How to do Sudoku Puzzles

怎样才能使数学变得更有趣英语作文全文共3篇示例，供读者参考篇1How to Make Math More FunMathematics is often seen as a difficult and boring subject by many students. The perception that math is hard and uninteresting leads to a lack of motivation and interest in learning it. However, there are ways to make math more enjoyable and engaging for students. By incorporating fun and interactive activities into mathematics lessons, teachers can help students develop a love for math and improve their understanding of the subject.One way to make math more fun is to use real-world examples and applications in lessons. By showing students how math is used in everyday life, such as in shopping, cooking, or sports, teachers can help students see the relevance of math and make it more engaging for them. For example, teachers can use examples of fractions in recipes or percentages in discount calculations to demonstrate how math is used in real life.Another way to make math more interesting is to incorporate games and puzzles into lessons. Games can help students learn and practice math concepts in a fun and interactive way. For instance, teachers can use math bingo, math dice games, or math board games to make learning math more enjoyable for students. These games not only help students practice their math skills but also make learning math more engaging and exciting.In addition, technology can be used to make math more interesting for students. There are many math apps and online resources available that make learning math fun and interactive. For example, students can use math apps to practice their math skills through games, quizzes, and interactive lessons. Teachers can also use online math resources to design engaging and interactive activities for students to help them learn math in a more enjoyable way.Furthermore, incorporating hands-on activities and manipulatives into math lessons can make learning math more engaging for students. Hands-on activities, such as using blocks, counters, or measuring tools, can help students visualize and understand math concepts better. For instance, students can use blocks to understand addition and subtraction or use measuringtools to learn about geometry and measurement. By incorporating hands-on activities into math lessons, teachers can make learning math more interactive and fun for students.In conclusion, there are many ways to make math more enjoyable and engaging for students. By incorporatingreal-world examples, games, technology, and hands-on activities into math lessons, teachers can help students develop a love for math and improve their understanding of the subject. With a bit of creativity and innovation, math can become a fun and enjoyable subject for students to learn and excel in.篇2How to Make Math More InterestingMath is a subject that is often met with a groan from students. Many people find math boring or difficult to understand. However, with the right approach, math can actually be a fun and engaging subject to learn. Here are some tips on how to make math more interesting:1. Make it relevant: One way to make math more interesting is to show students how it is relevant to their everyday lives. For example, you can teach them how math is used in cooking,sports, or even in music. By making math relatable, students are more likely to see its value and importance.2. Use real-life examples: Instead of just giving students abstract math problems, try using real-life examples to illustrate mathematical concepts. For instance, you can use shopping scenarios to teach students about percentages or fractions. By connecting math to the real world, students can see how it applies to their daily lives.3. Make it hands-on: Another way to make math more interesting is to make it hands-on. Let students use manipulatives, such as blocks or counters, to solve math problems. This can help make abstract concepts more concrete and easier to understand.4. Incorporate technology: Technology can also be a great tool for making math more interesting. There are many online resources and apps that can make learning math fun and engaging. For example, math games and puzzles can help make learning math more interactive and enjoyable.5. Encourage group work: Working in groups can make math more interesting by allowing students to collaborate and learn from each other. By working together, students can see differentapproaches to solving problems and can help each other when they are stuck.6. Make it fun: Lastly, don't be afraid to make math fun! Use games, challenges, and rewards to make math more engaging for students. By adding an element of fun, students are more likely to be motivated to learn and enjoy the subject.Overall, making math more interesting requires creativity, relevance, and a willingness to try new approaches. By incorporating these tips into your teaching, you can help make math a more enjoyable and engaging subject for students.篇3How to Make Mathematics More InterestingMathematics is often seen as a dry and dull subject by many students. The endless equations and calculations can be intimidating and boring to some people. However, with a little creativity and imagination, mathematics can actually be quite fun and fascinating. In this essay, we will explore some ways to make mathematics more interesting and enjoyable.1. Real-life Applications:One of the best ways to make mathematics more interesting is by showing students the real-life applications of the concepts they are learning. Mathematics is not just about numbers and equations; it is also about solving real-world problems. By showing students how mathematics is used in everyday life, they can see the relevance and importance of the subject.For example, teachers can use examples such as calculating the amount of paint needed to paint a room, or designing a budget for a shopping trip. By connecting mathematics to practical situations, students can see the value of the subject and become more engaged in learning.2. Hands-on Activities:Another effective way to make mathematics more interesting is by incorporating hands-on activities into the curriculum. Instead of just sitting at a desk and solving problems on paper, students can use manipulatives such as blocks, counters, or measuring tools to explore mathematical concepts.For example, teachers can use base-ten blocks to teach place value, or use geometric shapes to teach geometry. By allowing students to physically manipulate objects and visualize concepts, mathematics becomes more concrete and tangible.3. Games and Puzzles:Mathematics can also be made more interesting through games and puzzles. There are many math games and puzzles available that can help students practice their skills in a fun and interactive way. For example, teachers can use Sudoku puzzles to practice logical reasoning, or play math bingo to practice multiplication facts.By turning math into a game, students can develop their problem-solving skills and mathematical thinking in a playful and engaging way. Games and puzzles can also help students see the fun and challenging side of mathematics, rather than just the tedious side.4. Collaborative Learning:Collaborative learning is another effective way to make mathematics more interesting. By working together with peers on challenging problems or projects, students can discuss ideas, share strategies, and learn from each other.Collaborative learning can help students see different perspectives and approaches to solving problems, and can also foster a sense of camaraderie and teamwork. By workingtogether, students can build their confidence and motivation in mathematics, and see that they are not alone in their struggles.5. Multimedia Resources:With the advancement of technology, there are many multimedia resources available that can make mathematics more interesting and engaging. Teachers can use videos, simulations, animations, and interactive software to illustrate concepts and make learning more visual and interactive.For example, teachers can use Khan Academy videos to explain difficult concepts, or use Desmos graphing calculator to explore functions and equations. By incorporating multimedia resources into the curriculum, students can see mathematics in a new light and develop a deeper understanding of the subject.In conclusion, mathematics can be a fascinating and rewarding subject if taught in a creative and engaging way. By using real-life applications, hands-on activities, games and puzzles, collaborative learning, and multimedia resources, teachers can make mathematics more interesting and enjoyable for students. With a little imagination and effort, mathematics can be transformed from a dry and dull subject into a fun and exciting adventure.。

培养逻辑思维英语作文英文：Developing logical thinking is essential for problem-solving, decision-making, and critical analysis. It helps individuals to make sound judgments and draw valid conclusions. There are several ways to cultivate logical thinking, and I have personally found the following methods to be effective.Firstly, engaging in activities that require logical reasoning, such as puzzles, riddles, and brain teasers, can help to exercise the brain and improve logical thinking skills. For example, I enjoy solving Sudoku puzzles and playing logic-based games like chess and Scrabble. These activities challenge my mind and force me to thinkcritically and strategically.Secondly, studying formal logic and reasoning can also enhance logical thinking. Learning about logical fallacies,deductive and inductive reasoning, and syllogisms has helped me to understand how to construct and evaluate arguments. By studying these concepts, I have become more adept at identifying flawed reasoning and constructing coherent arguments.Additionally, practicing mindfulness and meditation can contribute to the development of logical thinking. By training the mind to focus and concentrate, individuals can improve their ability to think logically and analytically.I have personally found that incorporating mindfulness practices into my daily routine has helped me to approach problems and decisions with a clearer and more rational mindset.Furthermore, engaging in debates and discussions can sharpen logical thinking skills. By presenting and defending my own arguments while critically evaluating the points made by others, I have learned to think more logically and persuasively. Engaging in debates has taught me to consider multiple perspectives and weigh evidence and reasoning in a more systematic manner.In conclusion, developing logical thinking is crucialfor navigating the complexities of the world. By engagingin activities that challenge the mind, studying formal logic, practicing mindfulness, and participating in debates, individuals can cultivate their logical thinking skills and become more adept at reasoning and problem-solving.中文：培养逻辑思维对于解决问题、做决策和进行批判性分析至关重要。

Some people think that it is important to use leisure time for activities that improve the mind, such as reading and doing word puzzles. Other people feel that it is important to rest the mind during leisure time.Which one do you agree with? Discuss both views and give your opinion. Give reasons for your answer and include any relevant examples from your own knowledge or experience. You should write at least 250 words.Sample Answer 1:Many people believe that it is significant using spare time for things that enhance the mind, for instance, reading and doing word puzzles. The others think that it is vital to relax the mindduring spare time. In my opinion, I agree that people should use leisure time for activities which improve the mind.In one hand, some people have very exhausting jobs. Therefore, they would like to relax during leisure time, so they do not want to do anything within this time. They think they need to rest. Moreover, many people are lazy, for this reason, they do not keen to do activities to improve the mind. For instance reading a book and doing sudoku puzzles.On the other hand, many people have stressful jobs too, but they like to enhance the mind.Therefore, they do many things within leisure time such as reading a novel, solving puzzles, doing exercises which are useful for improving the mind. I think that people who utilise spare time doing activities that boost the mind are more successful than others who love to use spare time for relaxing.Finally, in my opinion, people should use their leisure time to improve the mind. For this reason, people should do anything which is useful for enhancing the mind. Moreover, ı would argue that people should be hard working not to be idle. So they always do something to be moresuccessful. Furthermore, I think some activities are so relaxing while they improve the mind. Such as doing sudoku puzzles.In conclusion, ı agree that people should do activities that improve the mind during leisure time. because, while that they are performing activities for improving the mind, the activities can contribute to job achievement. Moreover, I think, leisure time should be used effectively since it is so precious.Sample Answer 2:Nowadays people are divided on the issue how to utilise leisure time effectively. One line of thought believes there should be mind-twisting activities to develop mind but on the other hand, there are those who affirm with this notion that it is better to rest .I am partial accord with this ideology. In following paragraphs, I will shed light on arguments to supporting my views.To launch into the topic, as according to some people rest is necessary to keep our mind fresh and healthy and up to some extent it is right. Giving rest to mind revitalise it, helps to think better. Moreover, as our body is not a machine,so respite is mandatory. 24 hours use of mind can cause stress and many types of health maladies. So up to some extent ,it is mandatory to take a respite for some time.On the flip side, in the modern era, cut throat competition is going on in every domain, so developed minds can attain success. So these types of activities play a pivotal role in life. Activates like reading books can also play a role in broadening our knowledge, which will be helpful in later life. It can open new vistas of employment and opportunity. By solving puzzles and playing brain games, it escalates creativepower of our mind, a level of thinking also develop in an effective manner.At the end of the discussion, after analysing of arguments, both sides have their reasons for their arguments. Rests, as well as the development activities, are both are equally important. At last, I am in favour of both of these.I would suggest a good balance of taking rest and engaging in mind enhancing activities in leisure time for any individual.Sample Answer 3:Currently, what we should do in leisure time for the well-being has been a matter of public attention in this contemporary life. It is believed that doing some leisure activities will enhance people’s frame of mind. Diametrically, opposed to this, some people put forward the idea that the rest will be good for our mind. As far as I am convinced that the former opinion should receive more support.Central to this idea is the argument that for most people, the benefit of spending spare time for doing some recreational activities, such as reading and doing word puzzle is to promotebrain’s ability, which will enable them to become more intellectual and brilliant. The key explanation to this could be that evidence in favour of playing puzzles and answering IQ quizzes is found in much research which reveals that these activities will boost human mindset effectively and logically.Another noticeable justification is that leisure pursuits such as reading books and writing diary or novel will drive people further toward enriching their horizon and knowledge as well as cultivating their critical thinking. For example, in some developed countries, the governmentshave encouraged their citizens in creating the reading habit by establishing public libraries and e-libraries with a mixed variety of books.It is also true that refreshing mind by rest is particularly important for people’s living, which can make it possible for human to relax and unwind after long working time. If we do not pay enough attention to rest and sleep, it will be likely to make us feel exhausted and irritable. However, leisure activities are more important for most people.To recapitulate, the main advantage of spending spare time for doing leisure pursuits isto improve our frame of mind and thus make us develop IQ ability well. Meanwhile unwinding after hard work and study is not crushed aside.Sample Answer 4:It is generally accepted that we all require leisure time to recover from the hassle of works and everyday life. There are many ways individuals can spend their spare time. Few people believe the playing scrabble or reading a novel in free time is the best approach as it progresses mind, however, others believe that we should give rest to the brain. This essay will analyse both views before drawing a conclusion.On the one hand, people relax by playing sports, watching movies, playing scrabble and so on. They say that these physical and mental activities serve as a tonic and refresh them to face new challenges. For instance, people who are largely involved in indoor work like Computer Operator or Radar Operator spent most of their time in front of a computer screen and for them to go outside in fresh air, stretching their muscles, playing sports in spare time could ease the workloads and reduces mental pressures.On the other hand, few people believe that it is good to give rest to mind then involving in mindgames and other physical activities. They say that taking a power nap, relaxing on a rocking chair or spending sometimes at the beach as a silent observer would help them to energise their brain and to remain healthy. For instance, people who are involved in outdoor work like cargo loader, courier delivery and so on prefer to spend their leisure time relaxing on a sofa or bed.Although the above look is a solid evidence for both sides of the argument, I personally believe that it should be up to people to decide what isbest for them in leisure time that can help them to relax and remain healthy.Sample Answer 5:There is a great shift in people's lifestyle over the last few years. These days they involve themselves in hectic work routine and have very less time for leisure activities. However, leisure time is essential to ease tension and to get refreshed and energetic. Some people argue that leisure time can be utilised efficiently by doing mind booster type activities. Others state that after busier schedule, one should relax their mind. I believe that leisure time provides usopportunities to further enhance our knowledge and mind.Activities like reading, playing puzzle games etc. are the best use of spending leisure time. One cannot deny about the numerous benefits of reading. All the scientists and scholars used to read books in their leisure time. It not only increased our knowledge bur also enable us to draw our own observation under the light of reading materials. Furthermore puzzle games contribute a lot in making mind sharp and efficient.Nowadays, the lifestyle of people has dramatically changed because of the economic crisis and recession. People get involve in multiple jobs in order to earn their living. So it is mandatory to have peace and rest of mind by doing activities such as listening music, watching television and doing physical activities. It will help them in easing tension and make them energetic for their future work.In conclusion, I would like to say that constructive activities in leisure time could contribute immense benefits to us. By doing these activities one can gain plenty ofknowledge which would definitely help them in their coming life.。

做拼图的英文作文带中文英文回答：Puzzles are a popular pastime that people of all ages enjoy. They come in a variety of shapes and sizes, and they can be made from different materials. Some puzzles are designed to be challenging, while others are more relaxing.There are many different types of puzzles, including jigsaw puzzles, crossword puzzles, Sudoku puzzles, and word searches. Jigsaw puzzles are made up of small pieces thatfit together to form a picture. Crossword puzzles are made up of a grid of squares, and the goal is to fill in the squares with words that fit the clues. Sudoku puzzles are made up of a grid of squares, and the goal is to fill in the squares with numbers so that each row, column, and 3x3 box contains all of the numbers from 1 to 9. Word searches are made up of a grid of letters, and the goal is to find words that are hidden within the grid.Puzzles can be a great way to improve problem-solving skills, memory, and concentration. They can also be a fun way to relax and de-stress.中文回答：拼图是一种老少皆宜的流行消遣活动。

make a puzzle英语作文Title: Making a PuzzleCreating a puzzle is an engaging and rewarding activity that challenges one's creativity and problem-solving skills. Whether it's a jigsaw puzzle, a crossword, or a Sudoku, the process of making a puzzle requires careful planning and attention to detail. Below is a step-by-step guide on how to make a basic puzzle.1. Choose a ThemeThe first step in creating a puzzle is to decide on a theme. It could be a favorite book, a memorable vacation, or a special event. The theme will provide the foundation for the design and content of the puzzle.2. Decide on the Type of PuzzleThere are various types of puzzles to choose from, such as jigsaw puzzles, crosswords, word searches, and Sudoku. For this guide, let's focus on creating a simple jigsaw puzzle.3. Gather MaterialsTo make a jigsaw puzzle, you will need the following materials:- A photograph or picture that fits your chosen theme- A sheet of thick cardboard or puzzle-making foam- A utility knife or puzzle cutting blade- A ruler- Pencil and paper for designing the puzzle4. Design the PuzzleTrace the outline of your chosen picture onto the cardboard or puzzle-making foam. If you're using a photograph, make sure to secure it with tape to prevent movement during tracing.ext, decide on the number and shape of the puzzle pieces. You can opt for traditional jigsaw puzzle shapes or get creative with irregular shapes. Draw the puzzle piece outlines on the cardboard, ensuring that each piece interlocks with its neighbors.5. Cut the PuzzleUsing a utility knife or puzzle cutting blade, carefully cut along the outlines you've drawn. Take your time to make clean cuts, as this will affect the final look and ease of assembly of the puzzle.6. Test the PuzzleOnce you've cut out all the pieces, test the puzzle by assembling it. Make any necessary adjustments to the pieces if they don't fit together properly.7. Package the PuzzleTo complete your puzzle, package it in a suitable container or envelope. You can also create a custom puzzle box or include a printout of the completed picture as a reference for the assembler.ConclusionCreating your own puzzle is a fun and educational project that allowsyou to express your creativity. By following these steps, you can make a unique and personalized puzzle that will provide hours of entertainment for you and others. So, get creative and start making your own puzzle today!。

2007 年美国大学生数学建模竞赛M CM试题2007 MCM A: GerrymanderingGerrymandering The United States Constitution provides that the House of Representati ves shall be composed of some number (currently 435) of individuals who are elected f rom each state in proportion to the state’s population relative to that of the country as a whole. While this provides a way of determining how many representatives each sta te will have, it says nothing about how the district represented by a particular represen tative shall be determined geographically. This oversight has led to egregious (at least som e people think so, usually not the incumbent) district shapes that look ―unnatural‖ by some standards.Hence the following question: Suppose you were given the opportunity to draw congres siona l districts for a state. How would you do so as a purely ―baseline‖ exercise to cre ate the ―simplest‖ shapes for all the districts in a state? The rules include only that ea ch district in the state must contain the same population. The definition of ―simple‖ is up to you; but you need to make a convincing argument to voters in the state that yo ur solution is fair. As an application of your method, draw geographically simple congre ssional districts for the state of New York.2007 MCM B: The Airplane Seating Pro blemAirlines are free to seat passengers waiting to board an aircraft in any order whatsoev er. It has become customary to seat passengers with special needs first, followed by fi rst-class passengers (who sit at the front of the plane). Then coach and business-class passengers are seated by groups of rows, beginning with the row at the back of the p lane and proceeding forward.Apart from consideration of the passengers’ wait time, from the airline’s point of view, time is money, and boarding time is best minimized. The plane makes money for the a irline only when it is in motion, and long boarding times limit the number of trips that a plane can make in a day.The development of larger planes, such as the Airbus A380 (800 passengers), accentua te the problem of minimizing boarding (and deboarding) time.Devise and compare procedures for boarding and deboarding planes with varying numb ers of passengers: small (85–210), midsize (210–330), and large (450–800).Prepare an executive summary, not to exceed two single-spaced pages, in which you s et out your conclusions to an audience of airline executives, gate agents, and flight cre ws.An article appeared in the NY Times Nov 14, 2006 addressing procedures currently bei ng followed and the importance to the airline of finding better solutions. The article ca n be seen at: /2006/11/14/business/14boarding.html2008 年美国大学生数学建模竞赛M CM试题2008 MCM A: Take a BathConsider the effects on land from the melting of the north polar ice cap due to the pre dicted increase in global temperatures. Specifically, model the effects on the coast of Fl orida every ten years for the next 50 years due to the melting, with particular attentio n given to large m etropolitan areas. Propose appropriate responses to deal with this. A careful discussion of the data used is an important part of the answer.2008 MCM B: Creating Sudoku PuzzlesDevelop an algorithm to construct Sudoku puzzles of varying difficulty. Develop metrics to define a difficulty level. The algorithm and metrics should be extensible to a varyin g number of difficulty levels. You should illustrate the algorithm with at least 4 difficult y levels. Your algorithm should guarantee a unique solution. Analyze the complexity of your algorithm. Your objective should be to minimize the complexity of the algorithm a nd meet the above requirements.2009 年美国大学生数学建模竞赛M CM试题2009 MCM A: Designing a Traffic CircleMany cities and communities have traffic circles—from large ones with many lanes in t he circle (such as at the Arc de Triomphe in Paris and the Victory Monument in Bangk ok) to small ones with one or two lanes in the circle. Some of these traffic circles posit ion a stop sign or a yield sign on every incoming road that gives priority to traffic alre ady in the circle; som e position a yield sign in the circle at each incoming road to give priority to incoming traffic; and some position a traffic light on each incoming road (w ith no right turn allowed on a red light). Other designs may also be possible.The goal of this problem is to use a model to determine how best to control traffic flo w in, around, and out of a circle. State clearly the objective(s) you use in your model for making the optimal choice as well as the factors that affect this choice. Include a T echnical Summary of not more than two double-spaced pages that explains to a Traffic Engineer how to use your model to help choose the appropriate flow-control method f or any specific traffic circle. That is, summarize the conditions under which each type o f traffic-control method should be used. When traffic lights are recommended, explain amethod for determining how many seconds each light should remain green (which ma y vary according to the time of day and other factors). Illustrate how your model work s with specific examples.2009 MCM B: Energy and the Cell Phon eThis question involves the ―energy‖ consequen ces of the cell phone revolution. Cell pho ne usage is mushrooming, and m any people are using cell phones and giving up their l andline telephones. What is the consequence of this in terms of electricity use? Every c ell phone comes with a battery and a recharger.Requirement 1Consider the current US, a country of about 300 million people. Estimate from availabl e data the number H of households, with m members each, that in the past were servi ced by landlines. Now, suppose that all the landlines are replaced by cell phones; that is, each of the m members of the household has a cell phone. Model the consequences of this change for electricity utilization in the current US, both during the transition an d during the steady state. The analysis should take into a ccount the need for charging the batteries of the cell phones, as well as the fact that cell phones do not last as lon g as landline phones (for example, the cell phones get lost and break).Requirement 2Consider a second ―Pseudo US‖—a country of about 300 million people with about the same economic status as the current US. However, this emerging country has neither l andlines nor cell phones. What is the optimal way of providing phone service to this co untry from an energy perspective? Of course, cell phones have many social consequenc es and uses that landline phones do not allow. A discussion of the broad and hidden consequences of having only landlines, only cell phones, or a mixture of the two is welc omed.Requirement 3Cell phones periodically need to be recharged. However, many people always keep thei r recharger plugged in. Additionally, many people charge their phones every night, whe ther they need to be recharged or not. Model the energy costs of this wasteful practice for a Pseudo US based upon your answer to Requirement 2. Assume that the Pseudo US supplies electricity from oil. Interpret your results in terms of barrels of oil.Requirement 4Estimates vary on the amount of energy that is used by various recharger types (TV, DVR, computer peripherals, and so forth) when left plugged in but not charging the de vice. Use accurate data to model the energy wasted by the current US in terms of bar rels of oil per day.Requirement 5Now consider population and economic growth over the next 50 years. How might a ty pical Pseudo US grow? For each 10 years for the next 50 years, predict the energy ne eds for providing phone service based upon your analysis in the first three requirement s. Again, assume electricity is provided from oil. Interpret your predictions in term of b arrels of oil.2010 年美国大学生数学建模竞赛M CM试题2010 MCM A: The Sweet SpotExplain the ―sweet spot‖ on a baseball bat.Every hitter knows that there is a spot on the fat part of a baseball bat where maximu m power is transferred to the ball when hit. Why isn’t this spot at the end of the bat?A simple explanation based on torque might seem to identify the end of the bat as th e sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding.Some players believe that ―corking‖ a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) enhances the ―sweet spot‖ effect. Augm ent your model to confirm or deny this effect. Does this explain why Major League Baseball prohibits ―corking‖?Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major League Baseball prohibits metal bats?2010 MCM B: CriminologyIn 1981 Peter Sutcliffe was convicted of thirteen murders and subjecting a number of other people to vicious attacks. One of the methods used to narrow the search for Mr. Sutcliffe was to find a ―center of mass‖ of the locations of the attacks. In the e nd, th e suspect happened to live in the same town predicted by this technique. Since that ti me, a number of more sophisticated techniques have been developed to determine the ―geographical profile‖ of a suspected serial criminal based on the locations of t he crim es.Your team has been asked by a local police agency to develop a method to aid in their investigations of serial criminals. The approach that you develop should make use of a t least two different schemes to generate a geographical profile. You should develop a technique to combine the results of the different schemes and generate a useful predic tion for law enforcem ent officers. The prediction should provide some kind of estimateor guidance about possible locations of the next crime based on the time and locations of the past crime scenes. If you make use of any other evidence in your estimate, yo u must provide specific details about how you incorporate the extra information. Your method should also provide some kind of estimate about how reliable the estimate will be in a given situation, including appropriate warnings.In addition to the required one-page summary, your report should include an additional two-page executive summary. The executive summary should provide a broad overvie w of the potential issues. It should provide an overview of your approach and describe situations when it is an appropriate tool and situations in which it is not an appropriat e tool. The executive summary will be read by a chief of police and should include tec hnical details appropriate to the intended audience.2011 年美国大学生数学建模竞赛M CM试题2011 MCM A: Snowboard CourseDetermine the shape of a snowboard course (currently known as a ―halfpipe‖) to maxi mize the production of ―vertical air‖ by a skilled snowboarder.―Vertical air‖ is the maximum vertical distance above the edge of the halfpipe.Tailor the shape to optimize other possible requirements, such as maximum twist in th e air.What tradeoffs may be required to develop a ―practical‖ course?2011 MCM B: Repeater CoordinationThe VHF radio spectrum involves line-of-sight transmission and reception. This limitatio n can be overcom e by ―repeaters,‖ which pick up weak signals, amplify them, and retr ansmit them on a different frequency. Thus, using a repeater, low-power users (such a s mobile stations) can communicate with one another in situations where direct user-to -user contact would not be possible. However, repeaters can interfere with one another unless they are far enough apart or transmit on sufficiently separated frequencies.In addition to geographical separation, the ―continuous tone-coded squelch s ystem‖ (CT CSS), sometimes nicknamed ―private line‖ (PL), technology can be used to mitigate int erference problems. This system associates to each repeater a separate subaudible ton e that is transmitted by all users who wish to communicate through that repeater. The repeater responds only to received signals with its specific PL tone. With this system, two nearby repeaters can share the same frequency pair (for receive and transmit); so more repeaters (and hence more users) can be accommodated in a particula r area.For a circular flat area of radius 40 miles radius, determine the minimum number of re peaters necessary to accommodate 1,000 simultaneous users. Assume that the spectru m available is 145 to 148 MHz, the transmitter frequency in a repeater is eithe r 600 k Hz above or 600 kHz below the receiver frequency, and there are 54 different PL tones available.How does your solution change if there are 10,000 users?Discuss the case where there might be defects in line-of-sight propagation caused by mountainous areas.。

九宫数独大赛参赛流程英文回答：The Sudoku Grand Prix is a prestigious competition that brings together the world's top Sudoku solvers. The competition is held annually and consists of a series of rounds, each of which eliminates half of the competitors. The final round is a head-to-head battle between the two remaining competitors, with the winner being crowned the Sudoku Grand Prix champion.To enter the Sudoku Grand Prix, participants must first qualify by solving a series of online puzzles. The top qualifiers from each region are then invited to compete in the main event.The main event is held over a period of several days and consists of a series of elimination rounds. In each round, the competitors are given a set of Sudoku puzzles to solve. The competitors are ranked based on their speed andaccuracy in solving the puzzles. The top half of the competitors in each round advance to the next round.The final round is a head-to-head battle between the two remaining competitors. The competitors are given a set of Sudoku puzzles to solve, and the first competitor to solve all of the puzzles is crowned the Sudoku Grand Prix champion.The Sudoku Grand Prix is a challenging competition that requires both speed and accuracy. The world's top Sudoku solvers compete in the Grand Prix, and the competition is always fierce.中文回答：九宫数独大赛是一项著名的赛事，汇聚了世界顶尖的数独解谜高手。

15种免费活动提高生活质量1. Exercise 运动Exercise is a way to improve yourself both mentally and physically. If you don't want to cough up the money for a gym membership, personal trainer, or expensive in-home weight set, try this simple full-body workout by performing these exercises:运动是能从身心两方面提升自我的方法。

如果你不情愿掏钱去健身房，请私人教练，或者购买昂贵的运动器材，试试练习以下简单的全身锻炼：Push ups 俯卧撑Crunches 仰身Squats 蹲坐You can find a plethora of free workouts and workout tips online. For instance, don't have dumbbells? Use cans of soup, or a few old milk cartons filled with water.你可以在网上找到太多这种免费锻炼的方法和锻炼的诀窍。

举例说，没有哑铃，就用罐汤，或几瓶灌满水的旧牛奶盒。

2. Go to the Library 去图书馆The library is a great resource that is often under utilized - and its free! The library has books, CDs, tapes, DVDs, and free access to computers. Picking up a book is a great way to entertain yourself and also tune up your reading skills. Your local library holds a vast amount of knowledge just waiting to be acquired!图书馆是一种人们经常利用的丰富资源，而且是免费的！图书馆有书籍，光碟，录音带，数字影碟，还有免费上网端口。

Package‘sudoku’October14,2022Version2.8Date2022-04-19Title Sudoku Puzzle Generator and SolverAuthor David Brahm<**************.edu>and Greg Snow<*******************************>,with contributions from CurtSeeliger<*************************.gov>and Henrik Bengtsson<************.se>.Maintainer David Brahm<**************.edu>Suggests tkrplotDescription Generates,plays,and solves Sudoku puzzles.The GUIplaySudoku()needs package``tkrplot''if you are not on Windows.License GPLNeedsCompilation noRepository CRANDate/Publication2022-04-1914:30:02UTCR topics documented:fetchSudokuUK (2)generateSudoku (3)hintSudoku (4)playSudoku (4)printSudoku (6)readSudoku (7)solveSudoku (8)writeSudoku (9)Index1012fetchSudokuUK fetchSudokuUK Fetch the daily sudoku puzzle from https:///DescriptionFetches the daily sudoku puzzle from https:///or one of their archive from the previous31days.UsagefetchSudokuUK(day)Argumentsday Optional character string specifying the day of the puzzle to download.This is in European date format’dd/mm/yyyy’and needs to represent a date within thelast31days.ValueA9x9matrix representing a sudoku puzzle(blank squares have value0).NoteSee the website for copyright information.Don’t submit your solution for the prize contest if you used solveSudoku or playSudoku with solve=TRUE.This function requires a working internet connection.Author(s)Greg Snow<*******************************>Referenceshttps:///See AlsosolveSudoku,playSudoku,generateSudokuExamples##Not run:#todays puzzlepuz<-fetchSudokuUK()#puzzle from25Jan2006(if still available)puza<-fetchSudokuUK( 25/01/2006 )generateSudoku3playSudoku(puza)##End(Not run)generateSudoku Randomly Generate a Sudoku Puzzle GridDescriptionCreates a9x9Sudoku grid suitable for use by playSudoku.UsagegenerateSudoku(Nblank=50,print.it=FALSE)ArgumentsNblank Number of cells to blank outprint.it Logical.If true,print result to screen.DetailsThe basic algorithm is to start with a’primordial’Sudoku grid,swap around some rows and columns,then blank out some cells.ValueA matrix,representing a9x9Sudoku grid.Author(s)Curt Seeliger<<*************************.gov>>,Henrik Bengtsson<<************.se>>, and David Brahm<<**************.edu>>Referenceshttps:///ExamplesgenerateSudoku(print.it=TRUE)hintSudoku Give a Hint for a Sudoku CellDescriptionGenerates a text string containing a’hint’for cell(i,j)of Sudoku grid’z’.UsagehintSudoku(z,i,j)Argumentsz A9x9numeric matrixi Row indexj Column indexValueA character string,suitable for cat.Author(s)Greg Snow<*******************************>and David E.Brahm<<**************.edu>> playSudoku Interactively play a game of SudokuDescriptionInteractively play a game of9x9Sudoku with hints and undoUsageplaySudoku(z=NULL,hist.len=100,solve=TRUE,display=c("guess","windows","tk"),hscale=1.25,vscale=1.25,...)Argumentsz Either a9x9numeric matrix representing the Sudoku grid(with’0’represent-ing a blank cell),or0(zero)for an empty matrix,or afilename(passed toreadSudoku),or NULL to generate a puzzle randomly.hist.len Integer representing the number of history steps to remember(number of undo’spossible).solve Logical indicating if the solution should be computed(used for checking currentanswer or cheating).display Type of display.The default’guess’uses a windows graphics device if getOp-tion(’device’)==’windows’,otherwise it uses tk(requiring the’tkrplot’pack-age).hscale Passed to tkrplotvscale Passed to tkrplot...Arguments passed to generateSudokuDetailsTo play,move the mouse arrow over an empty cell and press the number key to enter the number in the cell.Typing’?’brings up a menu of additional commands:--a short help message1-9--insert digit0, --clear cellr--replot the puzzleq--quith--hint/helpc--correct wrong entries(show in red)u--undo last entrys--show number in cella--show all(solve the puzzle)ValueAn invisible matrix with the solution or current state of the puzzle.Save this if you stop part way through,and use it as the input for the function to start again where you left off(undo info is lost so make sure that everything is correct).Notedisplay=’windows’makes use of the getGraphicsEvent function,which currently only works on Windows.Author(s)Greg Snow<*******************************>and David E.Brahm<<**************.edu>>6printSudoku See AlsosolveSudokuExamples##Not run:puz1<-playSudoku(0)#Use as an editor to create a puzzle,then quitsol1<-playSudoku(puz1)#now play the puzzlepuz2<-edit(matrix(0,9,9))#Or use this editorsol2<-playSudoku(puz2)#now play the puzzleplaySudoku()#Play a randomly generated gameplaySudoku(fetchSudokuUK())#Play today s game##End(Not run)printSudoku Print a Sudoku Grid to the Terminal.DescriptionPrints a Sudoku grid(a9x9matrix)to the terminal.UsageprintSudoku(z)Argumentsz A9x9numeric matrix,with’0’representing a blank cell.ValueNone;used for side effect.Author(s)David E.Brahm<<**************.edu>>readSudoku7 readSudoku Read a File Containing a Sudoku GridDescriptionReads afile containing a Sudoku grid(a9x9matrix).UsagereadSudoku(fn,map)Argumentsfn Afilename.map Vector of unique puzzle elements(possibly longer than necessary).The default is c(1:9,letters),so an N=16puzzle should be encoded using’1’-’9’and’a’-’g’.DetailsThe inputfile should look like this:-6-1-4-5---83-56--2-------18--4-7--6--6---3--7--9-1--45-------2--72-69---4-5-8-7-Blank cells can be indicated with any character not in"map",such as the’-’used here.ValueA numeric matrix(usually9x9).Author(s)David E.Brahm<<**************.edu>>Examplesz<-readSudoku(system.file("puz1.txt",package="sudoku"))8solveSudoku solveSudoku Solve a Sudoku PuzzleDescriptionSolves a Sudoku Puzzle.UsagesolveSudoku(z,verbose=FALSE,map=c(1:9,letters),level=0,print.it=TRUE)Argumentsz Afilename(passed to readSudoku),or a numeric matrix.verbose If TRUE,report on progress.map Vector of unique puzzle elements(possibly longer than necessary).The default is c(1:9,letters),so an N=16puzzle should be encoded using’1’-’9’and’a’-’g’.level Recursion level(should not be set by user).print.it Logical:print the solution?DetailsA Sudoku puzzle consists of an NxN grid,where N is a perfect square(usually N=9).The grid issubdivided into N[sqrt(N)x sqrt(N)]boxes.You mustfill in the missing values so that each row, each column,and each box contains the integers1:N exactly once.The algorithm uses an NxNxN array of logicals,representing the NxN cells and the N possible elements.For example,if a[1,2,3]=TRUE,then z[1,2]is known to be’3’.If a[1,2,4]=FALSE,then z[1,2]is known not to be’4’.The basic rules of Sudoku are used tofill in FALSE’s,then elimination is used tofind the TRUE’s.If that approach runs out of steam,a guess is made and the program recurses tofind either a solution or an inconsistency.No attempt is made to prove a solution’s uniqueness.ValueInvisibly returns the solved(numerical)matrix,and prints the character version.Author(s)David E.Brahm<<**************.edu>>ReferencesExample"puz1"comes from https:///.writeSudoku9Examples##Not run:solveSudoku(system.file("puz1.txt",package="sudoku"),verbose=TRUE)##End(Not run)writeSudoku Write a Sudoku Grid to a FileDescriptionWrites a Sudoku grid(a matrix)to afile.UsagewriteSudoku(z,fn)Argumentsz A Sudoku grid.fn Afilename.ValueNone;used for its side effect.Author(s)David E.Brahm<<**************.edu>>See AlsoreadSudokuIndex∗arraygenerateSudoku,3hintSudoku,4printSudoku,6readSudoku,7solveSudoku,8writeSudoku,9∗dynamicplaySudoku,4∗miscfetchSudokuUK,2fetchSudokuUK,2generateSudoku,2,3,5hintSudoku,4playSudoku,2,3,4printSudoku,6readSudoku,5,7,8,9solveSudoku,2,6,8writeSudoku,910。

四宫格4个数的数独入门技巧口诀Solving a 4x4 sudoku puzzle can seem daunting at first, especially if you're new to the game. However, with some basic techniques and practice, you can improve your skills and start to enjoy the challenge of solving these puzzles.解决一个4x4数独难题在一开始可能看起来令人望而生畏，特别是如果你是初学者。

但是，通过一些基本技巧和练习，你可以提高你的技能，并开始享受解决这些难题的挑战。

One important technique to keep in mind when solving a 4x4 sudoku puzzle is to start by looking for obvious numbers. This means scanning the rows and columns for any numbers that can only fit in one cell. By focusing on these easy wins, you can start to build confidence and momentum in solving the puzzle.解决4x4数独难题时要牢记的一个重要技巧是要首先寻找明显的数字。

这意味着扫描行和列，找出只适合填在一个单元格的数字。

通过专注于这些易于获得的胜利，你可以开始在解决难题时建立信心和动力。

Another helpful tip is to look for pairs of numbers within the same row or column. If two cells in a row or column can only contain the same two numbers, this can help you narrow down the possibilities for the other cells in that row or column. By identifying and using these pairs effectively, you can make progress in solving the puzzle.另一个有用的提示是在同一行或列中寻找数字对。