Executive Summary With so much political and economic interest behind accurate results for the United States Census, the Census Bureau has implemented several strategies for dealing with the particularly pesky problem of undercounting, or the exclusion of certain individuals from the Census: These include sampling the population after the Census to gauge how many people were excluded, guessing values for missing data, and examining public records to estimate the breakdown of the population. Of these, we found that only the last two are sufficiently helpful to merit use, whereas the first strategy of post-Census sampling can lead to error greater than what it was intended to remedy. Of course, even with perfectly reliable Census results, proper political representation cannot be attained without a system that distributes seats in the House of Representatives in a manner that addresses the particularities of the population. Evaluating six methods that Congress has historically considered for dividing seats in the House, we found that the Hamilton-Vinton method surpasses the others in the arena of fair apportionment. After Congress, the next bearers of responsibility are the fifty states of the Union, which are constitutionally charged with drawing district lines that demarcate regions for their representatives. In regard to this process, we suggest that states commit to a system that impartially divides the state according to population density. Such a system, we hold, will serve the common good of the state by achieving the democratic goal that our representative democracy should reflect the sentiments of the American people. Note added in proof: We wrote this paper for the 2010 Moody's Mega Math Challenge. During the contest, we had 14 hours to independently conduct research and write a report, at which point we submitted our paper for judging. This article is an accurate depiction of what we accomplished during the 14-hour contest, meaning that we have limited our edits to minor stylistic and grammatical corrections; in terms of organization and statistical content, this paper reflects only the work completed during the contest. As we advanced in the contest, we were given the opportunity to update our research before giving a presentation. We were also allowed to discuss the problem with our coach and other teachers. At this point, we improved upon our computer algorithm for redistricting and corrected our statistical analysis for reapportionment. These important changes, along with our presentation slides, appendices containing code, and further information about …

Executive Summary With so much political and economic interest behind accurate results for the United States Census, the Census Bureau has implemented several strategies for dealing with the particularly pesky problem of undercounting, or the exclusion of certain individuals from the Census: These include sampling the population after the Census to gauge how many people were excluded, guessing values for missing data, and examining public records to estimate the breakdown of the population. Of these, we found that only the last two are sufficiently helpful to merit use, whereas the first strategy of post-Census sampling can lead to error greater than what it was intended to remedy. Of course, even with perfectly reliable Census results, proper political representation cannot be attained without a system that distributes seats in the House of Representatives in a manner that addresses the particularities of the population. Evaluating six methodsVinton) that Congress has historically considered for dividing seats in the House, we found one (Hamilton–Vinton) that surpasses the others in the arena of fair apportionment. After Congress, the next bearers of responsibility are the fifty states of the Union, who are constitutionally charged with drawing district lines that demarcate regions for their representatives. In regard to this process, we suggest that states commit to a system that impartially divides the state, according to population density. Such a system, we hold, will ultimately best serve the common good of the state and strengthen the faithfulness with which our representative democracy reflects the sentiments of the American people.

http://www.census.gov/prod/2001pubs/c2kbr01-7.pdf http://www.census.gov/dmd/www/pdf/Report21.PDF http://www.census.gov/prod/cen2000/doc/sf1.pdf http://www.census.gov/pred/www/rpts/L.2.pdf SIAM Society for Industrial and Applied Mathematics 3600 Market Street, 6th Floor Philadelphia, PA 19104 USA M3Challenge@siam.org M3Challenge.siam.org Moody’s Mega Math Challenge 2010 A contest for high school students

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