I think bentarm was offering "Combinatorics problems" as an example of the *opposite* of the phenomenon you describe. In particular the Four Colour Theorem is easy to formulate but hard to solve, and (as far as I know) the solution doesn't involve a reformulation.

Yes, upon re-reading I see that you are correct. I think there may be overlap between activities I consider part of the formulation and activities others may consider part of the solution.

To expand on my poker suggestion. When attempting to determine the probability of a hand in poker it is necessary to determine a way to represent that hand using combinations/permutations. I have found that for certain hands this can be rather difficult as you often miss, exclude, or double count some amount of possible hands. This process of representing the hand using mathematics is, in my mind, part of the formulation of the problem; or more accurately, part of the precise formulation of the problem. In this respect, the solution is reduced to trivial calculations once the problem is properly formulated. However, I can certainly see how one might consider this to be part of the solution rather than the formulation.

Thanks for pointing that out

Thanks FiftyTwo- I just looked up the article you refer to and it indicates that it may be a paraphrase of a longer quote. I heard this from Anthony Robbins, this quote is attributed to Einstein in some of his literature. It seems that the sentiment, if not the exact quote, seem to be attributable to Einstein

This is definitely true. General class of examples: almost any combinatorial problem ever. Concrete example: the Four Colour Theorem

General class of examples: almost any combinatorial problem ever

Yes! Combinatorics problems are a perfect example of this. Trying to work out the probability of being dealt a particular hand in poker can be very difficult (for certain hands) until you correctly formulate the question- at which point the calculations are trivial : )

The significant problems we face cannot be solved at the same level of thinking we were at when we created them.

-- Albert Einstein

But however you do it, focusing isn't as simple as saying no - or even as saying no to the right things. You'll exclude some things by default but knowing when to say 'let's see' and how strongly to say yes is also very useful.

Yes, agreed.

This reminds me of Steven Covey's idea of a coordinate graph with four quadrants where you graph importance on on axis and urgency on the other. This gives you for types of "activities" to invest your time into. 1. Urgent and Unimportant (a phone ringing is a good example): this is where many people loose a tremendous amount of time

Urgent and Important (A broken bone or crime in progress) hese immediately demand our "focus"

Not Urgent and Not Important: pure time wasters- not a good place to invest much energy

Not Urgent BUT Important. This is the area Steven made a point of saying that most people fall short. Because these things are not urgent, we tend to put them off and not invest enough enough energy into them, but since they are important this means we pay a hefty price in the long run. Into this category he puts things like our health, important relationships, personal development and self improvement to name a few.

When we choose what to focus our energy on, we would do well to direct as much of it as possible to these types of activites

At least sometimes the formulation is far easier than the solution.

In my experience it can often turn out that the formulation is more difficult than the solution (particularly for an interesting/novel problem). Many times I have found that it takes a good deal of effort to accurately define the problem and clearly identify the parameters, but once that has been accomplished the solution turns out to be comparatively simple.

(nods)

It continues to embarrass me that ultimately I was only "convinced" that the calculated answer really was right, and not some kind of plausible-sounding sleight-of-hand, when I confirmed that it was commonly believed by the right people.

One of my favorites for exactly that reason- if you don't mind, let me take a stab at convincing you absent "the right people agreeing."

The trick is that once Monty removes one door from the contest you are left with a binary decision. Now to understand why the probability differs from our "gut" feeling of 50/50 you must notice that switching amounts to winning IF your original choice was wrong, and loosing IF your original choice was correct (of course staying with your original choice results in winning if you were right and loosing if you were wrong).

So, consider the probability that you original guess was correct. Clearly this is 1/3. That means the probability of your original choice being incorrect is 2/3. And there's the rub. If you will initially guess the wrong door 2/3 of the time, then that means that when you are faced with the option to switch doors you're original choice will be wrong 2/3 of the time, and switching would result in you switching to the correct door. Only 1/3 of the time will your original choice be correct, making switching a loosing strategy.

It becomes more clear if you begin with 10 doors. In this modified Monty Hall problem, you pick a door, then Monty opens 8 doors, leaving only your original choice and on other (one of which contains the prize money). In this case your original choice will be incorrect 9/10 times, which means when faced with the option to switch, switching will result in a win 9/10 times, as opposed to staying with your original choice, which will result in a win only 1/9 times.

This was an excellent read- I particularly enjoyed the comparison drawn between our intuition and other potentially "black box" operations such as statistical analysis. As a mathematics teacher (and recreational mathematician) I am constantly faced with, and amused by, the various ways in which my intuition can fail me when faced with a particular problem.

A wonderful example of the general failure of intuition can be seen in the classic "Monty Hall Problem." In the old TV game show Monty Hall would offer the contestant their choice of one of three doors. One door would have a large amount of cash, the other two a non-prize such as a goat. Here's where it got interesting. After the contestant makes their choice, Monty opens one of the "loosing" doors, leaving only two closed (one of which contains the prize), then offers the contestant he opportunity to switch from their original door to the other remaining door.

The question is, should they switch? Does it even matter? For most people (myself included) our intuition tells us it doesn't matter. There are two doors, so there's a 50/50 chance of winning whether you switch or not. However a quick analysis of the probabilities involved shows us that they are in fact TWICE as likely to win the prize if they switch than if they stay with their original choice.

That's a big difference- and a very counterintuitive result when first encountered (at least in my opinion)

Hello LW community. I'm a HS math teacher most interested in Geometry and Number Theory. I have long been attracted to mathematics and philosophy because they both embody the search for truth that has driven me all my life. I believe reason and logic are profoundly important both as useful tools in this search, and for their apparently unique development within our species.

Humans aren't particularly fast, or strong, or resistant to damage as compared with many other creatures on the planet, but we seem to be the only ones with a reasonably well developed faculty for reasoning and questioning. This leads me to believe that developing these skills is a clear imperative for all human beings, and I have worked hard all my life to use rational thinking, discourse and debate to better understand the world around me and the decisions that I make every day.

This is what drove me towards teaching as a career, as I see my profession as providing me with the opportunity to help young people better understand the importance of reason and logic, as well as help them to develop their ability to utilise them.

I'm excited to finally become a member of this community which seems to share in many of the values I hold dear, and look forward to many intriguing and thought provoking discussions here on LW!

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