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The High School MathematicalContest in Modeling (HiMCM) continues to be an amazing and

rewarding experience for students, advisors, schools, and judges. A total of 938teams, with up to 4 students each, from256 schools competed. Congratulations toour 2017 Outstanding team winners andto all teams participating in our twentiethyear of HiMCM!

Outstanding TeamsCapital High School, Helena, MT,USA Advisor: Dennis Peterson

Columbus North High School,Columbus, IN, USA Advisor:Mike Spock

North Carolina School of Scienceand Mathematics, Durham, NC,USA Advisor: Daniel Teague (2 Teams)

Raffles Institution, SingaporeAdvisor: Haojin Yang

Shenzhen College of InternationalEducation, Shenzhen, Guangdong,China Advisor: Dan Li

Shenzhen Vanke MeishaAcademy, Shenshen, Guangdong,China Advisor: Wentao Zhang

St. Mark’s School, Southboro, MA,USA Advisor: Allyson Brown

YK Pao School, Shanghai, ChinaAdvisor: Jianping Yang

The solutions we received were trulyimpressive. Our participating teamsaccomplished the vision of ourfounders by providing unique and creative mathematical solutions to complex open-ended real-world problems. As in the past, studentschose from two problems, both repre-senting “real-world issues.” Thisyear’s problems allowed students toeither plan a drone aerial light showin Drone Clusters as Sky Light DisplaysProblem A, or design a ski resort inour Ski Slope Problem B. The finaljudging results and 2017 HiMCM statistics are shown in Figure 1.

OverviewThe HiMCM, now in its twentiethyear, continues to grow. As more highschools engage their students in mathematical modeling, we hope participation in the contest will follow. Figure 2 shows a plot of thegrowth over time. The trend continuesto reflect an increasing engagement ofhigh school students in mathematicalmodeling.

The 2017 contest had 1044 registeredteams resulting in 938 total submis-sions (89.8%), a registration increase ofabout 10% over last year. In total, 3922students competed in the contest, representing an increase of 10%. Awide range of schools/teams competedincluding teams from Australia,

Canada, Chile, China, Costa Rica,Germany, Hong Kong (SAG), Korea,the Philippines, Singapore, Thailand,Turkey, United Arab Emirates, theUnited States of America, and theUnited Kingdom. The 359 teams from the United States represented 28states. This number was up 8.8% fromlast year. Submissions included 579foreign teams, representing a 13.5%growth. China represented about 91%of the foreign entries.

Of the 3922 student participants thisyear, 1390 or about 37.4% werefemale, 2429 were male, and 103 didnot specify gender. Since the start ofHiMCM in 1999, females represent36.4% (10,985) of the 30,097 total participants. This participation hasbeen fairly consistent over a numberof years. We hope that all competingstudents will enjoy their contest experience and continue to pursuefurther STEM education.

Problem ChoiceWe congratulate all students and advisors for the creativity and ingenuity of their mathematicalefforts. Again this year, it appears thatteams are enjoying developing solutions to their chosen problem. Wecontinue to encourage all registeredteams to submit a solution in order toexperience the learning impact andsatisfaction of fully completing this

THE 2017 HIMCMDr. Kathleen SnooK, hiMCM DireCtor

Figure 1: 2017 HiMCM Statistics

Problem Outstanding % Finalist % Meritorious % HonorableMention % Successful

Participant % Unsuccessful % Total

A 6 1% 40 7% 77 13% 181 32% 255 45% 10 2% 569B 3 1% 25 7% 46 12% 113 31% 173 47% 9 2% 369

Total 9 1% 65 7% 123 13% 294 31% 428 46% 19 2% 938

The 2017 Contest

challenging contest. Of the 938 submissions, 569 completed ProblemA: Drone Clusters as Sky Light Displays,and 369 completed Problem B: SkiSlope. The 36 continuous hours towork on the problem provided forhigh quality papers for both problems.

JudgingThe mathematical modeling ability ofparticipating students and their faculty advisors continues to be evident in the problem solutions andprofessional submissions we receive.As you and your students engage inmathematical modeling at a higherlevel, we are happy and excited toassist your efforts. Let us know howwe might support your modelingactivities.

All submissions this year were elec-tronic which allowed us to expand ourjudging pool. In December 2017, weused two regional judging sites and athird group of remote judges. Theregional sites were located at FrancisMarion University in Florence, SC andCarroll College in Helena, MT.Remote judges were located inAlabama, California, Colorado,Georgia, Illinois, Massachusetts, NewYork, Ohio, Pennsylvania, SouthCarolina, and Virginia.

All judging was blind with respect toany identifying information about theparticipants or their schools. Judgesranked papers as Finalist, Meritorious,Honorable Mention, and SuccessfulParticipant. Each site judged papersfor both problems A and B. Judgessent all papers ranked as “Finalist” tothe National Judging in San Diego,California. This year, 74 papers wereforwarded to San Diego for elevenjudges, from both academia andindustry, to consider. As these 74papers were the best submissionsfrom the triage round, at final judgingthe judges chose the “best of the best”as Outstanding papers. Nine papersearned the Outstanding award. Thenational judges commended theregional judges for their efforts inselecting the high quality Finalistpapers forwarded to San Diego. Wefeel that the structure of Regional andNational Judging provides a goodprocess for identifying the winningpapers.

The FutureThe HiMCM attempts to give all highschool students an opportunity tocompete and achieve success in math-ematics. Our efforts are always towardmeeting this important goal.

Mathematical modeling is growingwithin the curricula at the high schoollevel. This contest provides a vehiclefor using mathematics to build modelsthat allow students to represent and to understand real world behavior in aquantitative way. It enables studentteams to look for patterns and thinklogically about mathematics and itsrole in our lives. We continue to striveto improve the contest, and want thecontest accessible to all students. Anyschool and team can enter, as there areno restrictions on the number ofschools or the numbers of teams froma school. A regional judging structureoffers flexibility to accommodate thenumber of teams.

Mathematics continues to be morethan just learning skills and opera-tions. Mathematics is a language thatinvolves our daily lives. Applying themathematical principles and conceptsthat one learns is key to individualand societal future success. The abilityto recognize problems, formulate amathematical model, use technology,and communicate and reflect on one’swork are important skills to develop.Students gain confidence by tacklingill-defined problems and workingtogether to generate a solution. We areexcited that in our contest applyingmathematics is a team sport!

Mathematical modeling is an art and ascience. Advisors need only be motivators and facilitators to encourage students to be creative andimaginative. Through modeling, students learn to think critically, communicate effectively, and be confident, competent problem solvers.Success is not only about the procedural technique used, but theconceptual understanding in discover-ing the role of assumptions and modeldevelopment in driving those techniques to a valid solution and conclusion. We encourage all highschool mathematics faculty to getinvolved, encourage your students to

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Figure 2: Number of HiMCM teams vs. contest year from 2008-2017

be problem solvers, make mathemat-ics relevant, and open the doors tofuture success.

2018 Contest DatesMark your calendars for the nextHiMCM to be held November 9 – 19,2018. Registration for the 2018HiMCM will open in September. Arule change in 2018 eliminates therequirement for teams to restrict contest work to a consecutive 36-hourblock. Teams will have an 11-day window to download and choose theirproblem, complete their modelingsolution, and electronically submittheir solution document. Also new in2018 will be awards for the topOutstanding teams selected at finaljudging. Teams can learn more andregister via the Internet at..............www.himcmcontest.com.

MathModels.orgPowered by COMAP content,Mathmodels.org is a wonderfulresource for students and teachers tomake math modeling a year roundactivity. Teachers and students mayuse the materials found on this site toenrich their classes and help prepare

students for mathematical modelingcompetitions. We encourage you tovisit www.mathmodels.org.

The InternationalMathematical ModelingChallenge, IM2C

We are several years into the newestinternational secondary school mathe-matical modeling competition: TheInternational Mathematical ModelingChallenge. The purpose of the IM2C isto promote the teaching of mathemati-cal modeling and applications at alleducational levels for all students inter-nationally. It is based on the firm beliefthat students and teachers need toexperience the power of mathematicsto help better understand, analyze andsolve real world problems outside of

mathematics itself – and to do so inrealistic contexts. Each country administers the contest for its studentsand then sends its top two teams tothe international final judging.

All USA teams that successfully compete in the HiMCM contest andare awarded a designation ofMeritorious or above (Meritorious,Finalist, and Outstanding) are invitedto compete in IM2C. From these participants, our USA IM2C judgeswill select the two top teams to represent the USA in the IM2C.......international round. The fourth annu-al IM2C is set to take place March 12th

through May 7th, 2018. To learn morevisit: www.immchallenge.org.

Regardless of the problem chosen,competitive papers include a compre-hensive summary, address all require-ments with reasonable responseswithin 30 pages, and write a clear letter or memo. Better papers do all ofthe above in an articulate, well- supported, well-organized, and well-presented manner. The bestpapers combine complete mathemati-cal and logical analysis in an organized presentation beyond simplyaddressing the requirements. Thesepapers are easy to read, flow logically,and they include sections thataddressed assumptions with justifica-tions, the modeling process(es), resultsof modeling and analysis, strengthsand weaknesses, conclusions, and references.

The specific problem discussionsbelow provide comments on howteams addressed each problem.

Inte

rna t i o n a l M a t h e m

atical M

odeling Challenge

2

2017 ProblemDiscussions and

Judges’Commentary

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AwardsThe award format has changed slightly in the past two years asthe contest continues to grow internationally.

After final judging, HiMCM papers are designated in the categoriesbelow. The top approximately 21% of submitted papers receives adesignation of Meritorious or above. The top approximately 1% isdesignated as Outstanding.

Successful Participant

Honorable Mention

Meritorious

Finalist

Outstanding

Following this section we break downthe various parts of a submission withexamples, and provide judges’ comments about the solutions and thepresentation of the solutions. To viewthe complete problems visit...............www.mathmodels.org.

Problem A: Drone Clusters asSky Light Displays

In Problem A, teams worked withtheir town’s Mayor to design an aeriallight show using clusters of drones.The Mayor asked the teams to investi-gate the idea of using drones to createthree possible sky light displays forthe city’s annual festival: a Ferriswheel, a dragon, and an image of theirchoosing.

The problem materials pointed teams tothe video of a 2016 light show createdby a cluster of 500 drones, controlledby a single laptop and one pilot. Theproblem also offered the Intel®Shooting StarTM drone as an exampleof a drone used for aerial light shows.

For each of the three required skylight displays, teams had to determinethe number of drones required andmathematically describe the initiallocation for each drone device. Theythen determined the flight paths of

each drone, or set of drones, thatwould animate each of their images,and describe their animations. Teamsestablished and discussed the require-ments for their 3-display light show toinclude required launch area, airspace, safety, and duration of theshow. Finally, teams wrote a two-pagememo to the Mayor to report theresults of their investigation and makea recommendation as to whether ornot to do the aerial light show.

Most teams conducted initial researchon drones and the use of drones inaerial light displays. This researchprovided teams specifications ofexample drones, particularly theShooting StarTM drone, to include size,speed, life of battery, and safe separa-tion distance. In addressing allrequirements, competitive papersdesigned the drone positions and animation actions for each of the threedisplays, determined and discussedrequirements for the light show, andwrote an articulate and clear memo tothe Mayor.

In designing the drone positions andactions, many teams first sketchedeach image using a “continuous” picture and then began to discretize itto account for the individual drones.Some teams used various graphingprograms such as DESMOSTM, whileothers simply sketched their design ongraph paper. Either method was fineif it resulted in an illustrative diagram.The best papers placed the drones inthe sky and determined an appropriatenumber of drones by explaining andincorporating a scale that accountedfor the distance from ground level(noting that the lights will be a distance away from the viewer). Thesepapers offered clear mathematicaldescriptions of locations and anima-tion paths. Additionally, these bestpapers addressed nuances such as theaudience viewing angle, 3D space perspective, local or governmentalrestrictions on drone usage, weatherconditions, and the geography of the

display and viewing areas. The bestpapers used more sophisticated mathe-matical descriptions, were more creative in their images and animations,and incorporated more of the actualdrone capabilities in their analysis.

Problem B: Ski Slope

By Pudelek (Marcin Szala) (Own work) [CC BY-SA 3.0 (https://creativecom-mons.org/licenses/by-sa/3.0)], via Wikimedia Commons

Problem B had the teams working forthe representative of a group ofwealthy winter sports fans interestedin finding a new mountain to developas a ski resort. Wasatch Peaks Ranchin Peterson, Utah, USA is for sale andMs. Mogul, the representative for theinvestors, asked the teams to identifypotential ski slopes and trails on theproperty in order to develop it as atop ski resort and a potential futureWinter Olympics location.

The problem materials included abrochure for Wasatch Peaks Ranch, atopographic map of the area, and apartial list of North American skislopes with comparison data. Theproblem statement also included adiscussion of how to measure anddetermine ski trail levels.

The problem required that teamsdesign the new Wasatch Peaks Ranchski area to have main slopes of varyinglengths, plenty of trails, at least a totalof 160 km of slopes and trails, and adistribution of 20% beginner, 40%intermediate, and 40% difficult ratedslopes. Teams then ranked their proposed ski area against existing skiareas/resorts in North America.

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Intel

Intel

Finally, teams wrote a two-pagememo to Ms. Mogul reporting theresults of their design and ranking.

We saw many viable and creativesolutions to our ski slope problem.Most teams conducted some initialresearch on ski slopes, and onOlympic venues and events. Whilehaving skied or visited a ski resortmay have provided some teams morefamiliarity with the scenario, the focuswas on terrain and slope. Teams usedthe materials and maps provided tobegin their design. In addressing allrequirements, competitive papersoffered a reasonable ski slope designfor the Wasatch Ranch. Specifically,these papers identified the actualslopes by difficulty rating on a mapand measured those slopes so thatthey met the design criteria, rankedtheir proposed ski area against existingski areas/resorts in North America,and wrote a clear letter to Ms. Mogul.

In designing their ski slopes, mostteams analyzed the terrain and slopesin various areas on the mountain.Some teams designed and drew theirslopes by hand on the provided mapsbased on analysis of the terrain. Otherteams used available terrain and satel-lite maps found on the Internet, alongwith terrain analysis to identify possibletrails based on the gradient (slope) ofthe mountain. Both methods providedreasonable solutions with betterpapers explaining how they analyzedthe terrain and determined where toplace the slopes and trails. Somegraphing programs assisted teams inelectronically marking the designedslopes and trails. The better papersdiscussed choices and decisions aboutslope placement over the entire moun-tain. The best papers also addressedintegration of the main ski runs withconnecting trails, as well as notingwhere ski lifts might be located.

In developing a ranking system, mostteams used the provided NorthAmerican ski slope comparison data,

along with web sites that rank skislopes. Competitive papers used thesedata to compare with their ski slopedesign and to rank Wasatch amongthese other slopes. Better papersaccounted for variables available forall ski slopes, explained and weightedthe variables, developed a comparisonmodel, and ranked the resorts. Thebest papers included a well-developedmathematical comparison modelwhich included the provided dataalong with additional informationsuch as average snow fall, snow making ability, lodging ability, localtransportation system, and ease oftravel to Wasatch (e.g. location of closest airport).

Judges’ CommentsWhile the Problem Discussions aboveprovide comments on the solutions tothis year’s problems, here we examinethe sections of a submission and provide comments about the solutionsand the presentation of the solutionsfrom a judge’s point of view. We have included examples from severalpapers at the end of this article.Mathmodels.org members can viewall the unabridged versions of theOutstanding papers online.

OverallParticipants must ensure their papersfollow the contest rules posted on thecontest site. Papers that are coherent,organized, clear, and well written provide a great impression to thejudges. The logic and mathematics ofthese papers are easy to follow. Teamsshould present their solution andanalysis in fewer than 30 pages usingat least 12-point font. While studentsmay want to include some back-ground research on the problem topic,this information should be brief. It isnot the number of pages, but the ability to complete all contest require-ments and communicate the solutionin a concise and articulate fashion thatwill merit recognition. Studentsshould use spelling and grammar

checkers before submitting a paper.Foreign papers should insure that allsymbols in tables and graphs are inEnglish.

Papers considered for Meritorious andOutstanding start with a clear summa-ry that describes the problem. Theythen preview their paper with anorganized Table of Contents. Theypresent assumptions with justifica-tions, explain the development of themodel and its solutions, support theresults mathematically and communi-cate them clearly, address strengthsand limitations, and finally, close bystating overall conclusions.

Executive SummaryJudges are best able to analyze a paperwhen students restate the problem intheir own words and clearly previewthe focus and organization of theirpaper. Teams should write these exec-utive summaries after they finish theirsolutions as they summarize the entirecontents of the paper. Teams shouldconsider a three to five paragraphapproach for their summary: a restate-ment of the problem and questions intheir own words, a short descriptionof their method and solution to theproblem (in words and not in mathematical expressions), and theconclusions providing the numericalanswers in context. The executivesummary should entice the reader, inour case the judge, to read the paper.Although written last, ensure youspend time on this important part ofyour submission. Your executive summary is the first page the judgeswill read and provides the firstimpression of your paper. Example 1is a good summary for Problem A:Drone Clusters as Sky Light Displays.

The comments above for summariesalso apply to any required memos orletters. Your letter might brieflydescribe your model or process, butdo so in a non-technical manner as thereader may not have a mathematical

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background. Your letter should focuson why your model and its results areapplicable and important to the reader. The key in your letter is toanswer the question(s), and to interpret and communicate the resultsclearly. Example 2 is a good 2-pagememo for Problem A: Drone Clustersas Sky Light Displays.

Assumptions withJustificationsModeling assumptions shouldinclude only those that come to bearon the solution (this assists with simplifying your model). Long lists ofassumptions that do not play directlyin the context of model developmentor its solution are not considered relevant and deter from a paper’squality. Assumptions that oversimpli-fy the problem too much do not allowfor a full solution. You should includea short justification to show theassumption is reasonable and neces-sary. Example 3 displays several goodassumptions for Problem A: DroneClusters as Sky Light Displays. Notethat within the paper, this team hasused in-text citations. The team thenlisted these sources in a Reference section at the end of their paper.

Mathematical ModelPapers should explain the develop-ment of the mathematical model(s),and define all variables. Teams thatmerely present a model withoutexplaining or showing the develop-ment of that model do not generallydo well. Presenting multiple models,without identifying the most appro-priate model to answer the questions,is detrimental to your paper’s success.Your team may have considered sev-eral models that did not work beforedeveloping a good model. Do notinclude all of your trials, but insteadclearly present the development andresults of your successful model. It ishelpful to the judges to have the definitions of your variables near

your model so they can follow thelogic of your model. If you include along list of variables in a table early inyour paper, consider reminding thereader of the variable definitions asyou present each equation in yourmodel.

Judges do value creativity and think-ing “outside of the box” in your modeling process. There is alwaysmore than one appropriate solutionmethod to our HiMCM problems, sobe creative. Example 4 is a descriptionof how one team developed a modelfor the dragon image of their lightshow for Problem A: Drone Clusters asSky Light Displays.

Strengths and LimitationsTeams should address strengths andlimitations in evaluating their modeland solution. Include model exten-sions or sensitivity analysis of yoursolution. Validate your model, even ifby numerical example or intuition.Example 5 displays a good strengthsand limitations section for Problem A:Drone Clusters as Sky Light Displays.Example 6 shows a nice sensitivityanalysis for Problem B: Ski Slope

ConclusionA clear conclusion and answers to thespecific scenario questions are keycomponents to an Outstanding paper.Attention to detail and proofreadingthe paper prior to final submission arevital as the judges look for excellencein your submission. Example 7 is agood conclusion from Problem B: SkiSlope.

Citations and References Citations are very important withinthe paper, as well as either a referencelist or bibliography page at the end.Teams that use existing modelsshould cite their source within thepaper at the point they present themodel and not just include a referencecitation in the back of the paper. Thisis also true for all graphs and tables

taken from the literature. Use “inline” documentation with footnotes orendnotes to give proper credit to outside sources. All data, figures,graphs, and tables that come fromoutside sources require documenta-tion at the point in the paper wherethey appear. Lack of documentationwill result in a lower score. We havenoticed an increase in use ofWikipedia. Teams need to realize thatalthough useful, information fromWikipedia might not be accurate.Teams should recognize andacknowledge this fact.

The quality of HiMCM submissionscontinues to improve each year. Weenjoy reading all of the papers submitted for review and are trulyimpressed by the work of the studentteams. We encourage teams to reviewthe comments and guidance providedin this article and to visit:............mathmodels.org in preparation fornext year’s contest.

List of Following Examples: 1. Summary (Problem A, Team 8040:Shenzhen Vanke Meisha Academy)

2. 2-Page Memo (Problem A, Team7359: YK Pao School)

3. Assumptions (Problem A, Team7879: Raffles Institution)

4. Model Development (Problem A,Team 8028: Shenzhen College ofInternational Education)

5. Strengths and Limitations (ProblemA, Team 8206: NC School of Scienceand Mathematics)

6. Sensitivity Analysis (Problem B,Team 8121: Columbus North HighSchool)

7. Conclusion (Problem B, Team 8201:NC School of Science andMathematics)

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Example 1: Summary from Team 8040: Shenzhen Vanke Meisha Academy.

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Example 2: Memo from Team 7359: YK Pao School

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Example 2 Continued: Memo from Team 7359: YK Pao School

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Example 3: Assumptions from Team 7879: Raffles Institution

2.2 Assumptions

The following assumptions are made:Assumption 1: The model of the drones used in this show is the Intel R©

Shooting StarTMdrone, with the following specifications (“Intel R© Shooting StarTMDronesFeatured in First-Ever Drone Integration during Pepsi Zero Sugar Super BowlLI Halftime Show”, 2017):

1 Each drone has a flight time of up to 20 minutes.2 Each drone can travel at the speed of 3ms−1 at all times.3 Each drone has a size of 384× 384× 93 mm

Justification: Intel R© Shooting StarTMdrones have already been used foraerial light shows in the past (as mentioned in the question) and are henceappropriate to be used as drones in the outdoor aerial light show for our citysannual festival.

Assumption 2: The show will be part of an annual festival of a city in theUnited States of America, and will have to comply with rules related to droneoperation stated by the Federal Aviation Administration (FAA) of the UnitedStates Department of Transportation (Federal Aviation Administration, 2017):

No. Rule Adherence

1Drone must flyunder the heightlimit of 121.92 m*

Not adhered to(refer to

Assumption 4)

2Drone must fly

during the day andnot at night*

Not adhered to(The show will becarried out at

night.)

3Drone must fly at

or below160 · 934kmh−1

Adhered to

4Drone must not fly

over peopleAdhered to

Justification: We chose United States of America as the country for ourmodel as they have clear and quantitatively specified rules on drone operationthat we can adhere to in our model. Note that items marked with * are nego-tiable.

Assumption 3: The show is carried out at night (Organizers need to obtainwaiver for rule 2, i.e. Drone must fly during the day and not at night), with aclear, dark sky as the background, and there will be negligible light pollutionin the vicinity of the show, such that the audience will be able to see a drone ifand only if the light of that drone is turned on.

Justification: Given that the annual festival of a large city should be amajor event that is carried out well, the show should be carried out underoptimal viewing conditions for the audience. Intel has managed to obtain the

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Example 3 Continued: Assumptions from Team 7879: Raffles Institution

waiver for flying drones at night (“Intel Receives Drone Waiver from the FAA”,2016). Since this show is similar in nature to the one organised by Intel, theorganizers should also be able to receive a waiver for rule 2 in Assumption 2.

Assumption 4: Minimum distance between drones is 1.5m, i.e. the distancebetween any two drones should not be within 1.5m at any point of time in air.It will not be a concern when the drones are on the ground.

Justification: A minimum distance between drones has to be set to preventany collision between drones, which may result in damage to the drones withthe drone or any broken parts of the drone becoming a falling-object hazardto the audience on the ground. In a previous aerial light show, five hundredIntel R© Shooting StarTMdrones were spaced at least 1.5m apart from each other(aeronewstv.com, 2016) (Kaplan, 2016). We will adhere to this minimum dis-tance as it has been tried and shown to be safe. It is reasonable to assume thisdistance need not be maintained by drones still on the ground as they are notinterfering with moving drones.

Assumption 5: The drones will remain in the sky throughout the show. Inother words, they will not return to the ground during the transition betweenconsecutive displays. They will only do so after they finish the third and finaldisplay - the double-headed eagle.

Justification: For simplicity of the model and to minimise transition timebetween consecutive displays, the drones shall remain in the sky throughoutthe show. Instead, the light on each drone can be turned on or off (refer toAssumption 3) to give the audience the illusion that the number of drones inthe sky is changing during the show. Furthermore, in Coachella Drone LightShow Delights Concert (KG, 2017), an aerial light show held in the past, thedrones stayed in the air throughout the show, showing that it is indeed feasibleto have such an arrangement.

Assumption 6: All drones will always fly in a straight line, except whentwo drones that are approaching each other are less than 2 metres apart, thosetwo drones will move around each other in a circular path.

Justification: It is reported that this drone is pre-programmed to avoidcollisions. (Glaser, 2016) For simplicity, we assume they can safely avoid anycollision.

Assumption 7: All drones function normally and are equipped with motionsensors that can sense presence of birds in the sky and fly around any bird inits path via a circular path.

Justification: For simplicitys sake, we assume so.Assumption 8: The safe distance between audience and drones will be

155m.Justification: This safe distance is used in Walt Disney Light shows. (“4

Things You Need To Know Before Using Drones At Your Event”, 2017)

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Example 4: Model description from Team 8028: Shenzhen College of International Education

Model Development

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Example 5: Strengths & Limitations from Team 8206: NC School of Science and Mathematics

10 Strengths & Weaknesses

10.1 Strengths

• Our model mathematically describes the linear path (including initial and final posi-tions) for each drone for both transitions and animations. We are able to calculatea definitive start and end point by optimizing each drone’s destination in the displayusing the Hungarian algorithm. This concept can be witnessed within Figure 7. Theblue drones, which start towards the left of the x-axis move to the left of the Ferriswheel. Similarly, the red drones, which start towards the right of the x-axis move tothe right of the Ferris wheel. The brown drones, which are closest to the center ofthe Ferris wheel, form the wheel’s center. This notion is also seen in the Dragon andBalloon models, indicating our optimization of shortest-distance linear paths is highlyfunctional.

• Our model outputs the velocity for each drone during all transitions and animations.Using our path functions and collisions avoidance program, a drone (such as a DJIPhantom 3) could be easily loaded with a matrix of velocity commands and can followthe path accurately.

• The pathing algorithm prevents drone collisions without sacrificing overall fluidity byoptimizing drone transition formations and finely tuning drone velocities such that alldrones will move to their target locations as efficiently as possible while not colliding.

• Safety analysis predicts the highest wind speed in which our drones may fly so thatevent organizers can plan a safe light show date based on weather forecasts.

• The collision avoidance algorithm to move the drones such that the interaction betweenthem is minimized. This algorithm is based on the idea that the drones move to theirtarget points such that the sum of the distances collectively traveled is minimized,otherwise known as the Extremal Principle.

10.2 Weaknesses

• Our model does not account for turbulence between drones. Turbulence is a significantreal word phenomenon that does occur between aircraft. Though the drones are spacedsignificantly away, there is a chance that turbulence could interfere with the droneflights when they are more densely packed (such as during takeoff or landing).

• Our model does not account for luminous flux; essentially, it does not account for thefact that objects get less bright the further away they are seen. We are assuming thatthe drones’ lights will always be seen, but should take into account factors like lightpollution and diminishing intensity.

• The pathing algorithm is very computationally expensive: the runtime increases rapidlywith increasing data set complexity, increasing collision radius, decreasing time stepsize, and decreasing change in velocity with each trial. However, if the time step orvelocity change factor is too large, the drones will choke up on each other and thesimulation will jam. If the collision radius is too small, the simulation may not besufficient to account for real-world drone collisions.

• The algorithm cannot calculate the collision avoidance routes for the drones’ transitionfrom the 14x14 grid to the Dragon, due to the program’s excessive runtime for complexdesigns. This could be remedied by using a more powerful computing platform orhaving a longer period of time for computations.

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Example 6: Sensitivity Analysis from Team 8121: Columbus North High School

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Example 7: Conclusion from Team 8201: NC School of Science and Mathematics

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