Extend our energy-balance model of global climate to include green-house effects, and other pieces of realism.

Explain the Laplace–Runge–Lenz vector as a conserved quantity in the theory of planetary orbits.

Write a paper explaining why dimensional analysis works.

Euler’s elastica

Recreate Newton, Buffon, Helmholtz, and others’ estimates of the age of the earth.

Before GPS, how was the old LORAN system used to calculation position at sea?

How long should we quarrantine people with Ebola?

What’s the most efficient way for analog generation of strong random passwords?

Floating cheerios and self-assembly

Bernoulli’s hanging chain, Bessel functions, n-pendulum systems, spinning hanging chains, …

Geodetic Coordinates (Keeler, 1998, SIAM review)

Standard cooking recommendations give a linear formula relating roast size and cooking time. How do such recommendations compare with data, and are they defendable, in light of the dimensional analysis recommending a nonlinear scaling of cook time to size?

Make a true-type font for Modern Carthaginian.

Consider the problem of designing a suspension system for a skiing robot (on one ski). What system would create the fastest robot skier?

Expand on Bressan’s work on plant growth stability by applying it to the growth of a Nautilus.

Monge’s development of the mathematics of “descriptive geometry” for precise representation of 3 dimensional objects on 2-d paper during the late 1700’s with applications to engineering and architecture, and it’s earliest appearances in American math and engineering curriculums.

Model the bomb distribution in London during the Blitz.

Transcribe Rorty’s seminal 1903 manuscript on probability modelling in telephony and explain it in modern terms.

How did Robert’s study Watt’s curve, and what were his results?

Rewrite Milton-Bradley’s game of life to be “realistic” from different people’s points of view.

Explain the uncontrollability of the half-bicycle, half-tricycle bricycle.

What’s the optimal way to parallel-park a car?

Develop Markov chains that can help determine authorship of the Federalist papers between Madison, Hamilton, and Jay.

Analyze Creighton’s plague data for London for years 1578 - 1583, 1603, 1625, or 1665.

Explain the speed of sound from first principles

Develop a 1 dimensional cellular automata model for the propagation of action potentials down the axon of a neuron.

Develop a simple compartmental model to explain seasonal phytoplankton and zooplankton bloom cycles.

Explain infrared video data of an object cooling using the heat equation.

Model the act of walking and determine your speed of walking on Venus.

Make a cellular automata model of the phase transition of a peaceful demonstration into a riot.

Review the works of White and Johnson, 1964 and the related Eggleston and Young, 1959. Explain the laws of motion in greater detail and recreate some of the authors calculations. Note that Katherine Johnson was featured in the book and movie “Hidden figures”.

Construct a model for pedal locomotion on mars that will predict how fast humans or some other animal will be able to walk based on physics and geometry.

Generalize the predator-prey model to model police and drug dealers.

Explain the theory of linkages, including Watt’s linkage and how we can use python to solve them.

Pick up a hardcover book with the length, width, and height are all unequal. Try flipping this book around each of it’s 3 principle axes. Explain the results of your experiments. Then present a set of differential equations that explain our experimental observations.

The KPZ equation is a recently discovered equation that many people are currently studying. Somebody one the Fields medal last year for related research. What is it, and why is it important. (include some math!)

The Ising model is a classic cellular automata model for magnetism consisting of mini-magnets of one of two orientations. Thermal fluctuations randomize the orientation of the magnets, while magnetic attraction tends to allign the mini-magnets. Use simulation to show that the two-dimensional Ising model exhibits a phase-transition similar to that of the percolation model.

How many times do you have to shuffle a deck of 52 playing cards to make sure the cards are randomly mixed? Discuss how you will measure randomness, how different shuffling methods add randomness, and use simulations to support your conclusions.

The BZ reactions are a family of chemical reactions that oscillate – a phenomena that was long believed to be impossible. The BZ reactions can be explained by the Oregonator model and it’s variations (note Tyson’s two-variable version, in particular). Discuss.

Fairy circles are a weird natural phenomena observed in the Namib Desert. Make a mathematical or computational model that explains why these circles occur there, and not other places in the world.

What 3d shape of a piece of paper would be least stable when you drop it?

When it was first discoverd, there was great controversy bout the nature of HIV. Some scientists argued HIV was a very slow virus because it too decades before people got sick; others argued that HIV was just as fast as other viruses, but was held in check by our immune systems. Nobody new how to test these hypotheses until somebody discovered antiviral treatments for it. Explain how Perelson et al. answered this question using a mathematical model and patient data.

Explain the mathematical model of the Karman vortex street.

Derive a distribution to predict how many gold medals the US will win in the next summer olympics.

In class, we built a differential equation model to describe Huffaker’s mite experiments. Build a Markov process model that generalizes this, and use it to estimate the extinction probability for the population over time.

Explain in modern language the ribbed arch calculations used to model and design Ead’s bridge over the Mississippi river in St. Louis, completed in 1874.

Balancing of a single-piston engine.

In our class analysis, we actually only solved half of the ballistics problem – the path of a simple projectile after it leaves a gun. This is call the exterior ballistics. The first half of the problem, called the “interior ballistics” is to predict the “muzzle velocity” of the projectile as it leaves the gun as a function of the gun’s shape, the cannonballs weight and the amount of gunpower used. Develop a model for the interior ballistics and combine it with our model of exterior ballistics to make predictions about cannon fireing.

Discuss and reproduce the calculations of Mason and Dixon in the surveying of 1 degree of lattitude or other parts of their project on the boundaries of Pennsylvania, Maryland, and Delaware.

Completely reanalyze Nathaniel Bowditch’s 1807 comet observations.

In November 1993, the state of Pennsylvania conducted elections for its state legislature. In the Senate election in the 2nd district (based in Philadelphia) was challenged in court, the Democratic candidate won 19,127 of the votes cast by voting machine, while the Republican won 19,691 votes cast by voting machine, giving the Republican a lead of 564 votes. However, the Democrat won 1,396 absentee ballots, while the Republican won just 371 absentee ballots, which more than offset the Republican lead based on the votes recorded by machines on election day. The Republican candidate sued, claiming that many of the absentee ballots were fraudulent. Consider data from 21 previous Pennsylvania Senate elections in seven districts in the Philadelphia area over the preceding decade. Was there fraud?

Before the Canada lynx was protected under the endangered species act, it was regularly trapped for its fur (trapping data from 1821-1934). The data shows interesting oscillations and peaks of dramatically different height. Can you create a population dynamics model of the Canada lynx that explains the data?

(2015 MCM) The world medical association has announced that their new medication could stop Ebola and cure patients whose disease is not advanced. Build a realistic, sensible, and useful model that considers not only the spread of the disease, the quantity of the medicine needed, possible feasible delivery systems (sending the medicine to where it is needed), (geographical) locations of delivery, speed of manufacturing of the vaccine or drug, but also any other critical factors your team considers necessary as part of the model to optimize the eradication of Ebola, or at least its current strain. FYI there’s data on github!

(2014 MCM) Sports Illustrated, a magazine for sports enthusiasts, is looking for the “best all time college coach” male or female for the previous century. Build a mathematical model to choose the best college coach or coaches (past or present) from among either male or female coaches in such sports as college hockey or field hockey, football, baseball or softball, basketball, or soccer. Does it make a difference which time line horizon that you use in your analysis, i.e., does coaching in 1913 differ from coaching in 2013? Clearly articulate your metrics for assessment. Discuss how your model can be applied in general across both genders and all possible sports. Present your model’s top 5 coaches in each of 3 different sports.

(2013 MCM) Fresh water is the limiting constraint for development in much of the world. Build a mathematical model for determining an effective, feasible, and cost-efficient water strategy for 2013 to meet the projected water needs of [pick one country from the list below] in 2025, and identify the best water strategy. In particular, your mathematical model must address storage and movement; de-salinization; and conservation. If possible, use your model to discuss the economic, physical, and environmental implications of your strategy. Provide a non-technical position paper to governmental leadership outlining your approach, its feasibility and costs, and why it is the “best water strategy choice.”

(2010 MCM) Explain the “sweet spot” on a baseball bat.

Every hitter knows that there is a spot on the fat part of a baseball bat where maximum 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 the 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. Augment 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) In 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 end, the suspect happened to live in the same town predicted by this technique. Since that time, a number of more sophisticated techniques have been developed to determine the “geographical profile” of a suspected serial criminal based on the locations of the crimes.

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 at 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 prediction for law enforcement officers. The prediction should provide some kind of estimate or 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, you 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.

Your report should include an overview of your approach and describe situations when it is an appropriate tool and situations in which it is not an appropriate tool.(2007 MCM) The United States Constitution provides that the House of Representatives shall be composed of some number (currently 435) of individuals who are elected from 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 state will have, it says nothing about how the district represented by a particular representative shall be determined geographically. This oversight has led to egregious (at least some 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 congressional districts for a state. How would you do so as a purely “baseline” exercise to create the “simplest” shapes for all the districts in a state? The rules include only that each 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 your solution is fair. As an application of your method, draw geographically simple congressional districts for the state of Pennsylvania.(2016 MCM) A person fills a bathtub with hot water from a single faucet and settles into the bathtub to cleanse and relax. Unfortunately, the bathtub is not a spa-style tub with a secondary heating system and circulating jets, but rather a simple water containment vessel. After a while, the bath gets noticeably cooler, so the person adds a constant trickle of hot water from the faucet to reheat the bathing water. The bathtub is designed in such a way that when the tub reaches its capacity, excess water escapes through an overflow drain.

Develop a model of the temperature of the bathtub water in space and time to determine the best strategy the person in the bathtub can adopt to keep the temperature even throughout the bathtub and as close as possible to the initial temperature without wasting too much water.

Use your model to determine the extent to which your strategy depends upon the shape and volume of the tub, the shape/volume/temperature of the person in the bathtub, and the motions made by the person in the bathtub. If the person used a bubble bath additive while initially filling the bathtub to assist in cleansing, how would this affect your model’s results?

- Derive the mobility equations for some linkages we haven’t studied, like https://synthetica.eng.uci.edu/mechanicaldesign101/Kempe-Straight-Line.pdf

- In the Conways life simulation steady-state features can be added or subtracted by flipping one cell. Show how to build more complicated things by clicking a few cells and letting the rules fill in the rest.