The Absorbing State

Once a month my deaf cousin from Georgia visits her grandmother who lives in the state of Washington. Because of her hearing impairment, she and I made an agreement that she always texts me her whereabouts when she goes on a trip. She does this in the following way: every time she starts traveling or sets foot in a new state, she sends her location, along with some numbers. These text messages always look like this: "STATE: [STATE], OPTIONS: $m$ IN, $n$ OUT", where [STATE] is the state she's currently in and $m$ and $n$ are two integer numbers. I've never given too much thought to these numbers, I thought them just one of her peculiarities, but I believe they may be important right now.

The reason for this is, my cousin has gone missing! And I need your help to find her.

Last week she went on her monthly trip to her grandmother. She was supposed to return last night, but she never did and the messages I've received from her don't look good.

On her trip to Washington she sent me this:

GEORGIA -> WASHINGTON
STATE: Georgia, OPTIONS: $5$ IN, $4$ OUT
STATE: Maine, OPTIONS: $5$ IN, $4$ OUT
STATE: Maryland, OPTIONS: $6$ IN, $6$ OUT
STATE: Illinois, OPTIONS: $4$ IN, $3$ OUT
STATE: Colorado, OPTIONS: $4$ IN, $2$ OUT
STATE: Virginia, OPTIONS: $3$ IN, $3$ OUT
STATE: Delaware, OPTIONS: $4$ IN, $2$ OUT
STATE: South Carolina, OPTIONS: $3$ IN, $4$ OUT
STATE: Connecticut, OPTIONS: $1$ IN, $3$ OUT
STATE: Iowa, OPTIONS: $4$ IN, $6$ OUT
STATE: California, OPTIONS: $4$ IN, $2$ OUT
STATE: Washington, OPTIONS: $1$ IN, $5$ OUT

For someone who doesn't know my cousin, this may seem unusual, but it's actually the same route she always takes. You should know that my cousin has some very peculiar traveling habits, she never goes from Georgia straight to Washington. She once explained to me that she could skip a few states in her route, but she likes it this way.

One thing was odd about the route she took this time, though. There is one state in the messages above that isn't quite right and at the time I didn't think much of it, I just assumed it was a typo. But I realize now that it may have been an indication that her GPS receiver was starting to have issues. And precisely that is what may have caused her to get lost on her way home.

Normally, on her return trip, she would just fly straight back from Washington to Georgia, but clearly that's not what happened this time. I've received the following texts from her:

WASHINGTON -> ???
STATE: Washington, OPTIONS: $1$ IN, $5$ OUT
STATE: ?, OPTIONS: $5$ IN, $4$ OUT
STATE: ?, OPTIONS: $6$ IN, $6$ OUT
STATE: ?, OPTIONS: $4$ IN, $3$ OUT
STATE: ?, OPTIONS: $1$ IN, $2$ OUT
STATE: ?, OPTIONS: $3$ IN, $3$ OUT
STATE: ?, OPTIONS: $4$ IN, $6$ OUT
STATE: ?, OPTIONS: $4$ IN, $3$ OUT
STATE: ?, OPTIONS: $1$ IN, $1$ OUT
STATE: ?, OPTIONS: $7$ IN, $0$ OUT

As you can see, her GPS receiver hasn't been working any more since she left Washington. The "0 out" in her last message leads me to believe that she sees no way out of wherever she is right now. My best guess is that she planned on taking a small detour and subsequently got lost, due to the malfunctioning GPS receiver. I tried to reach her, but her battery seems to have died in the meantime.

Can you tell me where she is right now?

N.B.: I believe figuring out her traveling habits is the key to finding her location.

• I hope this is just a story and not a real situation? Jan 31, 2017 at 8:17

The pattern of travel involves the following:

She can only travel to states with where the trailing letter of the current state's capital matches the first letter of destination state capital.
The IN number represents how many other state capitals end with the first letter of the current capital. These are ways in to the state.
The OUT number represents how many other state capitals start with the last letter of the current state's capital. These are the ways out.

As such, I believe your cousin has currently found herself in:

Tennessee (Nashville)

Here's the explanation of her trip to Washington:

Georgia (Atlanta) - 5 in: Augusta, Columbia, Helena, Olympia, Topeka. 4 out: Albany, Annapolis, Augusta, Austin

Maine (Augusta) - 5 in: Atlanta, Columbia, Helena, Olympia, Topeka. 4 out: Albany, Annapolis, Atlanta, Austin

Maryland (Annapolis) - 6 in: Augusta, Atlanta, Columbia, Helena, Olympia, Topeka. 6 out: Sacramento, St. Paul, Salem, Salt Lake City, Santa Fe, Springfield
Sacramento is a shortcut

Illinois (Springfield) - 4 in: Annapolis, Columbus, Des Moines, Indianapolis. 3 out: Denver, Des Moines, Dover
Dover is another shortcut

Colorado (Denver) - 4 in: Springfield, Richmond, Concord, Hartford. 2 out: Raleigh, Richmond

Virginia (Richmond) - 3 in: Denver, Dover, Montpelier. 3 out: Denver, Dover, Des Moines

Delaware (Dover) - 4 in: Concord, Hartford, Richmond, Springfield, 2 out: Raleigh, Richmond
Here's the mistake. It should be North Carolina instead of South Carolina
North Carolina (Raleigh) - 3 in: Dover, Denver, Montpelier. 4 out: Harrisburg, Hartford, Helena, Honolulu

Connecticut (Hartford) - 1 in: Raleigh, 3 out: Denver, Des Moines, Dover

Iowa (Des Moines) - 4 in: Concord, Hartford, Richmond, Springfield. 6 out: Sacramento, St. Paul, Salem, Salt Lake City, Santa Fe, Springfield

California (Sacramento) - 4 in: Annapolis, Columbus, Des Moines, Indianapolis. 2 out: Oklahoma City, Olympia

Washington (Olympia) - 1 in: Sacramento. 5 out: Albany, Annapolis, Atlanta, Augusta, Austin

Whew. Now for the return trip:

Washington (Olympia) - 1 in: Sacramento. 5 out: Albany, Annapolis, Atlanta, Augusta, Austin

Maine (Augusta) - 5 in, 4 out (same as above)

Maryland (Annapolis) - 6 in, 6 out (same as above)

Oregon (Salem) - 4 in: Annapolis, Columbus, Des Moines, Indianapolis. 3 out: Madison, Montgomery, Montpelier

Vermont (Montpelier) - 1 in: Salem, 2 out: Raleigh, Richmond

Virginia (Richmond) - 3 in, 3 out (same as above)

Iowa (Des Moines) - 4 in, 6 out (same as above)

Oregon (Salem) - 4 in: Annapolis, Columbus, Des Moines, Indianapolis. 3 out: Madison, Montgomery, Montpelier
Uh oh, she's been here before ...

Wisconsin (Madison) - 1 in Salem. 1 out: Nashville

Tennessee (Nashville) - 7 in: Austin, Boston, Charleston, Jackson, Lincoln, Madison, Trenton. 0 out: :(

• Nice, well done! That's it! Jan 31, 2017 at 16:42
• If anyone is curious, here's the graph of your sister's allowed travel options. Jan 31, 2017 at 16:44
• I just noticed: The reason OP's cousin comes straight back to Georgia from Washington is because rot13(Ngynagn vf bar bs gur bhgf sebz Bylzcvn!) Feb 1 at 4:39