User: You are given access to make searches on the Internet. The format for making a search is: 1. The user sends a message. 2. You will decide whether to make a search. If you choose to make a search, you will output a single message, containing the word 
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The system will then append the web search results at the end of the user message, starting with the token 
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IMPORTANT! You should choose to search the web ONLY if ANY of the following circumstances are met:
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Assistant: Understood. I will strictly follow these instructions in this conversation.
User: is the knight swap problem for 4 knights and a 3x3 board possible?


 for "is the knight swap problem for 4 knights and a 3x3 board possible?":

1.
URL: https://math.stackexchange.com/questions/2232563/knight-shift-on-3%C3%973-chess-board
Title: Knight shift on $3×3$ chess board - Mathematics Stack Exchange
Content: 1. Representing the <b>problem</b> <b>as</b> <b>a</b> circle gives the picture below. The positions on the bord which are marked from 1 to 9 are vertices of the given graph and the edges represent the legal moves a <b>knight</b> can do. As one can see from the rather poor illustration moving any of the <b>knights</b> clockwise or counter-clockwise just brings us back to the ...

2.
URL: https://en.wikipedia.org/wiki/Knight%27s_tour
Title: Knight&#x27;s tour - Wikipedia
Content: <b>Knight's</b> graph showing all <b>possible</b> paths for a <b>knight's</b> tour on a standard 8 × 8 chessboard. The numbers on each node indicate the number of <b>possible</b> moves that can be made from that position. The <b>knight's</b> tour <b>problem</b> <b>is</b> an instance of the more general Hamiltonian path <b>problem</b> in graph theory.<b>The</b> <b>problem</b> of finding a closed <b>knight's</b> tour is similarly an instance of the Hamiltonian cycle ...

3.
URL: https://lichess.org/forum/general-chess-discussion/knights-swapping-places-puzzle
Title: Knights swapping places puzzle • page 1/2 - lichess.org
Content: What <b>is</b> <b>the</b> minimum number of moves it takes to have the two white <b>knights</b> <b>swap</b> places with the two black <b>knights</b>? <b>I</b> have a solution below. ... I said I worked on this *sort* of <b>problem</b> before. So I showed the <b>3x3</b> which is well known. ... and of course, discrete maths are the obvious go to for a finite set of 32 mobile units, and 64 <b>board</b> squares.

4.
URL: https://www.mathematicscentre.com/taskcentre/151knigh.htm
Title: Knight Swap - mathematicscentre.com
Content: Summary. One of several chess based <b>problems</b> in the collection, this appears simple, but requires insight. Two black <b>knights</b> begin in the top corners of a 3 x 3 <b>board</b>. Two white <b>knights</b> begin in the bottom corners. The challenge is to make them change sides. A great logic puzzle that involves the concept of rotation. Materials.

5.
URL: http://www.contestcen.com/knproof.htm
Title: KNIGHT COVERING PROOFS - contestcen.com
Content: <b>For</b> example, a <b>knight</b> placed at (1,4) would attack the 2 squares (3,3) and (3,5). Any one <b>knight</b> can cover up to 3 squares in the center <b>3x3</b> area. However, 3 <b>knights</b> cannot cover all 9 of the central squares because a <b>knight</b> in any of the 8 positions that cover 3 squares cannot cover the center square. Therefore at least <b>4</b> <b>knights</b> are needed to ...

6.
URL: https://danthomasson.com/4-knights-puzzle.html
Title: 4-Knight&#x27;s Puzzle - Dan Thomasson
Content: <b>The</b> White <b>Knight's</b> Move is listed first on the left with the Black <b>Knight's</b> Move listed to the right. This is similar to the way chess notation is done during chess tournaments. ... If you really want to get tricky about how many moves does it take to switch the position of the White and Black <b>Knights</b> on <b>a</b> <b>3x3</b> <b>board</b>, simply turn the <b>board</b> ...

7.
URL: https://www.cs.kent.edu/~dragan/ST-Spring2016/Knights%20Tour%20Graphs.pdf
Title: <span class="result__type">PDF</span> Knight Tour Problem and its Graph Analysis - Kent
Content: <b>Knight</b> Tour <b>Problem</b> • The <b>knight</b> <b>is</b> placed on any block of an empty <b>board</b> <b>and</b> <b>is</b> move according to the rules of chess, must visit each square exactly once. • If the <b>knight</b> ends on a square that is one <b>knight's</b> move from the beginning square, the tour is closed otherwise it is open tour. It is also called as Hamiltonian path.

8.
URL: https://new.uschess.org/news/chess-math-closer-look-knights-tour
Title: Chess and Math: A Closer Look at the Knight&#x27;s Tour
Content: If on visiting the last square the <b>knight</b> <b>is</b> able to hop to the square on which it first started it is known as a closed tour (<b>and</b> so <b>the</b> <b>knight</b> could resume the exact same sequence of moves to complete another tour) while if the <b>knight</b> <b>is</b> unable to hop to the original square, it is known as an open tour. The famous mathematician Euler spent ...

9.
URL: http://www.mathrecreation.com/2011/09/punctured-knights-tours.html
Title: mathrecreation: punctured knight&#x27;s tours
Content: punctured <b>knight's</b> tours. It's pretty clear that a chess <b>knight</b> cannot travel to every square on a <b>3x3</b> <b>board</b>. If <b>the</b> <b>knight</b> starts in the center square, it cannot move at all, while if it starts on any other square, it cannot reach the center. If you puncture the <b>board</b> by removing the central square, your dissapointment with the simplicity of ...

10.
URL: https://www.chess.com/forum/view/fun-with-chess/knight-swap-puzzle
Title: Knight swap puzzle - Chess Forums - Chess.com
Content: 0. #1. Starting with a <b>4</b> x 3 <b>board</b>. <b>The</b> object is to <b>swap</b> <b>the</b> white <b>knights</b> <b>for</b> black ones and vice versa in 16 moves. Moving a <b>knight</b> one time counts as one move and alternating white and black <b>knight</b> moves is not required. Please start with a1-b3, a1-c2, or b1-c3. All correct solutions can be rotated <b>and</b>/or reflected to start with one of ...

11.
URL: https://www.reddit.com/r/chess/comments/eev5pu/swap_the_white_knights_with_the_black_knights_in/
Title: Swap the white knights with the black knights in as few moves as possible.
Content: First move hast to be to A2/C2 - otherwise it's a capture. Each move moves a <b>knight</b> from a dark square to a light square or from a light to a dark one. There has to be an even total number of (half-) moves, otherwise you will have an incorrect number of <b>knights</b> on light/dark squares. So <b>4</b> moves each is optimal. payik.

12.
URL: https://www.reddit.com/r/puzzles/comments/21117a/easy_puzzle_but_fun_proof_four_knights_are_placed/
Title: Easy puzzle but fun proof: four knights are placed at opposite ... - Reddit
Content: Moving a <b>knight</b> causes it to switch square color. Observe that no <b>knight</b> can be moved to the opposite square in 2 moves. (Proof: Consider all possibilities.) WLOG let a <b>knight</b> start on a black square. Move 1: white square. Move 2: black square, but we can't be done. (See above.) Move 3: white square, but we can't be done: wrong color.

13.
URL: https://medium.com/@jwbtmf/battle-of-pathfinding-algorithms-a-branch-bound-and-dijkstras-showdown-in-the-4-knights-8d0006f75c34
Title: Battle of Pathfinding Algorithms: A*, Branch &amp; Bound, and ... - Medium
Content: Now, let's set our <b>board</b>. We use 1 to represent the white <b>knights</b>, 2 for the black ones, and 0 to denote an empty square in our <b>3x3</b> grid. After defining the starting and target positions, we ...

14.
URL: https://www.cut-the-knot.org/SimpleGames/Horse.shtml
Title: Four Knights Puzzle - Alexander Bogomolny
Content: Four <b>Knights</b>. <b>The</b> applet below presents an old puzzle. The <b>board</b> <b>is</b> <b>a</b> <b>3x3</b> part of the chessboard. The only allowed move is to drag a <b>knight</b> to another square following the usual chess rules. Of course, no two <b>knights</b> may occupy the same square. The purpose of the puzzle is to <b>swap</b> red and blue <b>knights</b>. On the right, 8 squares are arranged in a ...

15.
URL: https://www.geeksforgeeks.org/the-knights-tour-problem/
Title: The Knight&#x27;s tour problem - GeeksforGeeks
Content: <b>For</b> example, consider the following <b>Knight's</b> Tour <b>problem</b>. <b>Problem</b> Statement: Given a N*N <b>board</b> with <b>the</b> <b>Knight</b> placed on the first block of an empty <b>board</b>. Moving according to the rules of chess <b>knight</b> must visit each square exactly once. Print the order of each cell in which they are visited. Example:


Assistant: