Following twin centuries that left even the most perceptive judges searching for superlatives, Steven Smith's skills have been summed up by Australia's coach Justin Langer as being those of the "best problem-solver in the game", who, at the same time, creates more problems for bowlers than anyone else on the planet.

Over the Australian summer, Langer was no stranger to greatness, given that his weakened team had to find ways to dismiss Virat Kohli, largely without success. But over five days at Edgbaston, on a pitch that started by offering seam and was spinning square by the end, Smith played contrasting but equally masterful innings that bumped him up to No. 3 on the Test rankings and set up Australia's 251-run victory, their first in Birmingham since 2001.

"I said during the summer that Virat Kohli is the best player I have ever seen but that [Smith's knocks] is just another level," Langer said. "You go through the periods, I remember being a young player on the fringe for a long time watching Steve Waugh on all the tours and he was a run machine, I wanted to be like Steve Waugh. Then I played with Punter and I wanted to be Ricky Ponting because he's a run machine.

"I played with Allan Border and in my third Test he made his 10,123rd Test run [becoming the highest scorer of all time] and I wanted to be Allan Border. You have in different teams, different eras, great players but for someone like Smudge [Smith], who is averaging over 60 and the way he played in this innings with all the pressure and everything that is on him, it was not only great skill but enormous character, enormous courage, very brave, unbelievable concentration, unbelievable physical stamina, unbelievable mental stamina, all traits of great players.

"You throw to him in the nets and you literally feel like you can't get him out and there's no batsman like that and you just end up throwing a million balls and you are just lost for answers. I guess you would have to ask England how they feel about that and they had some interesting tactics for him. They had the really short point which I had never seen before, they obviously had a tactic they prepared and Trevor Bayliss has seen a lot of Steve Smith since he was a kid. They would have studied him closely but he just has a knack, well, he is the best problem-solver in the game."

Langer's first sight of Smith had been back in 2009 when he was called into the Australian squad as a wristspinner and lower-order batsman, a truly modest platform from which Smith has built a career to be reckoned with. "I've got a good story but I'll leave it for my book I think," Langer joked about his first impressions.

"Let's face it; when he first came in, legspinner, unorthodox... [everyone thought] 'I'm not sure this kid's going to make it'. Then he goes away [and decides] 'I don't want to be a legspinner; I want to be the best batsman in the world'. Then he transforms himself and he is the best batsman in the world with Virat. It's a great credit to him.

"He's under no instructions. He can hit as many balls as he wants. I don't think he will because he won't have anyone to throw them to him. Unless Dani [Willis, Smith's wife] wants to put some in the bowling machine for him. She has done it before. He can prepare however he wants to prepare because we can have a theory on different players but he's just shown once again that he hits more balls.

"I said to Mark Ramprakash the other day, who used to hit more balls than anyone I've ever seen; Steve Smith hits double of what Mark Ramprakash used to hit. So, literally, he can prepare however he wants to prepare and we will keep encouraging that. But he's going to be that tired after this game, physically and mentally, so I'm sure he'll welcome the few days off."

As for Nathan Lyon, who claimed 6 for 49 in the final innings to seal victory and a 1-0 lead for the tourists in Birmingham, Langer conceded he had not realised the offspinner's true quality until he coached Australia. "Like a few of our players, like Steve Smith, I didn't realise how good Nathan Lyon was until I took over coaching the cricket team," Langer said.

"You see him on telly, yeah Nathan Lyon, offspin bowler, but the last 12 months or so I just can't believe how good a bowler he is. He's a brilliant bowler, such good control: he spins, he gets some bounce, great fielder, really good in the team, he's the song master, so that tells you something about his character. He's a gun bowler. Three hundred and fifty wickets - that's an incredible achievement, isn't it?"

Reminded of Australia's recent penchant for losing series after being 1-0 up, Langer said the team he mentors was demonstrating plenty of resolve. "We talk about our values and you guys will probably start yawning when I say it but I say it very seriously," Langer said. "Humility is one thing, very important, professionalism, mateship, you'll see plenty of mateship in the change room right now which is half the reason we play the game.

"[We] keep being honest about where we're at; we've still only won one Test match. And we've got a captain [Tim Paine] who is that focused in there and he's the best wicketkeeper in the world. You've just seen it. You've seen those catches today. He's the best wicketkeeper in the world. I thought the way he batted in the first innings, he freed up and he's worked really hard. He's a brilliant leader, a great bowler and I thought tactically he was superb. He is so focused in this series, so there is no way he is going to let it slip and I guarantee you, I'm not going to let it slip either."