I've seen many people asking things like: "How did Erwin figure out there were other Titan Shifters?", "How did Erwin figure out Reiner was a spy?", "How did Erwin figure out the Founding Titan could control people's memories?", etc.
Again, I don't blame people for not catching these details since the narrative isn't honest. Armin is just narrating Erwin's action/deductive work; this is priming the viewer to think that Armin is smarter than Erwin while in fact the opposite is true. Have you noticed that the show doesn't bother shedding light on what Erwin actually did behind-the-scene? The viewers don't see Erwin's deductions so most end up mistaking his deductions for Armin's, or thinking that he's "less smart" than Armin.
For instance, look at this thread where someone asked how Erwin knew there were other Titan Shifters. It is a common question among the fandom.
Armin is smart, but not to the level of "smarter than Erwin". The hype around so-called "genius intelligence" in many anime series is made up by the fandom because the anime says it instead of showing it.
Erwin came up with >90% of the strategies in S1-S3. He has been the brain of the SC all along.
Seasons 1-2: how he predicted the Titan Shifters and devised strategies to catch them:
Titans had never been smart enough to get over or through the Wall. But now there were a Colossal one that appeared out of nowhere, broke the wall and vanished TWICE, and an Armored one that did the same thing. When the Colossal Titan appeared out of nowhere to break the Wall the 2nd time in front of Eren & other cadets, none of them saw him run away, and he didn't engage in combat but vanished in thin air again, and there's no place that such a giant could've possibly hidden himself, so one possibility is that he was one of the people within the Walls all along.
Then Erwin learned that Eren has the ability to transform into a Titan and vanish at any point. Putting two and two together, he concluded that Eren wasn't the only human with the same ability.
Erwin knew Eren wasn't an enemy because he had done a background check on him [ extra info from Smartpass ].
One more crucial clue that Erwin knew they would come after Eren: The first time they broke the wall, the Colossal and the Armored Titans appeared together. We would expect the same behavior the 2nd time, but the Armored Titan never appeared the 2nd time. What does it mean? → Their motives must have changed halfway through: If they saw Eren transform, then their motive was probably to find him.
Erwin didn't know for sure that there would be a Female Titan but he knew that there was a high possibility that a spy had infiltrated their ranks during the fall of Wall Maria and that they would go after Eren because he's a Shifter working for the SC. What happened with Sawney & Bean (Hange's experimented Titans) confirmed his suspicion.
That is why only soldiers that Erwin knew before the fall of Maria could be trusted with his expedition plan.
After the deaths of Sawney and Bean, Erwin was able to narrow down the list of suspects, as only a soldier was able to kill the Titans with 3DMG. Because of this, he went around asking cryptic questions to try to weed out traitors. "Who do you think the enemy is?" Basically, he was looking for people who had the same suspicion as he did: that someone among them sneaked into the city during the fall of Wall Maria 5 years ago. If you understood the question, Erwin knew you had the same suspicion and therefore could be trusted, as a traitor would likely play along with the popular belief that a crazy Titan-hater must be the culprit that killed Sawney and Bean.
On the night of the induction ceremony when the cadets joined the Survey Corps, Erwin instructed Levi to take Eren out on a ride to survey the area. This was Erwin's way of narrowing down the suspects. By having Eren leave and the ceremony on the same night, he could deduct that if the enemy did not attack Eren or show up at the castle to kidnap him, then it must mean that they were somewhere else: the ceremony. If someone was missing from the ceremony, it would be suspicious, so they must be there. He specifically mentioned Eren and his basement in his speech to the cadets because he suspected it must have been one of those who saw Eren at Trost: the 104th. He believed that the spy would be immediately interested in joining the SC in order to go to the basement. So, whoever joined the SC at this point was ruled IN as suspects.
So, now Erwin knew 3 things: (1) there's a spy among their ranks, (2) they were a Shifter or had connection to one, (3) they were going after Eren.
During the expedition, everyone was told different sections where Eren was going to be. Some were told it's the right flank; others were told the center rear, the left flank, etc. Because the Female Titan attacked from the right flank, this put anyone who had been told that that was Eren's location under immediate suspicion (Reiner).
Erwin also predicted that the Female Titan would transform again and went after Eren, therefore he instructed Levi to stay behind to save Eren.
After the expedition [ extra info from official Smartpass ], Erwin did a background check on Annie and found that she came from the same village as Reiner and Bert. This is the reason Reiner and Bert ended up as suspects and were put under Mike Squad's supervision at Castle Utgard as far away from Annie and the Survey Corps as possible.
In Season 3, Erwin figured that the real reason that the government kidnapped Eren was because he possesses the Founding Titan, and from that, he deduced that the Founding Titan has the ability to alter memories and that's why the government was after him.
Yes, it is frustrating and not a lot of people realize what a great strategist Erwin is, because most people (including me) will not fully understand these details on our first watch (unless we take time to connect the dots). Actually, Armin was mostly reiterating the findings that the Survey Corps had found. The series tried to establish him as "a genius" by "showcasing" him in multiple ways while "forgetting" to credit Erwin.
Meanwhile, the narrative focuses a lot on moments where Erwin's plan has failed, thereby making people think that "his plans always failed" or "he's no good".
A lot of stuff about Erwin's leadership and what he does behind the scenes is truly... behind the scene. A lot of people don't know what his job as a Commander really entails. You can only find more about it in the semi-canon Smartpasses, which you can find here.
