FOXBORO -- The Patriots face a bit of a conundrum this week. 

They just went out of their way to prove to themselves and others that they can run the football, that they're equipped for the style of play required from teams who play in cold weather in November, December and January. Toughness, as Patriots have defined it many times, comes down to running the ball, stopping the run, and covering kicks. And Bill Belichick was certainly happy with the way his team did those things Sunday. 

"Definitely got the running game going," Belichick told players after the game. "That's a damn good job up front. Covering kicks better. Stopping the run better. Played better as a team. Keep improving, keep working alright. We're going to have a lot of work to do again this week against a football team that we don't know."

That team the Patriots don't know? The Vikings, one of the best run defenses in football, who allow 3.7 yards per carry (third in the NFL) and 93.6 yards per game (fourth). They have an athletic front-seven that features run-stuffers like Linval Joseph (6-foot-4, 330 pounds) on the interior and Everson Griffen (6-3, 275 pounds) on the edge.

Defensive linemen Danielle Hunter and Sheldon Richardson have been among the team's best players in all situations, run or pass. Linebacker Anthony Barr and safety Harrison Smith also do a good job of snuffing out opposing ball-carriers before they're able to pick up steam. 

The Vikings have been so effective against the run that the resulting third-and-long situations they've inflicted upon offenses has made them the best third-down team in football -- by a wide margin. They're forcing teams off the field almost three-quarters of the time on third down (27.6 percent success rate allowed). Meanwhile, the second-best third-down team in the league, the Jets, is significantly off that pace (34 percent success rate allowed). 

"They rank highly in every statistically category, deservedly so," Patriots offensive coordinator Josh McDaniels said Tuesday. "They have really good players at all three levels of the defense, they’re really well-coached, they’re extremely sound, and they really challenge you on third down and in the red zone – some schematically and just some based on their personnel and their rush and the way the cover and how they mix up their calls. 

"But, they have a great front, they have linebackers that are very athletic, the team speed in general is tremendous. You’re not going to out-run this group. Their scheme complements their players very well, they get excellent safety play, they don’t give up a lot of big plays in the run or the pass game, so you’re going to have to drive the football, you’re going to have to convert some third downs during the course of the drive and this is the best third-down team in all of football, easily. There’s a lot for us to wrap our minds around and we’ve really got to have a great week of preparation."

So what are the Patriots to do? Continue to do what they can to establish the run? Or bail on that aspect of their offense to avoid running into a brick wall?

The Vikings are 13th in football in yards per pass attempt allowed (7.5), which is good but not quite as suffocating as the Minnesota run defense. Though the Patriots passing game is at its best when they can run, they may be forced to chuck it more at Gillette Stadium on Sunday -- whether they can run it or not -- just to keep moving. 

Still, though, they're going to try to churn out yardage with their backs. That's what tough teams do, according to the Patriots, and they aren't going to abandon that identity after working so hard to get it established in Jersey last weekend.

"I think we can do it," Andrews said. "The games we've done it, we've been successful. The games we haven't, I think it's pretty telling. This game will always be a physical game. It'll always start up front. This will be a big challenge for us this weekend because they're very good against the run."

I asked Andrews if there's a psychological or emotional impact that running the football well (or not) can do to benefit (or hurt) a team.

"Oh yeah," Andrews said. "I mean, the games where you go out there and you're not the most physical team, you feel that. And sometimes you're fortunate to skate by and get a 'W.' A lot of times you're not. It's nice when you can go and finish the game like we did last week with the four-minute offense. There's something about that, getting that last first down when you need it. Ending the game running the ball is definitely kind of [contributes] to that emotional, psychological side you're talking about."

Running the ball has paid dividends for the Patriots in those intangible ways, but it's also helped protect the quarterback. It's helped receivers find separation. It's allowed them to maintain possession. 

But it won't be easy this week, and the Patriots know it.

"You always want to be balanced and maintain your balance if you can," McDaniels said. "It’s certainly a better way to play football if you can do that and not tell the defense what you’re trying to do on each play. But the flow of the game will determine many times whether or not you throw it more than you run it, run it more than you throw it, or somewhere in the middle. 

"Our intent is to do everything we’re trying to do in the game well. Certainly, that is not going to happen each week. I mean, the other teams that we play are really good, and this week, we’ve got an exceptional group on the other side of the ball. We’re going to try to have a great week of preparation in practice and be ready to play our best game on Sunday and go out there and really compete hard against a tremendous unit."

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