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Ohana The CATLounge Master the art of smart problem-solving This page is dedicated to mastering the art of problem-solving skills through puzzles and simple mathematical problems.

The contents of the page might use a lot of material targetted to, but no limited to CAT, GRE, GMAT and other tests that require aptitude and puzzles.

https://www.facebook.com/groups/188859575503667/learning_content/?filter=1092685977790696The Art of Handling Complexity-...
01/04/2020

https://www.facebook.com/groups/188859575503667/learning_content/?filter=1092685977790696

The Art of Handling Complexity
-----------------------------------

There are three aspects of problem-solving :

a) the solution
b) the solver
c) the solution process

We are mostly focussed on the "solution" aspect of a problem.

Given, a problem, we want to know what is the solution.
We look at the solution as a finished product.

Regardless of how many finished products do we look at, it doesnt teach us anything about the process of creating that product.

Follow this series of articles to explore an important component of creating great solutions - the art of handling complexity

https://www.facebook.com/groups/188859575503667/learning_content/?filter=1092685977790696

https://www.facebook.com/groups/188859575503667/learning_content/?filter=493072941357821When we start answering the ques...
14/03/2020

https://www.facebook.com/groups/188859575503667/learning_content/?filter=493072941357821

When we start answering the questions in a problem directly, it is not problem-solving.

For example, if we are asked to count the number of triangles in this figure, and you start counting - it is a random rambling, not problem-solving.

Problem-Solving requires a lot of preparatory steps before you can start answering.

The following series of 10 articles demonstrates the art of "constructing a solution" to a problem, through a discussion around a series of simple, but not easy problems.

https://www.facebook.com/groups/188859575503667/learning_content/?filter=493072941357821

27/02/2020

https://www.facebook.com/groups/188859575503667/learning_content/?filter=2879384468750938

A 10-article series on step-by-step thinking towards problem-solving.

Creative ideas do not come by taking artificial jumps and wandering around. Creative ideas come when we stick as close to the problem as possible, look at where the gaps are, and focus on those.

We miss out on a solution mostly because we are trying to pluck a solution out of the thin air.

We need to grow the solutions bottom up.

No matter how complex the problem is, there are very few instances when you need to take a flight or a jump.

For most of the duration of the problem, we just need to be on the ground, be with the problem, travel with every step and statement of the problem, and we would make it.

https://www.facebook.com/groups/188859575503667/learning_content/?filter=2879384468750938

20/02/2020

https://www.facebook.com/groups/188859575503667/learning_content/?filter=169894114316185

The above article series takes you further into answering the question :

"
What is the secret to repeatable confidence in solving problem?
"

There are times when we solve a lot problems, and yet feel nervous when we go on to solve another one in the same category.

Whether or not a problem-solving experience builds our confidence depends on a one critical factor -

Did I pluck the solution out of thin air ?

Or

Did I build the solution from grounds-up ?

If we plucked the solution from the thin air, chances exist that we may not always be able to catch hold of these floating ideas.

If we build it grounds-up, the chances are high that we will always get a good harvest. The components for building the solution comes from the problem itself.

Check out the article series to read further.

https://www.facebook.com/groups/188859575503667/learning_content/?filter=169894114316185

The Dictatorial Hat, Part 6-----------------------------------So, the last mathematician in the queue initiates the proc...
27/01/2020

The Dictatorial Hat, Part 6
-----------------------------------

So, the last mathematician in the queue initiates the process by announcing whether the total number of hats she sees is even or odd (encoded as "black" or "white").

Henceforth, each mathematician has two inputs :

a) the input received from the previous mathematician, cascaded to that point. e.g.

if the first mathematician signalled an "Even" count of black hats, and by the time my turn comes, 3 other mathematicians have called out "Black", it indicates that the parity right now is Even - 3, i.e. "Odd".

However, if only 2 have called out black till now, the parity would be Even - 2 i.e. "Even".

So, every mathematician gets the parity info cascaded from behind

b) each mathematician has her own count of what she sees ahead.

Based on these two info, each mathematician makes a conclusion about the hat she is wearing.

What's the logic each mathematician uses?

The Dictatorial Hat, Part 5-----------------------------------So, it boils down to these steps :a) The last mathematicia...
27/01/2020

The Dictatorial Hat, Part 5
-----------------------------------

So, it boils down to these steps :

a) The last mathematician checks the count of black hats.
b) She maps the count to a binary value. The most obvious such mapping is the even/odd parity.
c) Map this binary value to the set {black, white}

In simple words,
a) check the total count of black hats
b) if the count is even, say "Black"
c) if the count is odd, say "White"

This is what the last mathematician does.

What do the other mathematicians, up ahead in the queue do ?

The Dictatorial Hat, Part 4-----------------------------------Since M3 can either say 'black' or 'white', he can only pa...
26/01/2020

The Dictatorial Hat, Part 4
-----------------------------------

Since M3 can either say 'black' or 'white', he can only pass a 2-valued binary message forward.

So, although he knows the count of a certain color, e.g. black, he cannot say out the count.

He has to convert the count into a binary value, and then say either black or white based on the given value

The Dictatorial Hat Part - 3-----------------------------------When there are just 2 mathematicians, we need to focus on...
26/01/2020

The Dictatorial Hat Part - 3
-----------------------------------

When there are just 2 mathematicians, we need to focus on the following points :

a) It is obvious that the 2nd mathematician has the least information to start with, so he is the most likely to go wrong. Hence, we choose to sacrifice the 2nd mathematician.

b) The 2nd mathematician has to only focus on giving the 1st mathematician some info that he can use to save himself.

c) From the diagram, we see one such obvious option :

if M2 sees M1 as wearing black, he says black, M1 knows he is wearing black and he is saved.

if M2 sees M1 as wearing white, he says white, M1 knows he is wearing white and he is saved.

So, the strategy is :

M2 says what he sees.

A less-risky strategy could be to say the opposite of what he sees, and as long as both understand the strategy, that will work too.

Now, we can see that we cannot generalize the strategy for 2 mathematicians to 100 mathematicians. We need one more intermediate step.

Let us see how we generalize this for 3 mathematicians. That would give the solution.

In the given figure for 3 mathematicians, M3 is the one calling out first, and he is primarily giving his message to M2.

What does M3 say when he sees 0 black hats so that M2 knows what he is wearing ?

What does M3 say when he sees 1 black hats so that M2 knows what he is wearing ?

What does M3 say when he sees 2 black hats so that M2 knows what he is wearing ?

There are only 2 things that M3 could say - Black or White and there are 3 cases here.

What is the message here ?

The Dictatorial Hat Part - 2------------------------------------The problem looks very abstract. What are the factors th...
25/01/2020

The Dictatorial Hat Part - 2
------------------------------------

The problem looks very abstract. What are the factors that make it abstract, fuzzy, and difficult ?

1. There are a 100 mathematicians in the queue. It is too big a number for any reasonable analysis.

2. The number of black / white hats are not known. Hence, there is nothing certain about the number of black / hats that the mathematicians can communicate, even secretly. their secret communication has to be about something else.

What could be one such property that can be communicated ?

It must be obvious that the mathematician putting his life at risk would be the last one in the queue. It is he who starts the process without any input whatsoever.

What are the different ways to deconstruct the problem ?

One obvious, simple way is to look at a smaller data size.

What happens when there are just 2 mathematicians ?

Assuming that the last in the queue sacrifices himself, what strategy will ensure that the front guy lives no matter what ?

Considering the attached diagram,

what should M2 call that M1 knows what is on the top of his head ?

Problem Statement
--------------------------

A dictator who hates science has rounded up all 100 mathematicians in his country. (All women, it turns out). The next day, he says, he will line them up so that each one can see everyone in front of her, but nobody behind. He will then put either a black or a white hat on each mathematician’s head, but nobody will know what colour hat she herself is wearing. Starting at the back of the line—the mathematician who can see 99 hats in front of her—he will ask each mathematician in turn what colour hat she is wearing. If she’s right, she lives. If she’s wrong, she’s instantly and painlessly decapitated.

Prospects don’t look good. But being mathematicians, the 100 confer and find a strategy that, at worst, leaves just one of them headless (and at best, saves them all). Who is the mathematician who might die, and what is the strategy?

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