knight's tour 5 by 5

Exploring the Knight’s Tour Problem

The Knight’s tour problem is an interesting and challenging puzzle that has fascinated mathematicians for centuries. The problem is to find a sequence of moves for a knight on a chessboard such that the knight visits every square exactly once. This problem has been studied since the 9th century, and has been the subject of much research and speculation.

One of the most interesting aspects of the Knight’s tour problem is that it is not always possible to find a solution. This is because the knight’s movement is constrained by the geometry of the chessboard, and some configurations simply do not allow for a complete tour. However, it has been proven that a Knight’s tour is always possible on a rectangular board whose smaller dimension is at least 5.

For any m x n board with m ≤ n, a Knight’s tour is always possible unless one or more of thee three conditions are met: m = 1 or 2. m = 3 and n = 3, 5, or 6. This means that for most standard chessboards, a Knight’s tour is possible.

However, there are some exceptions to this rule. For example, if a square is removed from the chessboard, it may no longer be possible to find a complete Knight’s tour. This is because the removal of a single square can disrupt the geometry of the chessboard in such a way that it becomes impossible to visit every square.

Despite these challenges, the Knight’s tour problem remains a popular puzzle among mathematicians and puzzle enthusiasts. It is a fascinating example of how mathematics can be used to explore complex problems in a structured and systematic way.

The Knight’s tour problem is an intriguing puzzle that continues to fascinate mathematicians and puzzle enthusiasts alike. While there are some situations where it may not be possible to find a complete tour, the problem remains a challenging and rewarding pursuit for those who enjoy puzzles and problem-solving.

Is The Knight’s Tour Possible?

The Knight’s Tour is possible on a rectangular board whose smaller dimension is at least 5. Additionally, for any m x n board with m ≤ n, a Knight’s Tour is always possible unless one of the following conditions are met: – m = 1 or 2 – m = 3 and n = 3, 5, or 6.

It is important to note that a Knight’s Tour is a sequence of moves made by a knight on a chessboard, where the knight visits every square exactly once. The proof of the existence of a Knight’s Tour on certain types of boards is a well-studied problem in mathematics.

What Is The Knights Tour Theory?

The Knight’s tour theory is a problem in the field of mathematics and chess. The problem is to find a sequence of moves by a knight on a chessboard such that the knight visits every square exactly once. The knight is allowed to move two squares in one direction and one square in the perpendicular direction. The problem has been studied for centuries, and many different solutions have been proposed.

One of the main questions in the Knight’s tour theory is whether a knight’s tour exists for every possible starting position and board size. It has been proven that a knight’s tour exists on all chessboards with one square removed, except for some specific cases. For example, if the board size is even, or if the removed square is (i, j) with i + j odd, then a knight’s tour may not be possible.

Other specific cases whre a knight’s tour may not be possible include when the board size is 3 and any square other than the center square is removed, when the board size is 5, when the board size is 7 and any square other than square (2, 2) or (2, 6) is removed, and when the board size is 9 and a certain specific square is removed.

The Knight’s tour theory is a fascinating problem that has captured the imagination of mathematicians and chess players alike. It is a challenging problem that requires creative thinking and careful analysis, and it continues to be studied and explored by researchers today.

What Is The Knight’s Tour Problem?

The Knight’s Tour problem is a mathematical puzzle that requires a player to find a sequence of moves for a knight piece on a chessboard, such that the knight visits every square exactly once. In other words, the problem asks whethr a knight can move to every square on the chessboard without repeating any square. The knight is allowed to move only in an L-shaped pattern, i.e., two squares horizontally or vertically and then one square in the perpendicular direction. The problem has been studied for centuries and has fascinated mathematicians and chess players alike. It has many applications in computer science, such as in algorithm design and optimization. The Knight’s Tour problem is a classic example of a combinatorial optimization problem, and finding a solution to the problem requires a lot of creativity and strategic planning.

The Knight’s tour problem has been a topic of interest for mathematicians and chess enthusiasts for centuries. While the problem may seem simple at first glance, it presents a challenging puzzle that requires careful analysis and strategic thinking. Through the years, various algorithms and techniques have been developed to solve the problem, and it has been proven that a Knight’s tour is possible on crtain types of chessboards. However, some conditions must be met, and certain squares must not be removed for the tour to be achievable. Despite its difficulty, the Knight’s tour problem remains a fascinating and engaging puzzle that continues to capture the imagination of people from all walks of life.

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Knight's tour

A knight's tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once. If the knight ends on a square that is one knight's move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed , otherwise it is open .

The knight's tour problem is the mathematical problem of finding a knight's tour. Creating a program to find a knight's tour is a common problem given to computer science students. Variations of the knight's tour problem involve chessboards of different sizes than the usual 8 x 8, as well as irregular (non-rectangular) boards.

The knight's tour problem is an instance of the more general Hamiltonian path problem in graph theory. The problem of finding a closed knight's tour is similarly an instance of the Hamiltonian cycle problem. Unlike the general Hamiltonian path problem, the knight's tour problem can be solved in linear time.

The earliest known reference to the knight's tour problem dates back to the 9th century AD. In Rudraṭa's Kavyalankara (5.15), a Sanskrit work on Poetics, the pattern of a knight's tour on a half-board has been presented as an elaborate poetic figure ("citra-alaṅkāra") called the "turagapadabandha" or 'arrangement in the steps of a horse.' The same verse in four lines of eight syllables each can be read from left to right or by following the path of the knight on tour. Since the Indic writing systems used for Sanskrit are syllabic, each syllable can be thought of as representing a square on a chess board. Rudrata's example is as follows:

से ना ली ली ली ना ना ना ली

ली ना ना ना ना ली ली ली ली

न ली ना ली ली ले ना ली ना

ली ली ली ना ना ना ना ना ली

se nā lī lī lī nā nā lī

lī nā nā nā nā lī lī lī

na lī nā lī le nā lī nā

lī lī lī nā nā nā nā lī

For example, the first line can be read from left to right or by moving from the first square to second line, third syllable (2.3) and then to 1.5 to 2.7 to 4.8 to 3.6 to 4.4 to 3.2.

One of the first mathematicians to investigate the knight's tour was Leonhard Euler. The first procedure for completing the Knight's Tour was Warnsdorf's rule, first described in 1823 by H. C. von Warnsdorf.

In the 20th century, the Oulipo group of writers used it among many others. The most notable example is the 10 x 10 Knight's Tour which sets the order of the chapters in Georges Perec's novel Life: A User's Manual . The sixth game of the 2010 World Chess Championship between Viswanathan Anand and Veselin Topalov saw Anand making 13 consecutive knight moves (albeit using both knights); online commentors jested that Anand was trying to solve the Knight's Tour problem during the game.

Schwenk proved that for any m x n board with m ≤ n , a closed knight's tour is always possible unless one or more of these three conditions are met:

• m and n are both odd
• m = 1, 2, or 4
• m = 3 and n = 4, 6, or 8.

Cull et al. and Conrad et al. proved that on any rectangular board whose smaller dimension is at least 5, there is a (possibly open) knight's tour.

Number of tours

On an 8 x 8 board, there are exactly 26,534,728,821,064 directed closed tours (i.e. two tours along the same path that travel in opposite directions are counted separately, as are rotations and reflections). The number of undirected closed tours is half this number, since every tour can be traced in reverse. There are 9,862 undirected closed tours on a 6 x 6 board.

Finding tours with computers

There are quite a number of ways to find a knight's tour on a given board with a computer. Some of these methods are algorithms while others are heuristics.

Brute force algorithms

A brute-force search for a knight's tour is impractical on all but the smallest boards; for example, on an 8x8 board there are approximately 4x10 51 possible move sequences, and it is well beyond the capacity of modern computers (or networks of computers) to perform operations on such a large set. However the size of this number gives a misleading impression of the difficulty of the problem, which can be solved "by using human insight and ingenuity ... without much difficulty."

Divide and conquer algorithms

By dividing the board into smaller pieces, constructing tours on each piece, and patching the pieces together, one can construct tours on most rectangular boards in polynomial time.

Neural network solutions

The Knight's Tour problem also lends itself to being solved by a neural network implementation. The network is set up such that every legal knight's move is represented by a neuron, and each neuron is initialized randomly to be either "active" or "inactive" (output of 1 or 0), with 1 implying that the neuron is part of the final solution. Each neuron also has a state function (described below) which is initialized to 0.

When the network is allowed to run, each neuron can change its state and output based on the states and outputs of its neighbors (those exactly one knight's move away) according to the following transition rules:

Warnsdorff's rule

Warnsdorff's rule is a heuristic for finding a knight's tour. We move the knight so that we always proceed to the square from which the knight will have the fewest onward moves. When calculating the number of onward moves for each candidate square, we do not count moves that revisit any square already visited. It is, of course, possible to have two or more choices for which the number of onward moves is equal; there are various methods for breaking such ties, including one devised by Pohl and another by Squirrel and Cull.

This rule may also more generally be applied to any graph. In graph-theoretic terms, each move is made to the adjacent vertex with the least degree. Although the Hamiltonian path problem is NP-hard in general, on many graphs that occur in practice this heuristic is able to successfully locate a solution in linear time. The knight's tour is a special case.

The heuristic was first described in "Des Rösselsprungs einfachste und allgemeinste Lösung" by H. C. von Warnsdorff in 1823. A computer program that finds a Knight's Tour for any starting position using Warnsdorff's rule can be found in the book 'Century/Acorn User Book of Computer Puzzles' edited by Simon Dally (ISBN 071260541X).

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“I used to attack because it was the only thing I knew. Now I attack because I know it works best.” – Garry Kasparov

Knight’s Tour: The famous mathematical problem

Chess is a game of strategy and foresight, demanding players to think several moves ahead to outmaneuver their opponents. Among the numerous puzzles and challenges that arise from the complexity of chess, the Knight’s Tour Problem stands out as a particularly fascinating conundrum. The Knight’s Tour is a puzzle that involves moving a knight on a chessboard in such a way that it visits each square exactly once.

The Knight’s Tour Problem is a mathematical challenge that revolves around finding a specific sequence of moves for a knight on a chessboard. It has become a popular problem assigned to computer science students, who are tasked with developing programs to solve it. The variations of the Knight’s Tour Problem go beyond the standard 8×8 chessboard, including different sizes and irregular, non-rectangular boards.

If you’re looking to find your own solution to the Knight’s Tour Problem, there is a straightforward approach known as Warnsdorff’s rule . Warnsdorff’s rule serves as a heuristic for discovering a single knight’s tour. The rule dictates that the knight should always move to the square from which it will have the fewest possible subsequent moves. In this calculation, any moves that would revisit a square already visited are not counted. By adhering to Warnsdorff’s rule, you can systematically guide the knight across the chessboard and ultimately find a knight’s tour.

Modern computers have significantly contributed to the exploration of the Knight’s Tour Problem. Through advanced algorithms and computational power, researchers have been able to tackle larger chessboards and refine existing solutions. However, finding a general solution that works for all chessboard sizes remains elusive.

In conclusion, the Knight’s Tour Problem continues to captivate mathematicians, chess players, and computer scientists alike. Its unique combination of chess strategy, mathematical intricacy, and computational challenges makes it a fascinating puzzle to explore. While the search for a comprehensive solution is ongoing, the journey towards unraveling the mysteries of the Knight’s Tour Problem offers valuable insights into the world of mathematics, algorithms, and the boundless potential of human intellect.

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The Knight’s Tour

The  knight’s tour problem  is the mathematical problem of finding a knight’s tour, and probably making knight the most interesting piece on the chess board. The knight visits every square exactly once, if the knight ends on a square that is one knight’s move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed; otherwise, it is open.

The knight’s tour problem is an instance of the more general Hamiltonian path problem in graph theory. The problem of finding a closed knight’s tour is similarly an instance of the Hamiltonian cycle problem. Unlike the general Hamiltonian path problem, the knight’s tour problem can be solved in linear time.

Hamiltonian Path Problem

In Graph Theory, a graph is usually defined to be a collection of nodes or vertices and the set of edges which define which nodes are connected with each other. So we use a well known notation of representing a graph G = (V,E) where  V = { v 1 , v 2 , v 3 , … , v n }  and E = {(i, j)|i ∈ V and j ∈ V and i and j is connected}.

Hamiltonian Path is defined to be a single path that visits every node in the given graph, or a permutation of nodes in such a way that for every adjacent node in the permutation there is an edge defined in the graph. Notice that it does not make much sense in repeating the same paths. In order to avoid this repetition, we permute with |V| C 2 combinations of starting and ending vertices.

Simple way of solving the Hamiltonian Path problem would be to permutate all possible paths and see if edges exist on all the adjacent nodes in the permutation. If the graph is a complete graph, then naturally all generated permutations would quality as a Hamiltonian path.

For example. let us find a Hamiltonian path in graph G = (V,E) where V = {1,2,3,4} and E = {(1,2),(2,3),(3,4)}. Just by inspection, we can easily see that the Hamiltonian path exists in permutation 1234. The given algorithm will first generate the following permutations based on the combinations: 1342 1432 1243 1423 1234 1324 2143 2413 2134 2314 3124 3214

The number that has to be generated is ( |V| C 2 ) (|V| – 2)!

Schwenk proved that for any  m  ×  n  board with  m  ≤  n , a closed knight’s tour is always possible  unless  one or more of these three conditions are met:

• m  and  n  are both odd
• m  = 1, 2, or 4
• m  = 3 and  n  = 4, 6, or 8.

Cull and Conrad proved that on any rectangular board whose smaller dimension is at least 5, there is a (possibly open) knight’s tour.

Neural network solutions

The neural network is designed such that each legal knight’s move on the chessboard is represented by a neuron. Therefore, the network basically takes the shape of the  knight’s graph  over an n×n chess board. (A knight’s graph is simply the set of all knight moves on the board)

Each neuron can be either “active” or “inactive” (output of 1 or 0). If a neuron is active, it is considered part of the solution to the knight’s tour. Once the network is started, each active neuron is configured so that it reaches a “stable” state if and only if it has exactly two neighboring neurons that are also active (otherwise, the state of the neuron changes). When the entire network is stable, a solution is obtained. The complete transition rules are as follows:

where t represents time (incrementing in discrete intervals), U(N i,j ) is the state of the neuron connecting square i to square j, V(N i,j ) is the output of the neuron from i to j, and G(N i,j ) is the set of “neighbors” of the neuron (all neurons that share a vertex with N i,j ).

Code For Knight’s Tour

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Knight's Tours of Odd Squares

3×3 board (9 cells), 5×5 board (25 cells).

7×7 board (49 cells)

9×9 board (81 cells)

11×11 board (121 cells)

13×13 board (169 cells)

15×15 board (225 cells), 17×17 board (289 cells).

19×19 board (361 cells)

Wolfram Demonstrations Project

The knight's tour.

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A classic chess problem is to find a sequence of knight moves that visits each square exactly once. Start by placing the knight anywhere on the board, then drag the step slider to see a knight's tour, either as a broken line path or by numbering the squares in order.

Contributed by: Jay Warendorff   (September 2018) Open content licensed under CC BY-NC-SA

Based on and extending a program at Knight's tour .

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Jay Warendorff "The Knight's Tour" http://demonstrations.wolfram.com/TheKnightsTour/ Wolfram Demonstrations Project Published: September 5 2018

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Knight's Tour Challenge

How to play.

This "game" is basically an implementation of Knight's Tour problem.

You have to produce the longest possible sequence of moves of a chess knight, while visiting squares on the board only once. This sequence is called "tour". If your tour visits every square, then you have achieved a full tour. If you have achieved a full tour and from your last position you could move to your initial square, then you have achieved a closed full tour.

Currently occupied square is highlighted in pale blue, while possible moves are shown with pale green. Click on the currently occupied square to undo.

Knight's Tour

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Large Knight's Tours

I was at middle school when I first heard about the knight's tour problem: place a knight on any square on a chess board, and move it to every square on the board without visiting any square more than once. A chess knight moves by jumping two squares forward and one square sideways, so a big part of the challenge is getting into and out of all the corners. I remember filling up pages and pages of squared notebooks with attempted solutions, although my record from that time was a tour that covered only 61 of the 64 squares.

Enter the Computer

Well, it's one thing to look for knight's tours by hand, but quite another to search for them with a computer. It is actually a fun programming exercise to search for knight's tours: it is possible to use a standard backtracking algorithm, but it needs a couple of tricks to make it work. Firstly, if you have a choice of squares to jump to, then you should consider them in ascending order of accessibility (i.e., from how many unvisited squares could you jump to this square). This helps to ensure that you don't waste last chances to visit squares. However, even with this trick, it is possible to make mistakes early on in the search that you can never recover from, and end up lost forever in a maze with 10 51 turns and no exits. An ugly way to fix this is to assign a move budget—I chose 100 times the number of squares—and if you don't find a solution within this many jumps then give up and start again from an empty board.

Armed with a fast computer, a search algorithm and a couple of tricks, I was finally able to realize a childhood dream and find a knight's tour:

Note that we always begin our tours in the bottom right corner, so that the search program only has to work out how to get into and out of 3 corners: that time spent filling squared notebooks taught me something!

Building Up

Once the initial problem is solved, I naturally start wondering whether we can find knight's tours on larger boards. It is easy to modify the search program to change the board dimensions from an 8×8 square to an m × n rectangle, but as the board size increases the time it takes to find a knight's tour grows rapidly:

Extrapolating these results to a 100×100 board, it seems that my search program would take over a thousand years to find a knight's tour! Clearly I need a new approach to find knight's tours for large boards.

Our 8×8 knight's tour above looks like a little circuit, and drawn the same way a 100×100 knight's tour would look like a huge circuit. But let's take some inspiration from the way that huge circuits are designed in practice: not all at once, but rather by wiring together smaller circuits. Consider this special 6×6 knight's tour:

One reason this knight's tour is special is because it has no start and end: it is a loop! There are probably clever ways to find knight's tours that are loops, but I took a simple-minded approach. I modified my program to keep searching for knight's tours until it found one where the end was a knight's jump away from the start, and then I connected the ends of the tour to make a loop.

There is another reason that this knight's tour is special, which is not so easy to see. You can put two of them together:

then switch two of the jumps around to connect the tours together:

and the result is a knight's tour on a 12×6 board. To see that the result is indeed a tour, let's visualize the two separate knight's tours as loops using a schematic diagram:

And now when we perform the connection step the schematic diagram makes it clear that we have created one big loop from the two separate loops:

The same thing also works by vertically stacking the special 6×6 knight's tour:

and connecting them like so:

Assembling Large Tours

I've shown how to connect special 6×6 knight's tours together, and the next step is to connect them in an m × n grid to make a knight's tour of a 6 m ×6 n board. Connecting them together in a grid requires a little bit of care to avoid fragmenting the knight's tour into disconnected paths. Here's a schematic diagram for a 2×2 grid illustrating how tours can be fragmented into disconnected paths:

One way to avoid tour fragmentation is to connect along some path through the grid that visits every 6×6 knight's tour. I chose a spiral, starting at the bottom right of the grid, travelling anti-clockwise and ending in the middle. Here is the schematic diagram for a spiral path on a 3×3 grid:

And here's the resulting knight's tour on an 18×18 board:

Filling in the Gaps

At this stage we have the machinery to construct knight's tours for arbitrarily large boards of dimension 6 m ×6 n . But what about boards of other dimensions, such as the 100×100 board that we set as our target? These boards can be handled by searching for more special tours of dimensions m × n , where 6 ≤ m,n ≤ 11, and using these in place of 6×6 knight's tours on the bottom and right edges of the grid. For example, here is a knight's tour of a 23×23 board:

This has the same 3×3 grid pattern as the 18×18 knight's tour, but with larger pieces around the right and bottom edges. Also, the knight's tour in the bottom right corner is not a loop—you can see one of its ends in the bottom right corner—so the schematic diagram for the assembled tour ends up looking a little different:

In fact, it is impossible to find a knight's tour that is a loop on this board, because it has an odd number of squares. To see why, imagine the squares of the board painted light and dark colors in a checkerboard pattern:

As a knight jumps around the board it alternates between light and dark squares, and a tour of a board with an odd number of squares will contain an even number of jumps. This means the start and end squares of the tour will have the same color, and so there's no way to jump from the end square of the tour back to the start square to close the loop.

Using these techniques you can easily assemble knight's tours for large boards: you can view a tour of a 100×100 board or even assemble your own knight's tour:

width     (between 6 and 100)

height     (between 6 and 100)

But these techniques only work for boards where the width and height are at least 6: there are also an infinite number of thin boards that might contain knight's tours:

Can you think of techniques for assembling knight's tours on thin boards? ♘

The performance results were gathered using a MacBook Air with a 1.8GHz Intel Core i7 processor and 4Gb of RAM, running OS X 10.7.5.

Large knight's tours by Joe Leslie-Hurd .

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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Warnsdorff’s algorithm for Knight’s tour problem

Problem : A knight is placed on the first block of an empty board and, moving according to the rules of chess, must visit each square exactly once. 

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

We have discussed Backtracking Algorithm for solution of Knight’s tour . In this post Warnsdorff’s heuristic is discussed.  Warnsdorff’s Rule:  

• We can start from any initial position of the knight on the board.
• We always move to an adjacent, unvisited square with minimal degree (minimum number of unvisited adjacent).

This algorithm may also more generally be applied to any graph. 

Some definitions:   

• A position Q is accessible from a position P if P can move to Q by a single Knight’s move, and Q has not yet been visited.
• The accessibility of a position P is the number of positions accessible from P.

Algorithm:   

• Set P to be a random initial position on the board
• Mark the board at P with the move number “1”
• let S be the set of positions accessible from P.
• Set P to be the position in S with minimum accessibility
• Mark the board at P with the current move number
• Return the marked board — each square will be marked with the move number on which it is visited.

Below is implementation of above algorithm.  

Output:  

Time complexity: O(N^3) Space complexity: O(N^2)

The Hamiltonian path problem is NP-hard in general. In practice, Warnsdorff’s heuristic successfully finds a solution in linear time.

Do you know?   “On an 8 × 8 board, there are exactly 26,534,728,821,064 directed closed tours (i.e. two tours along the same path that travel in opposite directions are counted separately, as are rotations and reflections). The number of undirected closed tours is half this number, since every tour can be traced in reverse!”

If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to [email protected]. See your article appearing on the GeeksforGeeks main page and help other Geeks.  

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HILTON HEAD ISLAND, S.C. – A sly smile emerged as Brian Harman sat Tuesday for his pre-tournament press conference at the RBC Heritage. A reporter was starting to ask about Scottie Scheffler, the hottest golfer in the world.

“Is any part of you surprised that Scottie is playing this week with everything going on, and is there a part of you that hopes he doesn't?” the report asked.

Harman chuckled. “Yeah, there's lots of parts of me that hope he doesn't play,” he said.

Scheffler is the new benchmark with which the rest of the PGA TOUR measures itself. The rest of the TOUR knows it. Harman’s response was part joke, part honest admission. Scheffler only needs his “B” game to contend and win, Harman continued. That’s a scary proposition.

Scheffler has won three times this year, including THE PLAYERS Championship and the Masters. He’s the first since Tiger Woods to win those two high-profile tournaments in the same season and the first since Woods to win them both multiple times.

The Scheffler-Woods comparisons are growing more frequent with every tournament Scheffler wins. It’s not that Scheffler is as dominant as Woods. He’s just the most dominant since Woods.

Players talk about Scottie Scheffler's dominating season

And it doesn’t seem to be slowing down. As Scheffler looks to build on his historic season with more victories at Signature Events and major championships, it’s an excellent time to take stock of the history in the making. Here are The Five stats that explain Scheffler’s dominance.

1. Locked at the top of the world ranking

Scheffler has already spent the last 47 weeks atop the Official World Golf Ranking, and he’s not going anywhere anytime soon.

With his win at the Masters, Scheffler further stretched his lead to a point that is likely insurmountable in 2024. The win elevated Scheffler’s average point total to 13.99, with more than a six-point lead over No. 2 Rory McIlroy.

The minutiae of the point totals are hard to follow, so here’s the necessary context to understand the feat: Scheffler’s gap over No. 2 is larger than any world No. 1 has had since Tiger Woods in 2009 when Woods held an eight-point lead. Woods amassed that lead during his record 281 consecutive weeks at world No. 1.

Scheffler has a long way to go to reach that sustained level of dominance, but what he’s doing right now is on par with any stretch of golf played in the modern era.

Calculating OWGR projections can be a bit fickle. There are so many variables week-to-week about field strength and eventual outcomes, but world ranking expert Nosferatu put it in the context of what is needed to dethrone Scheffler. Nosferatu wrote on social media this week that McIlroy would need to win “three majors plus some extra” to overtake Scheffler. That also assumes Scheffler isn’t playing well.

So, get used to Scheffler at the top. It’s going to stay that way for a long time.

2. It’s been years since Scheffler’s last poor iron performance… literally

Scheffler’s ball-striking is his superpower on the golf course. There have been weeks when the putter has failed him or his driver has been wayward. Finding a week when Scheffler’s iron play has abandoned him is nearly impossible.

It’s what makes him special. Every top player has a superpower, but Scheffler’s is more consistent than any other player’s.

In fact, the last time Scheffler lost strokes to the field with his approach play was the 2022 FedEx St. Jude Championship. That’s a streak of 36 events spanning more than 600 days.

Scottie Scheffler's top eight shots since 2023 Masters

To put it in perspective, McIlroy is widely considered the best off-the-tee game on TOUR. He lost strokes off-the-tee at THE PLAYERS last month. Collin Morikawa, one of the players thought to rival Scheffler’s approach play, has lost strokes with his approach play in three events this season. Hideki Matsuyama, the TOUR leader in SG: Around the Green, has lost strokes in that category twice this year. Taylor Montgomery, the TOUR leader in SG: Putting, has had two weeks where he lost strokes on the green.

No player more consistently gets more out of their best skill than Scheffler. It’s the reason he has such an incredibly high floor.

3. Best of both worlds

Scheffler leads the PGA TOUR in both birdie average (5.43 per round) and bogey avoidance (1.6 per round). It’s a lethal combo that further explains why Scheffler’s scoring average (68.8) is a full shot lower than No. 2 Xander Schauffele (69.9).

It’s made possible because of the specific type of dominance Scheffler deploys. He is the best iron player in the world. He also has one of the best short games, a borderline unfair combination. Scheffler hardly ever puts himself in a spot of bother. When he does, he is better equipped than any to get up and down for par.

Scheffler upends the widely accepted notion that a byproduct of making birdies is running into bogeys along the way. Scheffler is the only player on TOUR inside the top 10 in both birdie average and bogey avoidance. While others need to be aggressive to notch five birdies a round, Scheffler runs into birdies by the sheer, consistent force of his ball-striking.

Scottie Scheffler's eagle hole-out from 92 yards is the Shot of the Day

Sure, Scheffler could attack pins that others don’t dare take on. That happens occasionally, but Scheffler is making the smart play more often than not. He’s hitting the fat sides of greens, keeping the ball underneath the hole and assuring he doesn’t hit it near the trouble. It is death by 1000 paper cuts. He leads the PGA TOUR in Greens in Regulation and is sixth in Proximity to the hole. Putting statistics be damned, with those numbers Scheffler finds birdies regardless of whether the putter is cooperating.

4. Always in the red

The Masters cumulatively played 568 over-par across four rounds. Scheffler didn’t contribute. He hasn’t at any point this season, not for a tournament or a round. Scheffler has carded zero over-par rounds this season. His second-round 72 at Augusta National Golf Club was his worst score of his season. He has more wins (three) than rounds over par (zero).

Here’s how Scheffler stacks up to some of his peers:

McIlroy – Eight rounds over par

Schauffele – Four rounds over par

Wyndham Clark – Six rounds over par

Hovland – Five rounds over par

It’s an encapsulation of the Scheffler experience. He doesn’t have bad rounds often and when he does, by his standards, he knows how to manage them.

5. FedExCup frontrunner

Scheffler finished second in the 2022-23 FedExCup Regular Season standings, amassing 3,146 points before the Playoffs last August. Second only to Jon Rahm, Scheffler held an 800-point advantage over McIlroy in third. That was unquestionably a dominant season, and somehow, it’s not even close to as impressive as this season.

Scheffler has already outpaced his 2022-23 season-long points total. After his Masters victory, Scheffler has amassed 3,215 points. He’s accumulated in nine events what it took him 20 events to do a year ago. Then there’s how it looks compared to the rest of the FedExCup. Scheffler has more than double the FedExCup points than No. 2 Clark. Scheffler has more than triple the points of No. 8 Byeong Hun An.

Clark would need to win multiple more Signature Events (and then some) to reach Scheffler’s current point total.

At this pace, it won’t be the race to the FedExCup; it will be a slow march to King Scheffler's coronation.

Five Nights at Freddy’s: Help Wanted 2 coming to PS5 on June 20

Steel Wool Studios will release Five Nights at Freddy’s: Help Wanted 2 for PlayStation 5 on June 20, the developer announced .

Five Nights at Freddy’s: Help Wanted 2 first launched for PlayStation VR2 on December 14, 2023. Users who already own the PlayStation VR2 version will receive the PlayStation 5 version at no extra charge.

Get the latest details below.

Have you been feeling some FOMO at Freddy’s recently? Has your lack of a PlayStation VR2 left you unable to play Five Nights at Freddy’s: Help Wanted 2 ? Well, I come (Faz)bearing great news: Freddy and the Gang heard you loud and clear, and they’re coming to your PlayStation 5 on June 20. Welcome to the team, superstars. This adaptation of Help Wanted 2 seamlessly transitions the terrifying gameplay experienced in the PlayStation VR2 game onto the flat screen utilized by the power of the PlayStation 5. No matter how you prefer to play, be it from your couch or wandering around your living room with a headset, Fazbear Entertainment is happy to welcome you to the Pizzaplex. As a new hire, your job comes down to a series of tasks, the most important of which is clocking out of your shift alive. Stave off thirsty bots with a refreshing Fizzy Faz, now made using point-and-click controls for speedier service, happier customers, and less danger. window.top.__vm_add = window.top.__vm_add || []; //this is a x-browser way to make sure content has loaded. (function(success) { if(window.document.readyState !== "loading"){ success(); } else { window.document.addEventListener("DOMContentLoaded", function(){ success(); }); } })(function(){ var placement = document.createElement("div"); placement.setAttribute("class", "vm-placement"); if (window.innerWidth > 1000) { //load desktop placement placement.setAttribute("data-id", "5f806c91d0b2e22f7c503a7a"); } else { //load mobile placement placement.setAttribute("data-id", "5f806cb5d0b2e22f7c503a7c"); placement.setAttribute("data-display-type", "hybrid-banner"); } document.querySelector("#ad-container-2").appendChild(placement); window.top.__vm_add.push(placement); }); You know what they say though, all work and no play makes for an awfully dull shift. When you’re not trying to avoid the nearest animatronic, you’re encouraged to explore your new job and see all it has to offer. Cool off on Captain Foxy’s Log Ride, which now operates with first-person shooter -style controls that allow gamers to aim and fire with the swivel of a joystick and pull of a trigger. The game contains all the joy of Freddy’s VR dark ride, but with more opportunities to impress your boss with that high score you’ve been shooting for (pun intended). The fun doesn’t stop there. All of the games in Help Wanted 2 have been carefully adapted to accommodate the PlayStation 5, allowing a wider range of new hires to come join the fun. If you’re already an employee via the PlayStation VR2, the standard PlayStation 5 version is included at no extra cost to you. Our team has taken great care to ensure that every step you take is filled with the chills and thrills only Freddy’s can provide.

Shakira announces 2024 U.S. tour. Get tickets today

The “She Wolf” is coming out of her cage.

For the first time since 2018, Shakira will officially tour North America as part of her upcoming ‘LAS MUJERES YA NO LLORAN WORLD TOUR’ (which translates to “women no longer cry”).

Midway through the 14-concert run, the three-time Grammy winner is scheduled to stop in at Brooklyn’s Barclays Center on Thursday, Dec. 5.

At all shows, fans can expect to hear cuts from the Colombian icon’s recently released album that shares a name with the tour.

Hits like “Hips Don’t Lie,” “Whenever, Wherever,” and “Dia De Enero” will likely sneak into the setlist as well.

For those who can’t wait to get their hands on tickets, they can be secured as soon as today.

Although inventory isn’t available on Ticketmaster until Monday, April 22, fans who want to ensure they have tickets ahead of time can purchase on sites like Vivid Seats before tickets are officially on sale.

Vivid Seats is a secondary market ticketing platform, and prices may be higher or lower than face value, depending on demand.

They have a 100% buyer guarantee that states your transaction will be safe and secure and will be delivered before the event.

A complete calendar including all tour dates, venues, and links to buy tickets can be found below.

On March 22, Shakira released her 12th studio album “Las Mujeres Ya No Lloran.”

Comprised of 17 tracks featuring big-name collaborators like Cardi B, Bizarrap, Grupo Frontera, Rauw Alejandro, and Ozuna among others, the 47-year-old songstress sticks to español this time around. However, whether or not you speak Spanish is beside the point; Shakira’s music this time around is so catchy that you won’t mind looking up what the lyrics mean on Google Translate.

Highlights include the dance floor-ready hit “Shakira: Bzrp Music Sessions, Vol. 53,” the infectious mariachi-inspired “El Jefe” with Fuerza Regida, and the icy “TQG” featuring Karol G.

While the whole album slaps, we’d be remiss if we didn’t shout our favorite track: the disco-flavored “Cohete.” We can’t stop listening.

Want to check it out for yourself? You can find “Las Mujeres Ya No Lloran” here .

Prior to announcing the tour, Shakira surprised the crowd at at Coachella on Friday, April 12 when she performed “La Fuerte” and “BZRP Music Sessions, Vol. 53” alongside Bizzarap.

A raucous fan video can be seen here:

Many Latin icons are on the road all year long.

Here are just five of our favorites you won’t want to miss live.

•  Jennifer Lopez

•  Fuerza Regida

• Gloria Trevi

Who else is out and about? Take a look at our list of the 50 biggest concert tours in 2024 to find out.

This article was written by Matt Levy , New York Post live events reporter. Levy stays up-to-date on all the latest tour announcements for your favorite musical artists and comedians, as well as Broadway openings, sporting events and more live shows – and finds great ticket prices online. Since he started his tenure at the Post in 2022, Levy has reviewed Bruce Springsteen and interviewed Melissa Villaseñor of SNL fame, to name a few. Please note that deals can expire, and all prices are subject to change.

Vivid Seats is the New York Post's official ticketing partner. We may receive revenue from this partnership for sharing this content and/or when you make a purchase.

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Author Interviews

Two nights before the attack, salman rushdie dreamed he was stabbed onstage.

Terry Gross

Salman Rushdie says writing Knife allowed him to change his relationship to the attack. "Instead of just being the person who got stabbed, I now see myself as the person who wrote a book about getting stabbed," he says. Rachel Eliza Griffiths/Penguin Random House hide caption

Salman Rushdie says writing Knife allowed him to change his relationship to the attack. "Instead of just being the person who got stabbed, I now see myself as the person who wrote a book about getting stabbed," he says.

Two nights before he was stabbed onstage at a literary event in 2022, Salman Rushdie had a nightmare. In his dream, Rushdie was in an ancient Roman amphitheater, rolling around on the ground while a gladiator with a spear stabbed down at him.

"It certainly felt very vivid and very actual and very scary," Rushdie says. "I was rolling about in bed and thrashing around, and my wife had to wake me up."

His immediate impulse was to cancel his upcoming appearance at the Chautauqua Institution in upstate New York, but then he rationalized his fear away: "People have dreams. You don't run your daily life because of having a bad dream. And so I decided I would go," Rushdie says.

Rushdie is no stranger to death threats. After his 1988 novel The Satanic Verses was published, Iran's leader, the Ayatollah Khomeini, declared the book blasphemous in its treatment of Islam and the Prophet Muhammad, and issued a fatwa calling for Rushdie's death. Faced with the threat, Rushdie, who grew up in India in a secular Muslim family, stayed out of public view for years.

But after years of being surrounded by security, Rushdie had resumed normal life. Then, on Aug. 12, 2022, a 24-year-old man in the audience rushed the stage at Chautauqua. In an attack lasting 27 seconds, the man stabbed Rushdie multiple times, severing all the tendons and most of the nerves in one hand and wounding his neck, chest, thigh and eye.

5 takeaways from Salman Rushdie's new memoir 'Knife'

Lying in a "lake of blood," Rushdie believed he was about to die. He was hit with a feeling of loneliness: "Dying in the company of strangers — that was what was going through my mind," he says.

Rushdie says he's disinclined to believe in miracles, but he struggles with how else to explain his survival.

"Many of the doctors who I have been involved with in the last year and a half are not only surprised that I survived — which they are — but they're surprised that I have recovered to the degree that I have," he says. "Miracles are all around me, it seems."

Why Salman Rushdie's work sparked decades of controversy

In his new book, Knife , Rushdie writes about the attack, the damage to his body (including the loss of sight in one eye) and more existential questions about facing death and finding his identity in an altered body and state of mind. Rushdie says he was initially reluctant write about the Chautauqua incident, but he's glad he did.

"It changed my relationship to the event," he says. "Instead of just being the person who got stabbed, I now see myself as the person who wrote a book about getting stabbed. And so it feels like it's back in my own authorial space, and I feel more in charge of it. And that feels good."

Interview highlights

On how he had stopped worrying about the fatwa prior to the attack

I'd been living in New York City for close to 24 years, and during that time I'd done hundreds of literary events, readings, lectures, festivals, etc., and they had never been the faintest trace of a problem. So I'd kind of told myself that that time had gone, but sadly, I was wrong. All I ever wanted to do, Terry, was to write stories. And if I've got a few more years left to write a few more good stories, then that'll do for me.

On not holding onto anger about the attack

Becoming 'Anton,' Or, How Rushdie Survived A Fatwa

One of the things that I think has been very strange for me is that the emotion that I haven't really had in the aftermath of all this is anger. And it's as if something in my head tells me that anger would be a way of being stuck in the moment. It would be a way of not being able to get past it. And so I don't have anger. I guess somewhere deep down I am pretty furious with various people, notably the gentleman with the knife, but it doesn't seem productive to me to linger on anger.

On writing about both violence and hatred and healing and love

When I started thinking about writing this book, you know, I asked myself, "OK, there's obviously this attack and I want to talk about that. But beyond that, what's the book about?" And I came to feel that it was about myself being in between two forces. One is a force of violence and hatred, and the other is the force of love and healing. ...

The first force was obviously embodied in my assailant, and the second force was embodied in my wife, Eliza, the writer Rachel Eliza Griffiths. And I mean, since I didn't die, I'm able to say that the force of love and healing overcame the force of violence and hatred. But I felt that that triangle was what the book was about, that the book was about three people. It was about me and him and Eliza. And so I wanted to write about love, and in a more open and direct way than maybe that I've ever done before.

On the hallucinations he experienced while on a ventilator

I was hallucinating palaces made out of alphabets. ... I was seeing architecture, palatial architecture, of which the building blocks were all letters. ... There were letters floating in the air between me and other people in the room. I remember when the ventilator was taken out I said to Eliza and her family, I said, "Why are all these letters on your clothes?" ... It does indicate to me the extent to which the world of books is the world in which I live.

On how being attacked changed his understanding of death

I think what it did is two things that it, first of all, gave me a kind of familiarity with death. I kind of know how it goes now. I didn't get to the final note of the music, thank goodness. But I kind of understand how the tune goes. But also what it did, what it has done, is to give me an enormously increased appreciation of life. The reason I quoted, one point, a poem by Raymond Carver , written after he was told he had almost no time to live, and then he lived another 10 years and did some of his best work. And he said he felt like all that time that he wasn't supposed to have, he describes it as gravy. "Every day is gravy." And I kind of feel like that now. I feel like these are days I wasn't supposed to have, and yet here I am, having them, and every day is a blessing.

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Sam Briger and Joel Wolfram produced and edited this interview for broadcast. Bridget Bentz, Molly Seavy-Nesper and Meghan Sullivan adapted it for the web.

Chicano Batman are not who you expect them to be

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In the summer of 2017, Chicano Batman played a free show at One Colorado, the shopping complex in the heart of Old Pasadena, as part of KCRW’s Summer Nights concert series. The band had just released “Freedom Is Free,” its third album.

Organizers had anticipated a large turnout for the event and warned fans for a week that they expected to reach capacity early.

The place was packed an hour before the band hit the stage. Fans arriving late, many of them Latino, climbed over and crawled under temporary fences installed for crowd control to try to get in to see the hometown heroes play their signature blend of Latin psychedelic, Chicano old-school soul and Tropicalía.

“That was some Beatles s—,” Eduardo Arenas, the band’s bass player, recalled with a chuckle.

“It was awesome because the power went out on the last song,” added lead vocalist Bardo Martinez over food at Highland Park’s breakfast taco joint HomeState.

“Power outages always make for a moment,” said Arenas.

Arenas and Martinez remember that evening of loosely controlled chaos fondly, but are quick to point out that it was a different era for the band, which recently released the new album “Notebook Fantasy” via indie label ATO Records.

Onstage, the band members have abandoned their matching three-piece ruffled suits — a nod to old-school Latin acts on both sides of the border— in favor of a “wear what you feel” dress policy. The quartet also became a trio; longtime drummer Gabriel Villa left the band after the release of 2020’s “Invisible People.”

Entertainment & Arts

Self-isolating with Chicano Batman

Chicano Batman’s new record, “Invisible People,” is coming out during a pandemic. But they aren’t worried.

April 22, 2020

Notably, that experimental album, influenced by krautrockers Can and Nigeria’s William Onyeabor, was the first Chicano Batman album to not feature any songs in Spanish, an artistic decision that puzzled some longtime fans who expected the band to stay loyal to the “Chicano” part of its name.

Arenas sees it differently, arguing that the band was always meant to be an experimental project.

“We’re always trying to see where we can go next,” he said. “This is just part of the natural journey we’re on. I think it’s dope to have a fan base that believes in what we’re doing, and we can’t be the same people for 16 years, you know?”

Even the band’s name, which Martinez says he came up with while enrolled at UCLA in the early 2000s, allowed for interpretation and growth.

“I just put the two names together and thought, ‘Damn, that’s a dope band name,’” he said, adding that he drew inspiration from his own involvement in the school’s MEChA chapter. It was this political awakening that made him realize that there was no one way to be Chicano. “That was the whole point, that the band could be whatever we decided it to be.”

If “Invisible People” was a departure from the familiar, “Notebook Fantasy” is the arrival of what Chicano Batman was always meant to sound like.

Working with producer John Congleton, the band recorded at Sunset Sound, the legendary Hollywood studio used by some of the biggest names in rock history.

Commentary: Sunset Sound is one of the last great L.A. recording studios. To lose it would be a disaster.

The legendary Sunset Sound studios in Hollywood is in trouble. If we care about music — its history and its future — we must take notice.

Feb. 20, 2024

“We were in the same place where all these big bands — the Doors, Van Halen — recorded their albums,” said Arenas. “We were in the company of legends and the walls were talking to us.”

The ghosts of some of these legends show up in the recordings. The album’s first single, “Fly,” channels Prince — a Sunset Sound alumnus — through its upbeat funk and sexually charged lyrics. The track’s music video is also a visual throwback to the Purple One’s 1979 eponymous sophomore LP, which featured the musician shirtless on the cover.

“Notebook Fantasy” also marks the return of Spanish-language songs on Chicano Batman albums. “Era Primavera” is a slick, springtime ode to Latin America’s pop balladeers of the ‘70s and ‘80s that starts to bloom by the time it hits the chorus. Arenas takes over lead singer duties in “Hojas Secas” and gets self-reflective in this down-tempo track about the end of a relationship. His song of lament is accompanied by Carlos Arévalo’s wailing guitar, and crescendos with a full-on choir.

“We’re not trying to shy away from them,” Arenas said of their Spanish tracks. Rather, they’re just one of their many musical offerings. “It’s like ‘Yo, here’s your coconut chocolate, here’s your almond with sea salt and here’s your Carlos V [Mexican chocolate bar], we got you bro.”

“We were just trying to have fun in the studio, just geeking out on what potential sounds we could make,” said Martinez.

On Friday, Chicano Batman will take its vast musical samplings on tour, kick-starting a run of 32 U.S. cities in Las Vegas and ending with a homecoming show on June 29 at the historic Kia Forum in Inglewood.

“This journey to the Forum, it’s kind of a mammoth journey that involves everyone,” Arenas said before listing several local bands that came up with them and helped Chicano Batman become better musicians, including Brainstory, Chola Orange and Tropa Magica.

The other group that helped propel their journey: the fans, opinions and all.

“Chicano Batman at the Forum is not an individual act,” he added. “It’s a collective, a movement. You don’t play at the Forum without the people’s power, and so I’m thankful for that.”

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Fidel Martinez writes the Latinx Files, a weekly newsletter that focuses on the American Latinx experience. He started at The Times in 2018 as an audience engagement editor, focusing on sports. Previously he worked as politics editor for Mitu, as a social storytelling producer for Fusion Media Group and content curator and managing editor for Break Media. He is a proud Tejano who will fight anyone who disparages flour tortillas.

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Germany’s Leader Walks a Fine Line in China

Chancellor Olaf Scholz tried to promote German business interests while delivering warnings from Europe about trade and geopolitical tensions.

By Alexandra Stevenson and Melissa Eddy

Chancellor Olaf Scholz of Germany tried to strike a delicate balance on a trip to China this week, promoting business ties with his country’s biggest trading partner while raising concerns over its surge of exports to Europe and its support for Russia.

Mr. Scholz met with China’s top leader, Xi Jinping, at the Diaoyutai State Guesthouse in Beijing on Tuesday, the culmination of a three-day visit with a delegation of German officials and business leaders. He also met with Premier Li Qiang as the two countries navigate relations strained by Russia’s war in Ukraine and China’s rivalry with the United States, Germany’s most important ally.

Throughout his trip, Mr. Scholz promoted the interests of German companies that are finding it increasingly hard to compete in China. And he conveyed growing concern in the European Union that the region’s market is becoming a dumping ground for Chinese goods produced at a loss .

But Mr. Scholz chose a conciliatory tone over a combative one in his opening remarks before sitting down with Mr. Xi on Tuesday morning, in a meeting that stretched over three hours and turned into a walk and lunch.

It was the German leader’s first visit to China since his government adopted a strategy last year that defined the Asian power as a “partner, competitor and systemic rival ,” calling on Germany to reduce its dependency on Chinese goods.

Germany’s economy shrank last year, and its weaknesses have exposed a reliance on China for growth. Energy prices have risen because of the war in Ukraine, which has been facilitated by Beijing’s support for the Kremlin. German companies have pushed for more access to China and complained that they face unfair competition .

The chancellor visited German companies with extensive investments in China and met with trade representatives and officials in the sprawling industrial metropolis of Chongqing in China’s southwest and in Shanghai and Beijing.

At a talk with a group of students in Shanghai on Monday, Mr. Scholz fielded a question from a student who planned to study in Germany this year, who said he was “really worried” because the country had partially legalized cannabis. “When you study in Berlin, you can run around the whole time and never meet anyone who does that,” the chancellor assured him.

But he also used the platform to push more serious messages about trade. “Competition must be fair,” Mr. Scholz told the students. “We want a level playing field,” he said.

Mr. Scholz’s trip was an example of the difficult dance that Germany is trying to do: maintaining economic ties with China while managing U.S. pressure to align itself more closely with Washington against Beijing.

In his meetings, Mr. Scholz highlighted Germany’s commitment to doing business with China, but he also warned that Beijing had to curb the flood of Chinese goods into Europe. At the same time, he expressed reservations about the European Union’s investigations into China’s use of subsidies for green technology industries, saying that any discussion about trade must be based on fairness.

“This must be done from a position of self-confident competitiveness and not from protectionist motives,” Mr. Scholz told reporters on Monday.

China’s manufacturing push in green sectors like electric cars and solar panels has touched off trade disputes with Europe and the United States, where such industries have also received government support. But with 5,000 German companies active in the Chinese market, Germany stands to lose more than many of its European partners would if Beijing were to retaliate against the European Union.

“If the E.U. goes too hard against China, we could expect countermeasures and this would be a catastrophe for us,” said Maximilian Butek, the executive director of the German Chamber of Commerce in China.

“For us it’s extremely important that the Chinese market remains open,” he said.

In his meeting with Mr. Xi, Mr. Scholz indicated that Russia’s war on Ukraine and its arms buildup were high on his agenda. “They directly affect our core interests. Indirectly, they damage the entire international order,” he said in opening remarks at the meeting, a transcript of which was provided by Mr. Scholz’s office.

Despite pressing the issue with Mr. Xi, he did not appear to win the commitment that he had sought from the Chinese leader to take part in an international conference for Ukraine planned in June. Germany had hoped that China could use its influence over Russia to help work toward a peace agreement.

Germany would also like China stop selling goods to Russia that have potential uses on the battlefield, and Mr. Scholz told reporters that he had brought up the issue in his meeting. “The point has been made,” he said. “There can be no misunderstanding about how we see things.”

China is hoping to drive a wedge between Europe and the United States by courting leaders such as Mr. Scholz. State media reports depicted his visit as demonstrating the strength of China’s relations with Europe, playing up its economic ties with Germany.

In his opening remarks to Mr. Scholz, Mr. Xi said cooperation between China and Germany, which have the second- and third-largest economies, was beneficial to the world, a remark that could be read as directed at those who have urged Berlin to distance itself from Beijing.

“The two countries should view and develop bilateral relations from a long-term and strategic perspective and work together to inject more stability and certainty into the world,” Mr. Xi told Mr. Scholz, emphasizing the importance of seeking “common ground.”

Beijing is sure to welcome the message that German businesses are committed to China. The Asian giant is trying to court foreign investment to reinvigorate its economy, which has faltered because of a housing slowdown. Some Western businesses and investors have also been rattled by Mr. Xi’s emphasis on national security, which they regard as making it riskier to operate in the country.

From China’s perspective, Germany may be its best hope of delaying or watering down any trade restrictions from Europe, said Noah Barkin, a senior adviser in the China practice at the Rhodium Group, a research firm.

German carmakers have invested billions of dollars in China, and much of their revenue comes from there. Many worry that if the European Commission imposes higher tariffs on Chinese exports, and Beijing retaliates, German businesses will suffer most.

Chinese officials “know that German companies are heavily invested and they use that politically to influence political decision making in Berlin,” Mr. Barkin said.

Germany’s biggest companies, including BMW, Mercedes-Benz and BASF, have large operations in China and strong, effective lobbies in Berlin, Mr. Barkin added. Executives from those companies, along with several others, traveled with Mr. Scholz to China.

“The supply chain in China is stuffed with German goods,” said Joerg Wuttke, a former president of the E.U. Chamber of Commerce in China. “If China has a price war with Germany, then no one will make money anymore.”

Mr. Scholz also brought along the German ministers for agriculture, the environment and transportation, officials who experts said would be particularly interested in working with China.

“You set an agenda with these three ministers, the tonality is overall a cooperative one, these are areas that we want to work on,” said Janka Oertel, director of the Asia Program at the European Council on Foreign Relations.

Chinese officials, for their part, have brushed off the European accusations of unfair trading practices, calling them groundless and an act of “ typical protectionism .” They have hinted that they could retaliate for any actions taken by the European Union, saying that China was “strongly dissatisfied with and firmly opposes” its investigations.

In an interview with the German newspaper Handelsblatt, Wu Ken, China’s ambassador to Germany, said the competitive edge of Chinese electric vehicles “relies on innovation, not subsidies.”

“The challenge faced by developed countries lies more in the fact that Chinese companies are more efficient,” the ambassador said.

Vivian Wang contributed reporting from Beijing and Zixu Wang from Hong Kong.

Alexandra Stevenson is the Shanghai bureau chief for The Times, reporting on China’s economy and society. More about Alexandra Stevenson

Melissa Eddy is based in Berlin and reports on Germany’s politics, businesses and its economy. More about Melissa Eddy

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NikeCourt Air Zoom Vapor 9.5 Tour

Men's tennis shoes.

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Etch your name into greatness with the Vapor 9.5 Tour. It has a breathable mesh upper that moves with your foot, and a Zoom Air unit provides stability and a lightweight, responsive feel on the court.

• Shown: Hot Punch/Volt/White/Racer Blue
• Style: FB2664-600

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Grab'em while they're still making them

btg44 - Apr 10, 2024

Best tennis shoe by far. I keep buying 2-3 pairs at a time only because you never know when they're going to stop making them. Last few colors leave something to be desired but the fit is perfection. Looks like they're sold out in multiple sizes so maybe we'll get more colors soon! ...

Best tennis shoe ever

AnteP379217777 - Mar 01, 2024

Great shoe, best tennis shoe ever made in my opinion. We just need some nicer colour options. Thanks

For me, still the best tennis shoe by Nike.

indoor tennis player - Feb 24, 2024

Looks sweet and fits true to size.

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When the shoe is laced up, a band tightens around the midfoot to provide a snug fit to provide a glove-like fit. Mesh and synthetic material provide a breathable, lightweight fit.

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A Zoom Air unit in the heel provides low-profile, responsive cushioning. A foam midsole gives you lightweight, yet firm cushioning and optimal responsiveness.

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• Molded midfoot plate provides lateral support and stability.
• Rubber outsole with an extra-durable rubber compound on the heel and forefoot gives you multi-surface traction and durability.