>> What is the hardest technical problem YOU have run into?
I have solved about ten "hard" problems in my career, most of which has been in R&D. Each one of these had multiple prior failed attempts, and in some cases took me months of thinking before I could find a solution.
1. Qualcomm wanted me to devise a computer vision solution that was more than two orders of magnitude power-efficient than what they had then. There was a clear justification existing as to why such a drastic improvement was needed. Nobody had a solution in spite of trying for a long time. Most laughed it as impossible. I started by looking for a proof as to why it could not be done if it indeed could not be done. After some three months of pulling my hair, I started getting glimpses of how to do it. Some three months later, I could convince myself and a few others that it is doable. Some three months later, the local team was fully convinced. Some three months later, the upper management was convinced. You can read the rest here: https://www.technologyreview.com/s/603964/qualcomm-wants-you...
2. I wanted to solve a specific machine learning and Artificial Intelligence challenge. I would code for a day or so, and then again run into days of thinking how to proceed further. E.g., coded a specific parser algorithm for context-free grammars, including conversion to Chomsky normal forms, in 1.5 days including basic testing. However, what's next. Woke up with new ideas for about ten days in a row. Conceived Neural Turing Machines back in 2013, about a year before Google came up with their paper on the subject. (Unsurprisingly, I did not had that name in mind for it back in 2013.) I also did not get an actual opportunity to work on it, as a result of which I am still not sure if I could have actually done it.
3. Needed to make a very sensitive capacitance measurement circuit, trying to get to atto Farad scale floating capacitance even with pF scale parasitic capacitance to ground. The noise and power requirements were very challenging. After about three months of seeking inputs from the team lead without hearing a solution, I ended up coming up with a solution. I later discovered that the technique was already known in RF circles, though only a few were aware of it. Capacitance measurement circuits with such sensitivity did not show up in the market for several years. (My effort was target at using inside a bigger system.)
4. I was working on measuring bistable MEMS devices. The static response of these was well understood. However, so far, the dynamic response was only measured by the team; there was no theoretical explanation behind it. We invited several professors working in the field to give seminars to us, and asked questions for this, but never heard back a good answer. A physicist colleague found an IEEE paper giving the non-linear differential equations behind it, which worked, but yet provided no insights into the device behavior, and took time to solve numerically. I wanted a good enough analytical solution. I kept on trying whenever I had time-opportunity, while the physicist colleague kept on telling me to give up. Six months later. I woke up with a solution in mind, and rushed to the office at 7 am to discuss with whosoever was there at work at that time. The optics guy I found did not fully understand it, but did not find it crazy either. A few hours later, the physicist friend confirmed my insight by running some more numerical solutions. I could then soon find tight enough upper and lower bounds, and the whole thing fit the measurements so well that most people thought it was just a "curve fit". (It was pure theory vs. measurements plotted together.)
5. I proposed making pixel-level optical measurements on mirasol displays using a high-resolution camera to watch those pixels after subjecting them to complex drive waveforms. Two interns were separately given the task (surprisingly without telling me), and both failed to develop algorithms for pixel-level measurements. Later a junior employee worked on it, was unable to develop pixel-level measurements still, though was able to get it to work at lower resolutions. The system took about 40 minutes of offline processing in Matlab. Later, a high-profile problem came up where pixel-level measurements were a must, and I was directly responsible for solving. Solved in one day. Processed images taken in real-time, not 40 minutes. The system stayed in deployment for years to come.
6. We had bistable MEMS devices, and there was a desire to make tri-stable MEMS devices. Several people at the company attempted it, including a respected Principal Engineer, but no one could figure how to even start. I could not figure either at the outset, but started bottoms up from Physics and using Wolfram Mathematica to create visualizations around the thing. And bingo. In a few days, I had not only figured how to make these tri-stable MEMS devices, but also multiple schemes of driving them. My VP's reaction was "Alok, you should patent that diagram itself", given the clarity it had brought on the table.
7. We were creating grayscale/color images using half-toning. A famous algorithm, Floyd Steinberg, works very well for still images but has lots of artifacts for videos. An PhD student working in the field was brought in as an intern, nevertheless, the results were not great. The team also tried binary search algorithms to find the best outputs iteratively, however, it was not implementable in real-time as needed. I was interested in the problem, but was not getting time to give it a fresh thought that it needed, until one day. A few days later, the problem was solved. I developed some insights into it and just had the solution coded, to the surprise of people who had spent months working on it.
I could go on writing about more cases.