The best way to write a clever person engaging in a battle of wits is to research the situation thoroughly and then let your characters work through the problem at a speed that is minutes compared to hours of research.
Consider the following scenario:
A modern-day Sherlock Holmes and Watson are in a sinister trap by Professor Moriarty. Moriarty has tricked the pair into entering a room with a Nuclear Bomb, a timer counting down from ten minutes and a gun with only one bullet. Moriarty, from a safe location, presents the following challenge: When the clock hits zero, the door will open and the men are free to go. However, if the gun is not fired, the nuclear bomb will go off, and London will be a crater. And if the gun is shot and at least one person in the room is not dead before the clock hits zero, then Moriarty will remotely detonate the Nuclear Bomb. Because they suspected that the case was nuclear in nature, Sherlock and Watson are both wearing radiation proof suits, so they won't get irradiated and die shortly after and the room they're locked in is shielded, so the radiation will remain inside, but they still have quite a problem on their hands. Who will take the gun and what will his target be? Or more importantly, how do they get out of this alive?
Well, that's where your research needs to play in. We know that, because this is Sherlock Holmes with Nukes, Watson lives, because Watson is the narrative voice of all Holmes stories, so by dint of story existing, he lives. Watson would never shoot Sherlock, and Sherlock would see the logic in shooting Watson over himself because they still need to beat Moriarty, and that's his job. And neither will nuke London, a major city and seat of their nations government nor through inaction let London be nuked.
Sherlock's answer depends on what research I, the real writer, am free to set up the scenario so there was a way out of the death trap... and the solution has already been presented in my scenario, but it requires you to know what I did and how Holmes can use it to his advantage.
While Holmes is thinking it out, he and Wattson argue over which of them will die and who will sacrafice himself. Watson would naturally make the argument that Holmes needs to live. Holmes calls him an idiot for not seeing the obvious, Wattson insists he can't beat Moriarty. The fight goes on, the clock gets closer and closer to zero. Drama and tension ensue. Finally down to the wire, Holmes convinces Watson to give him the gun and let him do what they both know he needs to be the one to at least shoot the gun... He cannot watch his friend commit suicide. Watson agrees, and stands in the line of fire. Holmes aims the gun at Watson and hesitates... Watson now starts to get upset because there are now seconds left on the clock. It's in the final moments, and if Holmes doesn't act now, the whole arguement will be for naught. Holmes still hesitates. As we get to the final five, Holmes takes a deep breath and...
Yells "Duck!" Watson does so, Holmes pulls the trigger, the bullet flies over Watson's head and hits the Nuke. It explodes... but with a much less dramatic and definately not "London Destroying" blast. In fact, the room is intact. The door opens, Holmes grabs Watson and they run out, closing the door. Moriarty is irrate and presses the detination button... and the bomb explodes but only with enough force to destroy the bomb. It doesn't make a mushroom cloud... it doesn't even break the shielded room.
Of course, we halve to have Watson ask Holmes what the hell, as the readers are doing. It's then we get the solution to the puzzle: It's a Nuclear Bomb!
Or to explain, Nuclear Bombs are very sensative and difficult devices. One of the many reasons is that, to get that detonation that you want, you have to implode the Uranium... and in order to do that, the fissile material is surrounded with conventional explosives that are rigged to go off with precision timing. If any of these explosives are even the slightest bit off time (with a second between detonations being way too late) then you go from having a Nuclear Bomb to a very expensive Dirty Bomb.
Knowing this, the answer is elementary. If you detonate even one conventional explosion early, the Nuclear Bomb is just a Bomb. London is saved, both heroes now have a survivable chance. Sherlock wouldn't tell Watson this because Moriarty is watching them from a far so if Holmes told Watson this, then Moriaty would have ample time to remotely detonate the bomb. Holmes needed to wait until the last second before he enacted his plan as he figured that Moriarty would be so caught in the drama of the final second disarm that he would not know what was happening until it was too late, and if the remaining packets exploded they will still die. And what Hollywood movie would have a ticking time bomb that was Disarmed with 9:30 minutes to spare and cut to the Bomb Squad having cake in the office while celebrating the Squads new disarmament record.
Since the gun firing disarms the bomb and the clock opens the door, if Holmes would additionally risk Moriarty pressing the button anyway, which is still leathal while they waited for the clock to hit zero. And of course, the shielded room and proper radiation gear was just writing out the problem in that the radioactive core is going to be spread by the blast. Which did happen, but again, it was minimally reduced.
I did nothing clever. I just researched many aspects of the Cold War including the weapons systems and hey, it's unlikely that disarming a nuclear bomb will ever be important in my entire life. But it hurts no one to know the basic principle of it and if you don't show it, someone who does know this trick will show you.
And, to be meta, even the use of Holmes was an allusion to a trait of the character. Holmes knows what is important for him to make his deductions because he specifically studies the things he needs to know that many other people will over look. And only those things. There's a cannon story where Watson lists Holmes' areas of expertise and among them, it's not astoronomics. In fact, Watson was shock to learn that Holmes could identify where a body came from by the dirt on his boots, but literally did not know that the Earth revolved around the Sun (yes, you fans of the BBC series, that exchange was from the books. They weren't just making it up).
As Holmes reasoned, knowing all the soils of London was more important because it was useful for solving London based crimes... but knowing the earth revolved around the sun was useless when solving London Crimes. At best, he would only need to know that it rose in the east and set in the west, but why it was always that way was useless and retaining that fact took up too much memory in his brain that could be used for knowing something useful.
And that's really the trick. It's not about being witty, clever, smart, or stratigic the moment you write the plan down. It's about researching the things that make you say "I never knew that" and then writing about characters that did. Just because they Have 10 minutes, doesn't mean you should write the entire scene just as quickly.
There are two rules about the difference between a good Lawyer and a Great Lawyer:
"A Good Lawyer will only ask questions when he wants the answer. A Great Lawyer will only ask questions when he knows the answer."
"A good lawyer knows the law. A great lawyer knows the judge."
Translation: Know the outcome before you do it. Works for clever characters and the clever people who write them.