43 American Cities for Nerdy, Trivial Mathematicians

43 American Cities for Nerdy, Trivial Mathematicians


43 Special American Cities for the Trivial Math Nerd.

I like this chart because it combines a few services that are available on the Internet and shows the flexibility of a Mash-Up. 

Alledonia, Ohio  43902
Arbovale, West Virginia  24915
Ashley, North Dakota  58413
Burnips, Michigan  49314
Canova, South Dakota  57321
Coaldale, Colorado  81222
Cross Timbers, Missouri  65634
Cuney, Texas  75759
Cynthiana, Ohio  45624
Dora, Missouri  65637
East Wenatchee, Washington  98802
Glenmora, Louisiana  71433
Great Falls, Montana  59403
House Springs, Missouri  63051
Humble, Texas  77346
Kansas City, Kansas  66105
Kensal, North Dakota  58455
Las Vegas, Nevada  89145
Lawrence, Mississippi  39336
Madison, Alabama  35757
Mandan, North Dakota  58554
Mcbrides, Michigan  48852
Miami, Florida  33144
Millcreek, Illinois  62961
Minneapolis, Minnesota  55446
Monticello, Minnesota  55581
Nephi, Utah  84648
North Hollywood, California  91605
North Richland Hills, Texas  76182
Oneill, Nebraska  68763
Rantoul, Illinois  61866
Reno, Nevada  89523
Renton, Washington  98055
Saint Louis, Missouri  63129
San Jose, California  95154
Scott, Mississippi  38772
Shawnee Mission, Kansas  66276
Tilden, Nebraska  68781
Tilden, Texas  78072
Tolleson, Arizona  85353
Trafford, Alabama  35172
Tucson, Arizona  85743
Waterloo, Iowa  50706

I've been to several of these cities. Renton, Miami, San Jose, Saint Louis, Las Vegas, Kansas City, and Reno, certainly. Tilden, NE is almost exactly 1000 miles north of Tilden, TX, though I suspect that's a coincidence. 

What is this Map?
Before identifying how these cities were chosen, it's interesting to see if it's a Rorschach test of some sort. 

Is it political? Our society is so inundated with political messaging that one first action would be to think that this is some sort of political map such as Congressional races that are at risk. This map covers a lot of "Republican" territory, though Miami and the San Francisco Bay area stand out as less likely. However, if I was told this was related to an election, it could pose as an electoral communication aid. But it's not.

Is it a transportation map? The points appear to be regularly spaced. If someone told me these were the proposed locations for refueling stations for commercial electrical trucking, I'd believe it. The Northeast would represent a market that had not yet been penetrated, and the population-inverted presence in the midwest mirrors places where wind energy is strong. The stations in Colorado, Montana and Utah seem like plausible points where intermediate charging stations had to go to allow connectivity with the west coast.


Significance
Each of these Zip Codes is the perfect square root of a perfect pan-decimal number, that is, a number with each digit only once. For example,  St. Louis, MO, 63129 * 63129 = 3985270641. Each digit 0 - 9 occurs only once.

Step 1
In all, there are 3265920 perfect pan-decimal numbers. The smallest perfect pan-decimal number is 1023456789, and the largest is 9876543210. A leading 0 seems inappropriate for this exercise because every integer can be represented by an arbitrary number of leading 0s.

I determined that there are 87 perfect pan-decimal numbers that are perfect squares. 

I started with every possible integer square root, squared it, and determined if it was a perfect pan-decimal number. On the low side, the value is 31991 would be the smallest square root, and on the high side, the value is 99380 the highest. This provides a relatively small number of numbers to process, 67390. A simple java program collected all 87 candidates, squaring the number, and then testing to assure that each digit value '0' through '9' occurs once.

Step 2
Using USPS Lookup Software, I looked up which post offices matched to the zip codes. The on-line package I used at:
http://www.geonames.org/postalcode-search.html?q=76182&country=US
provided a means to identify not only the name of the city, but also the latitude and longitude coordinates of the city.

Step 3
Having the latitudes and longitudes of each of these cities, I was then able to use this tool to plot the map:
http://www.copypastemap.com


Analysis and Questions
  • It makes sense that the Northeast is not covered in the map because most of the Northeast has a leading 0 in the zip codes. A square of those numbers would result in at least one leading zero, which I exclude.
  • It is perhaps only by chance that Alaska and Hawaii did not have an appropriate Zip Code. 
  • 87 of the 3265920 perfect pan-decimal are perfect squares. I'm curious if this rate is high or low for a selection of any equal-sized collection of numbers in that number range. Does the restriction of being a perfect pan-decimal number make it less likely to be a perfect square? 
  • Educationally, I wonder if there would be any P.R. benefit or synergy for the High School math departments to form affinity groups. Numerology is not a technical reason to collaborate on mathematical-educational research, but it is an excuse. Miami could be voted the Perfect Pan-Decimal Root Capital of the country. What kid in North Dakota wouldn't study math a little harder for a chance to get to Miami for a conference? 

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