Brute Force vs Clever Methods

[jamonwindeyer]

This is something that popped up in the 3U Question/Answer Thread, links nicely to a discussion I had with someone at uni before my last exam.

Rui presents here what I'd call a "clever" method. This method is definitely more difficult to spot, requires a cunning eye, but would you say that it makes it easier?

Basically, I'm interested to hear some thoughts on this idea: Where is the turning point between brute force and clever workarounds?

I'd say I lean slightly towards "brute force" methods, within reason. In my opinion, the time you spend looking for shortcuts is, in some cases, better spent just executing the typical calculation.

This doesn't include of course, just immediately knowing the shortcut (as I'm sure Rui just immediately did because he is God), but actually investing effort in going, "Hmm, is there an easier way to do this?" I suppose it is really a matter of payoffs. Where do you guys think the turning point lies, how many lines of working is worth the price of expending the processing power to find a shortcut?

I actually love this debate, keen to hear what everyone thinks 

PS - Sorry for using your work as an example Rui 
PPS - Yes I debate about Mathematical Reasoning for leisure, I'm a nerd 

[RuiAce]

Wow really Jamon


Observation of these shortcuts comes with practice and intuition. You don't necessarily see them the first time. It's exposure to the tricks that help you know what to do.

[jakesilove]

Wow really Jamon


Observation of these shortcuts comes with practice and intuition. You don't necessarily see them the first time. It's exposure to the tricks that help you know what to do.

I found that I could pick up these tricks by doing billions of past papers, and looking at the solutions (not just the answer, but the way that they got there). Often, there will be nifty little tricks that you can pick up in the answers, and apply them in other, similar questions. Just smash out a billion questions and you'll realise how helpful repetition is!

[RuiAce]

I found that I could pick up these tricks by doing billions of past papers, and looking at the solutions (not just the answer, but the way that they got there). Often, there will be nifty little tricks that you can pick up in the answers, and apply them in other, similar questions. Just smash out a billion questions and you'll realise how helpful repetition is!

Tis why solutions exist and why maths gets a massive emphasis on past papers compared to every other subject

[jamonwindeyer]

Tis why solutions exist and why maths gets a massive emphasis on past papers compared to every other subject

Absolutely! Do you guys find then, with uni, that you go less with the "look for the most efficient method" approach and instead for the "get the answer" approach? Purely because the content is "newer?"

I acknowledge that at its core this conversation is really at its core pointless banter.

[RuiAce]

Absolutely! Do you guys find then, with uni, that you go less with the "look for the most efficient method" approach and instead for the "get the answer" approach? Purely because the content is "newer?"

I acknowledge that at its core this conversation is really at its core pointless banter.

Haha that bold text.

I find shortcuts at uni as I...
a) Ask for help
b) Do my homework
c) Read past paper solutions
d) Realise that shortcuts I learnt in high school can be re-used/improved

As for just plain faster methods instead of the brute (e.g. t=tan(x/2) for a nasty integral) that just requires seeing ahead. (Or trying out something new but pref. do that out of the exam, not during)

[wyzard]

Relying just on brute force method is not good idea, as mathematics will then become pretty mechanical, and tedious.

This is not what mathematics is really about. Mathematics is filled with plenty of clever tricks to work out really difficult problems, (Some of my favorites are Feynman's integral trick, using Taylor series to quickly approximate a complicated function, and Residue theorem from Complex Analysis) and understanding how these clever trick are just really smart exploitation of the same fundamental ideas.

I'd say it's a lot better to explore around possibility of working around problems, and seeing if there's any possible tricks to get around them. Even if you manage to get the answer by any means, it's good practice to ask yourself "Is there any other way of arriving here?" This really trains you to think outside the box, to look for alternatives instead of just sticking to conventional methods. This will usually prompt on a discovery quest, and you'll be surprised as there are almost always alternatives.

Of course another great way is to look at other people's clever method and include that in your own bag of tricks. This is why mathematics is so much fun, by understanding the fundamental concept and then having your own bag of tricks which you can employ to solve a myriad different problems, or who knows, maybe you'll invent a new method