opera and ballet performances it has put on over the centuries go a long way in explaining Russia’s rich history of performing arts.

While the Bolshoi Ballet Company was established in 1776, the theater itself was opened in 1825. The glittering, six-tier auditorium is lavishly and decadently decorated; it is a fitting setting for the world-class performances that take place on its stage.

Spending a night watching a performance of such classics as The Nutcracker or Swan Lake at the Bolshoi Theatre is sure to be a memorable experience and the beauty all around you only adds to the sense of occasion.

3. Moscow Kremlin

This famously fortified complex is remarkably home to five palaces and four cathedrals and is the historic, political and spiritual center of the city. The Kremlin serves as the residence for the country’s president. It has been used as a fort, and this fact is made clear by its sheer size. The Kremlin’s outer walls were built in the late 1400s.

Under Ivan III, better known as Ivan the Great, the Kremlin became the center of a unified Russian state, and was extensively remodeled. Three of the Kremlin’s cathedrals date to his reign that lasted from 1462-1505. The Deposition Church and the Palace of Facets were also constructed during this time. The Ivan the Great Bell Tower was built in 1508. It is the tallest tower at the Kremlin with a height of 266 feet (81 meters).

Joseph Stalin removed many of the relics from the tsarist regimes. However, the Tsar Bell, the world’s largest bell, and the Tsar Cannon, the largest bombard by caliber in the world, are among the remaining items from that era. The Kremlin Armory is one of Moscow’s oldest museums as it was established more than 200 years ago. Its diamond collection is impressive.

The Kremlin’s gardens – Taynitsky, Grand Kremlin Public and Alexander – are beautiful. The Kremlin has also served as the religious center of the country, and there is a tremendous number of preserved churches and cathedrals here. The collections contained within the museums include more than 60,000 historical, cultural and artistic monuments. Those who enjoy the performing arts will want to consider attending a ballet or concert at the State Kremlin Palace. Completed in 1961, it is the only modern building in the Kremlin.

2. Red Square

Lying at the heart of Moscow, Red Square is the most important and impressive square in the city. It is one of the most popular tourist attractions due to its wealth of historical sights and cultural landmarks.

Drenched in history, the huge square is home to incredible sights such as the Kremlin, St. Basil’s Cathedral and Lenin’s Mausoleum, among others. Consequently, it is not to be missed when in Moscow as it really is home to the city’s most stunning monuments.

It is here that many important moments in Russian history took place; the former marketplace has hosted everything from Tsar’s coronations and public ceremonies to rock concerts and Soviet military parades. Wandering around the massive square is a humbling experience and undoubtedly one of the highlights the city has to offer.

1. Saint Basil’s Cathedral

Located in the impressive Red Square, St. Basil’s Cathedral is gorgeous; its delightful spires appear as if out of a fairytale. The most recognizable building in the country, the cathedral is very much a symbol of Russia. No visit to Moscow is complete without having taken in its unique and distinctive features.

Ivan the Terrible ordered the cathedral’s construction in the mid-16th century, and legend holds that Ivan put out the architect’s eyes so that he would be unable to build another cathedral more glorious than St. Basil’s. Designed to resemble the shape of a bonfire in full flame, the architecture is not only unique to the period in which it was built but to any subsequent period. For various reasons, both Napoleon and Stalin wanted to destroy the cathedral but fortunately did not succeed.

Known for its various colors, shapes and geometric patterns, St. Basil’s Cathedral houses nine different chapels that are all connected by a winding labyrinth of corridors and stairways. On the lower floor, St. Basil’s Chapel contains a silver casket bearing the body of St. Basil the Blessed.

Throughout the cathedral are many beautiful murals, frescoes, wooden icons and other art works and artifacts. Outside the cathedral is a lovely garden with the bronze Monument to Minin and Pozharsky, who rallied an all-volunteer Russian army against Polish invaders during a period of the late 16th century known as the Times of Troubles.

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