Many times I was awed by some talented people solving convoluted problems in a clever cunning way, wondering how do they do it. Then, I realize if grasp the right methods and tools, I can solve lots of puzzles too.
I summarize the methods into the following kinds.
- Break down to simple
- Representation vividly by writing down or drawing a diagram
- Representation by making or modeling things
- Reversely deducing
- Self contradict
- Break down to simple – this technique is hugely used in solving all sorts of problems. For example, to calculate the turnover of a large portfolio, I start from a 2-stock portfolio and manually plug in stock price at each time point to get the sense of all intricacies might be embedded. Once 2-stock is straight clear, 3-stock, 10-stock, 1000-stock portfolio computation all follow the same rule.
- Representation vividly by writing down or drawing a diagram – It’s futile to keep thinking and thinking without a slight effort to use aids. Drawing to writing down steps or painting/drawing is so powerful whereas people tend to ignore simply due to laziness. I found it greatly helpful to unclutter my mind by writing blogs or keep diaries. It is consistent with another statement (I forgot which famous person said so) that ‘language forms the way you think’. It’s damn right. So not only remember to write down our thoughts, in the process of writing down, we will recognize how inadequate we are in term of thinking ability, and it comes hand in hand with our inability to summon the right words or phrases to just express.
- Representation by making or modeling things – it takes one step further from just writing down summaries or steps of approaching a problem, rather, this means we should just roll up the sleeves, do things, build models to figure out how to do it. I got great inspiration from watching Patrick Winston teaching Artificial Intelligence at MIT. On the very first class, he emphasized the importance of “representation” by bringing a wheel to the classroom. He asked the students which way the wheel will go if blown on a part of the wheel. He teased so-called “right-hand screw rule” that chemical engineers tend to solve this problem, instead, he took out a duck tape and stick onto the wheel and then deduce, then check by flipping it rolling. The video is here, pay attention at time 5:10. I also think the approach that Einstein tackles the greatest myth of time by imagining he is riding on the beam of light, or dwelling on an elevator going up or down is an adoption of representation method.
- Reversely deducing – lots of complex math problems are solved in this way. For example, the lottery sequence problem, if the order of draw lottery affects the outcome or not? In this video, Yongle Li guided through the mathematical way to deduce the answer. If the rule of game changes as is illustrated in that second part of that video, the sequence of joining the game does matter. However, it would be diabolically difficult to jump to figure out a scenario of 5-people game, so, we start from two-people, then three-people, note, the three-people scenario calls forth 2-people variable to be plug-in. Note he took a very clever way to move A forward/after B, and hence assume a probability of P22 as if A becomes B.
- Inferring – this technique is also vastly used in solving all sorts of problems. For example, we are asked to solve chicken rabbits problems since we are children. The question is that there are 15 heads and 40 legs in total, now you figure out how many are chicken, and how many are rabbits, given everybody knows a chicken has two legs, one head, a rabbit four legs, one head. We don’t have to lay out an algebra equation to calculate x, y value, instead, infer from heads/legs: if 15 heads, supposedly there are 15X4 = 60 legs if all rabbits, but in reality, only 40 legs are there, so additional 20 legs are from surplus legs of rabbits. for every one rabbit-chicken pair, there is two legs surplus, hence, a delta of 20 can be translated to 20/2 = 10 additional chicken, hence, the number of rabbits is 5. Problem is solved. The tricky part is to translate the delta as the additional number of chicken. Similarly, we can infer from the reverse direction:
supposedly there are 15X2 = 30 legs if all chicken, now there are 40-30 =10 additional legs, these 10 legs have to be from rabbits, so the number of rabbits has to be 10/2 = 5. Another way to think reversely is to imagine the 40 legs subtracted by 15 heads twice, all chickens are gone, the rest (40-15-15 = 10) has to be 2-leg rabbits, hence the number of rabbits is 5. Assigning/interpreting a meaning for every math step is crucial to solving the problem