I wanted to give everyone an alternate solution to this achievement that I feel is less time consuming, and also viable for people who may not have 4 controllers.
To recap, you need to land on *OR* own every property on every board in the game. After playing for some time you will likely have unlocked several boards on your own without going for this.
Dogfather76's solution is valid and quite effective on boards where you do not have any stamps in your passport, and I recommend it if you are filling in a new board.
Later on, when you have only a few stamps on a board left, relying on the game to randomly deal a property you need can be frustrating, even with 4 players. You'll find that sometimes round one dice will land as 1,1,2,3 or some other low combo that deals out less than 10 of the 28 properties.
What I did was go into my passport book and for a given board, look at what properties I needed. So, say for example I needed Boardwalk only on the classic board.
Start a richest game using the board with the stamp you need on Tycoon level (12 rounds) with 3 computer players and play normally. At some point in the first three rounds the property will likely get dealt to a player. If that player is you, then congrats you have your stamp and you can move on. If not you have more options in tycoon:
- Chance cards that allow you to take one property
- Community Chest cards that allow you to take three properties
- Go space lets you buy an unowned property
- Free Parking lets you move a piece to another space
Even with these extra ways to gain properties, my primary goal was to use the trade intermission after round three to trade ALL my properties to the computer player that had the property that would get me the stamp. So I might trade 3-6 properties for just one property, which even when breaking up monopolies is practically a guaranteed trade for the computers. This will grant you a stamp for the property, even though you have not technically landed on it.
If for some reason the property you need is not owned at the end of round three, you can quit the game and then resume it at the beginning of round, giving you another shot at the property you need.
After round three its up to you if you want or need to continue on the same board, I found it easier to quit and start a new game, continually playing until the end of round three. Note comments below which indicate you must at least start round four for your stamps to save.
Using this method I was able to get the 14 stamps I needed to unlock the final board in about 45 minutes, and then filled in the final board in another 30 minutes or so.