It has happened once. The stats given in this thread indicate as much.
I wonder if a significant gambling edge can be gamed here…
Yup, darned unlikely…though, as long as Bill Belichick is coaching, I wouldn’t say it’s impossible. He does seem to like to run up the score, given the opportunity.
Septimus said it has never occurred.
My thought is that, sure, a two-safety score is extremely unlikely, but every time a game is played its a possibility, and there have been a LOT of foobtall games played. To put it in gaming terms, sure it’s a 45,000-sided die, but the number of football games played, well … that’s a LOT of rolls.
Septimus said 4 never won a game. 4 was, however, the losing score in 1 game.
Sorry, correction. Septimus said 4 never occured as a total score. The second half of my post above stands, the losing team scored 4 one time.
This thread reminds me of a game in college…well into the fourth quarter, the score was 2-0. I was hoping it would end that way, just so I could say that I saw the lowest scoring game possible that ended in a win. Someone scored a touchdown, though.
So I almost have a story to tell.
Well you almost tell it very well.
Typo on the zero?
There has not been a 0-0 tie in the NFL since 1943:
Oops. :o These were total points, so divide by two for the correct ordered list for points by tieing team:
0, 7, 10, 17, 20, 13, 24, 28, 14, 21, 6, 3
(I should have included my usual disclaimer: Output is not warrantied or recommended for use in life-threatening applications. 
)
That page shows 73 games with 0-0 score between 1920 and 1943, which places 0-0 ahead of second most common tie score, 7-7 (31 games).
The main page shows 75 games with 0-0 score, including one in 2012. Procedural bug by them?
Relevant thread that contains the 4 point game.
http://blogs.wsj.com/numbersguy/more-on-rare-nfl-scores-491/
Hello: I know this is a five year old thread but I came upon it looking for optimal numbers for “squares”. To answer the question about using the fact that 3 is the most common scoring difference in the NFL from my fellow Jacksonville person…If you bet with a professional sportsbook, you will have to pay “double juice” to move an NFL game onto or off of 3. In other words, an NFL team is favored by 3…you can lay the normal 110 to win 100 and if it lands on 3 you push. You can buy a 1/2 point for 120 to win 100; HOWEVER,
If it’s an NFL game and you are moving the line to 3, 2.5, or 3.5, favorite or underdog, you will have to pay 130 to win 100. Hence the “double juice”.
This does not apply to ANY other number(not even 7), and it doesn’t apply to college football either. Just NFL, just the number 3. In all other cases, buying a 1/2 point, you would be laying 120 to win 100.
On the fun side…I once crushed a local bookie for several weeks buying the 1/2 point for 120 to win 100 on NFL games(it comes up a lot); until he said “we ain’t doin” that shit no more". There are places and people who will let you do it for 120. If you can find them, it works and it is powerful, I promise.
I would have thought the Canton Bulldogs could have beaten the Washington Glee Club, but apparently not.
It should be noted that the scores by the two teams are not independent events, so you can’t assume that the most common square will be the intersection of the two most common individual team scores. How close the game is will influence whether the coaches make risky decisions like going for it on 4th down (and possibly getting a touchdown) or settling for a field goal, or going for 2 vs. 1 after a touchdown.
Even though it’s a zombie I’d like to point out that a team scoring 1 point is possible.
If team A gets the touchdown, and goes for the conversion, and screws it up so badly that they lose the ball all the way back to their endzone where it either goes though the end zone, or is recovered by one of their players who is tackled, it goes as a 1 point safety for team B, resulting in a 6-1 score. It’s never happened of course, and very likely never will, but it is slightly more likely now that the kicking team only has to lose 85 yards, rather than 98 
My high school lost a game 2-0. It was rainy, the field was a mud pit and neither team could even hold on to the ball, much less advance it.
I remember this trivial factoid not because of the unusual score, but because I was on the high school newspaper, the sports reporter missed the game, and it was the only sports story I wrote the entire year.
It has happened in the NCAA, as recently as 2013.
Dennis
On the topic of getting bookies to give you an edge, I have a friend who was a big basketball bettor and got some bookies to let him place bets that might seem sensical but where the bookie is giving up a lot:
The bookie offers spread and over/under bets on the same game, both at 100 to 110. My friend asked for (and received!) parley bets like
“I bet $1210 on the Warriors but if it wins just keep the whole $2310 and bet it for me on the Over.”
If the two wagers were uncorrelated this would be a normal way to bet. But if not — you know that If the Warriors win Then it’s likely to be a high-scoring game — you’ve just tricked the bookie! (If the mathematical principle isn’t obvious, take the extreme case, where you take the $2310 and re-bet it on … the Warriors’ win!)
The likelihood of a high-scoring game if the Warriors win isn’t the big advantage in this example. The real advantage is the higher payout than a normal parlay gives.
Back in the day when I used to bet, a two-team (or team + over/under) parlay paid 12:5. So if I bet $50 and lost I’d owe $50 but if I won I’d get paid $120. (No vig on a parlay, mainly because the payout itself makes it a sucker’s bet.)
If instead I got to place $50 on a team, and then if that won, I could place the whole $100 on a second bet to get $200 if that also won…yeah, that’s massively better.