If I have enough time, and there’s a prize making it worth it, I might write a computer program to try many possible route combinations.

I might just have fun studying the maps, and looking for things like corners between several states with roads that let you hit several at once, etc.

How do prove your visits? Pictures of ‘welcome to’ of every state? Electronically?

This is a thought puzzle, your route is the answer. For a real race there would be lots of ways to verify you arrived in each state.

I start by researching the traveling salesmen problem and end with a Fields Medal.

@Zaku “I might write a computer program to try many possible route combinations”

It’s an NP-hard problem with a O(n!) time factor.

@gorillapaws wow!

Since there is no restriction on visiting a state more than once, the shortest path would involve travelling the westerns from Four Corners up through the Northwest and the mountain states, and the travelling along the Mississippi south and then following it north, across the top tier and then from Maine taking I 95 all the way to Florida and Dixie.

@gorillapaws In other words, it’s at least a somewhat interesting problem.

I’d start in Maine and proceed as follows:
ME, NH, VT, MA, RI, CT, NY, NJ, DE, MD, DC, VA, WV, PA, OH, MI, IN, IL, WI, IA, MN, ND, MT, WY, SD, NE, CO, UT, NV, ID, WA, OR, CA, AZ, NM, TX, OK, LA, MS, AL, FL, GA, SC, NC, TN, KY, IL (2nd time), MO, and I’d end up in the middle of the country KS. BOOM!

Can I fly?

I would go in a zigzag throughout the country and hit D.C one way or another.