# How many borders between states are there in the United States?

How many borders between states are there in the United States?

Sure, you could get out a map and count them. Or you could estimate.

There are 48 contiguous states. The average state has six borders [citation needed], so that’s 288 borders, but we double-counted, so that’s 144. But we need to apply a bit of a haircut for those states that are around the edge. How many of those are there? Figure the US is roughly a 5-by-10 rectangle of states, so there are 30 states around the edge. 144 minus 30 is 114.

There are actually 109. In 1998 Thomas Holmes constructed a data set of those borders for a paper, The Effect of State Policies on the Location of Industry: Evidence from State Borders. I haven’t read the paper. It appears that it shows that there was more manufacturing activity on the “pro-business” (anti-union, has so-called right-to-work laws) side of a state border than on the “anti-business” (pro-union, doesn’t have so-called right-to-work laws) of the state border.

This method could probably also be applied with, say, mask mandates and COVID case rates. Early on in the pandemic there was some coverage of how Tennessee was doing much worse than Kentucky, although that may have been overly politicized (Kentucky has a Democratic governor, Tennessee a Republican) and may have been due to higher testing rates in Tennessee. (See Andrew Gelman’s post on the topic; it appears that data on deaths didn’t show the same gap.)

Some people like counting the borders they’ve crossed, as in this post at Twelve Mile Circle. That post includes a map by Jon Persky that gives 138 borders, but that includes 16 land crossings between contiguous US states and Canadian provinces; 8 between US states and Mexican states; two between Alaska and Canadian provinces; and three borders that can only be crossed by water (Maine – Nova Scotia, New York – Rhode Island, and Ohio – Ontario).

As for that fact that “the average state has six borders”, this is really a statement about planar graphs. From the map of the US, construct a planar graph by taking the 48 states as vertices and the state borders as edges. (You have a problem at Four Corners, which we’ll ignore.) Let E be the number of edges in the graph, and F its number of faces. Here a “face” corresponds to a place where three states meet, such as Pennsylvania-Maryland-Delaware or Georgia-Alabama-Tennessee. Then every edge meets two faces and, except for around the perimeter of the graph, every face has three edges, and thus $2E \approx 3F$. Euler proved $V - E + F = 2$, which we’ll approximate as $V - E + F \approx 0$. Thus $V - E + 2E/3 \approx 0$, or rearranging $E \approx 3V$; the number of edges (state borders) is about three times the number of states.

This is all brought to you by Colin Beveridge’s kids asking the same question about national borders.