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Skeleton in the Problem Solving Closet
by Quant Ghost - Wednesday, 7 March 2007, 01:11 AM
 

Quant Ghost

How many of you often wonder why you can’t ‘see through’ a problem? Why is it that you know all the formulas and have gone through a lot of problems associated with a particular chapter, and yet you are unable to solve a new problem when it comes across to you? What is it that those math geniuses possess that you don’t?

There are two differences between a mathematical mind and an ordinary mind:

· A mathematical mind keeps its armory of formulas and theorems very handy and in an assorted manner, ready to be recalled at an instant notice.

· A mathematical mind identifies the uniqueness to a problem and subconsciously searches through its repository of formulas to find one that fits to that uniqueness.

Which brings me to the most essential part of problem solving-

Problem Solving principle

Identifying that uniqueness can often help us solving the problem. Here is a specimen:

alphametic problem 1

Where will you start?

What is unique about this problem?

Would this problem make sense if it was something like ?

Yes. As the unique part of this problem is the number DDD, we start with this number itself.

· The number DDD = D × 111 = D × 3 × 37 = 3D × 37.

· 37 is a two digit number and since it cannot be reduced further it can be one of the numbers. Let it be AB.

· Then CB is a number ending in 7 because both AB and CB have the same unit digit.

· CB = 3D, i.e. CB is a multiple of 3.

number CB

Now let’s check our solution- Multilying 37 and 27, we get 37 Ã— 27 = 999. Therefore, our logic is correct and A = 3, B = 7, C = 2, and D = 9. And A + B + C + D = 21.

Broadly speaking, here are the steps that you should try in solving a problem, regardless of the topic that problem came from:

 problem solving steps

Here is another simple problem:

cryptarithm problem 2

Where will you start?

What is unique about this problem?

There are two unique points about this problem- first that the digits of the number are getting reversed and second that the number is being multiplied by 4. Notice that ABCD Ã— 4 = DCBA is different from ABCD Ã— 4 = EFGH or ABCD Ã— 7 = DCBA.

selecting uniqueness

Right now you can handle that the number is multiplied by 4 because you know something about properties associated with number 4. Let’s start with that.

· Any number multiplied by 4 will give us an even number. Hence, the digit D when multiplied by 4 will give us an even number. Since A is the unit digit of the product it is even. Hence, A = 2, 4, 6 or 8 (It cannot be 0. Why?)

· A is also the first digit of the multiplicand and if A = 4, 6 or 8 the product will become a 5 digit number. Hence A = 2. Writing the value of A we get

                               cryptarithm step 1

· What can be the value of D? looking at the first and last digits of the multiplicand, we can see that 4 Ã— D gives the unit digit of 2 and 4 Ã— 2 gives the first digit of D. Yes, you got it right. D = 8. Writing the multiplication again with the value of D we get:

                             alphametic step 3

· What can be the value of B? From your repository of formulas associated with 4 recall the one about divisibility of 4. A number is divisible by 4 if the number formed by the last two digits is divisible by 4. Since the number 8CB2 is a multiple of 4, the number B2 should be divisible by 4. Or, the number B2 = 12, 32, 52, 72 or 92. Hence the original number ABCD is 21C8, 23C8, 25C8, 27C8 or 29C8. But the last 4 numbers when multiplied by 4 will not give you the first digit of 8 in the product! Therefore B = 1 and the original number is 21C8. We write the multiplication again:

                                step 4

· Can you identify C now? Notice that when you multiply 8, the unit digit of 21C8, by 4 you write 2 in the unit digit of the product and carry 3. The tenths digit of the product is 1. Therefore, 4 Ã— C + 3 (carryover) gives a unit digit of 1. Hence, C is 2 or 7. You can easily check by the hundreds digit in the product (which is C again) that C = 7.

Therefore, our answer is final answer

Notice how we grabbed a uniqueness of the problem, started unraveling it, and slowly discovered the complete solution to the problem.

The beauty of the above two problems is that they require elementary knowledge of numbers but they will confuse many of you because you do not know where to start looking. Now you have your answer:

                    problem solving principle

 

Quant Ghost

Re: Skeleton in the Problem Solving Closet
by Anupam Agarwal - Sunday, 11 March 2007, 05:55 PM
  never thought like this man...after reading ur article atleast i cn say dat next time i wont get disheartened wen it comes to solving the questions...thnx total.. smile
Re: Skeleton in the Problem Solving Closet
by saurabh kaushik - Monday, 30 April 2007, 09:36 PM
 

hi

ghost sir!!

u hv done a  really good job.. but i think a job is better if u continue to do that..

so i request u to please continue writing these sort of articles.. we are desparately waiting for that..

so plz do write those more  articles..

Re: Skeleton in the Problem Solving Closet
by Diya Biswas - Thursday, 3 May 2007, 01:37 PM
 

This really made me look at problems from a different perspective..

Thanks!!

 Long Live Quant Ghost!!

Re: Skeleton in the Problem Solving Closet
by ankit tulsyan - Sunday, 6 May 2007, 02:44 PM
  @ mr ghost....
we're ready  for your next(4th??) piece.
ready n waiting.
Re: Skeleton in the Problem Solving Closet
by mohammed javed - Thursday, 17 May 2007, 03:41 PM
  Commendable job......Got a Skeptre feel after reading this article!
Re: Skeleton in the Problem Solving Closet
by priyank daga - Wednesday, 23 May 2007, 09:04 AM
  it would be great if you post more articles like this one
Re: Skeleton in the Problem Solving Closet
by Brijesh Mishra - Tuesday, 29 May 2007, 07:44 PM
  This is the awesome approach man keep it up
Re: Skeleton in the Problem Solving Closet
by ramesh akula - Wednesday, 30 May 2007, 11:11 AM
 

we need some more awesome articles...

smile

Re: Skeleton in the Problem Solving Closet
by mayank gera - Monday, 4 June 2007, 09:20 PM
 

You do make quant seem a lot more simpler. Thanks for giving the confidence.

 

Re: Skeleton in the Problem Solving Closet
by rahul g - Thursday, 14 June 2007, 12:24 AM
  Great Job!
Skeleton in the Problem Solving Closet
by malvika agarwal - Thursday, 28 June 2007, 01:04 AM
 

hello ghost

it is good but dont u think that can we spend this much time in paper to solve such a question.i hope solving using options given will b more easier

the article is really very nice.

Re: Skeleton in the Problem Solving Closet
by Anubhav Jain - Tuesday, 3 July 2007, 09:57 PM
  hey now this is an art n throoo ur article we can acquire it..great work!!! thanxa tonn!!!
Re: Skeleton in the Problem Solving Closet
by pranesh sharma - Tuesday, 10 July 2007, 12:48 PM
  Time to throw the Quant Ghost of my back has come and the first step taken while reading ur article,.,,,, great job.....
Re: Skeleton in the Problem Solving Closet
by reva g - Thursday, 26 July 2007, 03:38 PM
 

@TG-- really smart work

 

Re: Skeleton in the Problem Solving Closet
by dheeraj sunkavalli - Wednesday, 31 October 2007, 11:44 PM
  Simply superb!!!!!!!! No words to describesmile
Re: Skeleton in the Problem Solving Closet
by vivek kumar - Sunday, 12 October 2008, 01:58 AM
 

awesome article ... it comes as close to finding a pearl in an ocean ....

i think i am going to find a new direction for my preparation with just one month to go for CAT !

Re: Skeleton in the Problem Solving Closet
by Shubham Agarwal - Thursday, 13 August 2009, 08:32 PM
  This article hits the nail on the head i.e. to develop the problem solving skills. Nice !!!
Re: Skeleton in the Problem Solving Closet
by sidharth nayak - Friday, 18 September 2009, 02:57 AM
  wta does the symbol A`  stand for?

am new to tg

                            
Re: Skeleton in the Problem Solving Closet
by Rahul Adwani - Friday, 18 September 2009, 10:57 AM
  Hats Off Man!!
Re: Skeleton in the Problem Solving Closet
by Rajesh Kumar - Wednesday, 30 September 2009, 12:11 AM
 

Hi Buddy ... A'~ stands for multiplication symbol.

Re: Skeleton in the Problem Solving Closet
by Sanjeev Kumar - Wednesday, 30 September 2009, 11:04 AM
  Thanks 4 such a nice article. Can we have some more mind openers.
Re: Skeleton in the Problem Solving Closet
by afroz m - Saturday, 22 May 2010, 08:17 PM
  nice.......thanks a lot
Re: Skeleton in the Problem Solving Closet
by Abhijit Tambe - Tuesday, 6 September 2011, 04:11 AM
  NIce....Approach...but there cannot be Uniqueness in approaching Numbers....."QUANT GHOST" THERE ARE OTHER WAYS TO LOOK AT THINGS...