Learn Farsi: ما برای حل کردن این مشکل به تعقل و تفکر نیاز داریم. - We need reasoning and thought to solve this problem.

Both follow the Arabic tafa'ol pattern (ta-C1-a-C2-C2-o-C3), which often expresses a deliberate, intensive, or step-by-step mental action. For تعقل (ta'aqqol, reasoning), the root is '-q-l (intellect). For تفکر (tafakkor, deep thought), the root is f-k-r (think). If you plug those root consonants into the C1, C2, and C3 slots, you get these exact nouns.

Are we supposed to write a tashdid (the doubling symbol) over the middle letters in تعقل and تفکر?

In everyday written Farsi, no. The tashdid is almost always left out. Because you now recognize the tafa'ol pattern, you are expected to know you must pronounce the doubled middle consonant (ق in تعقل and ک in تفکر) even without the visual cue.

Why is there a به right before تعقل?

It is required by the verb at the end of the sentence. The standard structure for "to need" is به ... نیاز داشتن (be ... niaz dashtan). Even though we just say "we need reasoning" in English, Farsi literally says "we have need to reasoning."

Could we just say برای حلِ این مشکل instead of using the full infinitive حل کردن?

Yes! Both are very common and natural. () uses the infinitive acting as a gerund ("for this problem"), while just () uses the noun form ("for the of this problem"). Using the infinitive slightly emphasizes the physical or mental action of solving it.