Saturday, 26 March 2016

[R] Reasoning Tricks - 3 [Counting squares, rectangles, triangles]

Many of you wanted shortcuts to count the number of triangles. So I am writing this post which will cover every possible scenario for counting squares, rectangles and triangles. If the question doesn't fall in any of the below category, you will have to count manually :)

1) Counting squares within a square

How many squares are there in the above figure?
Solution : There are 4 rows in the above figure. Hence n=4
Apply the formula [n(n+1)(2n+1)/6]
= 4*5*9/6
Answer : 30

2) Counting squares within a rectangle



How many squares are there in the above figure?
Solution : This rectangle is a 4 x 3 grid [4 rows and 3 columns]
So total number of squares is m(m+1)(2m+1)/6 + (n-m)*m(m+1)/2.where n is the larger dimension
Here n = 4, m = 3
Answer : 20

3) Counting rectangles within a rectangle



How many rectangles are there in the above figure?
Solution : For a 'n x m' grid ['n' is the number of rows and 'm' is the number of columns]
Number of rectangles = (m)(m+1)/2 (n)(n+1)/2 = m(m+1)(n)(n+1)/4
In the above figure
n = 4, m=6
Put the values
Answer : 210

4) Counting rectangles in a square
The formula to count the number of rectangles within a square can be obainted by putting n = m in the above formula (3rd case)
So, No. of rectangles = [n(n+1)/2]^2

Ques. How many rectangles are there in a chess board?

Solution : We know that the chess board is a 8 x 8 grid
Put n = 8 in the above formula
Answer : 1296

Note : From the above formulae, you only have to mug two, i.e., 2 and 3. Formula 1 can be obtained by putting n=m in formula 2 and formula 4 can be obtained by putting n=m in formula 3.

5) Counting triangles within a triangle



How many triangles are there in the above figure?

The formula is given by - 

You should note 3 things about this formula-
1. It is only applicable for matchstick arrangement (like the figure above)
2. n = number of unit triangles in a side. In this question n=4 (I have counted the triangles for you)
3. Those are not brackets enclosing the formula; they denote floor function. So after applying the formula, if you are not getting an integer, you should completely reject the decimal numbers. E.g. 17.12 will become 17 and 56.98 wil become 56
Caution : You don't have to round off the number, so 77.9 will be equal to 77 not 78.

Put n=4 in the formula
Answer : 27

Next question-

Here n=3
Apply the formula and you will get 13.125. Reject the decimal places-
Answer : 13

Keep Reading :

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25 comments:

  1. couldn't undetsyand d logic behind these formulae
    coz i lv to find out d logic behind evry formula, which makes it easy for me mug them up

    anyways thanx for d info
    waiting for ur next post
    and please if it possible for u to speed up a li'l bit
    that will b highly appreciated

    Thanx n regards

    ReplyDelete
  2. couldn't undetsyand d logic behind these formulae
    coz i lv to find out d logic behind evry formula, which makes it easy for me mug them up

    anyways thanx for d info
    waiting for ur next post
    and please if it possible for u to speed up a li'l bit
    that will b highly appreciated

    Thanx n regards

    ReplyDelete
  3. if a=9 b=8 and c=-17 then find the value of (a^4+b^4+c^4)/(a^2*b^2+b^2*c^2+c^2*a^2)=???

    ReplyDelete
  4. A B C can do a piece of work in 6 hours working together. If A and C working together works four times of B's work and A working alone works 5 times of C's work then only A will finish the work in??

    ReplyDelete
    Replies
    1. ur answer is right any short cut of this u knw then post the solution

      Delete
    2. 1/a + 1/b + 1/c = 1/6 ...(1)
      1/a + 1/c = 4/b ...(2)
      1/a = 5/c or c=5a ...(3)
      Put (2) in (1)
      5/b = 1/6
      b = 30 days ...(4)
      Put (3) and (4) in (2)
      1/a + 1/5a = 4/30
      6/5a = 2/15
      or a = 9 days

      Delete
    3. Plz tell us how to prepare Idiom and phrases..? Can you provide us list of important Idioms and phrases

      Delete
  5. plz provide solution of this

    ReplyDelete
  6. Pls provide solution of hours question

    ReplyDelete
  7. plz provide solution of a b c question

    ReplyDelete
  8. Non of the sites explained such topics so easily.ur the first. Thanks a lot sir.

    ReplyDelete
  9. Please need method of solving syllogism .

    ReplyDelete
  10. Pls post How to aprroach syllogism..

    ReplyDelete
  11. Thnku so much sir ....... M Msc maths but still I was not able to solve the sums in less time..
    . u made it easy.... Thnku sir...

    ReplyDelete
  12. how many zeroes in the following expression
    1. 40!^40!= ?
    2. 2!^2!+4!^4!+6!^6!+8!^8!+10!^10!= ?

    ReplyDelete
  13. thanks a lot for the above formula

    ReplyDelete
  14. I think formula for model 3 is wrong

    ReplyDelete
  15. here is the video for counting of figures: https://youtu.be/4RM8Fh5Jx0M

    ReplyDelete
  16. We can't remember every formula.. You should provide some better ways to solve those (not formulae)

    ReplyDelete
  17. How would you count rectangles in the chess board if the centre 4x4 grid were cut out of the middle of it?

    ReplyDelete
  18. Thank you for proving good information to count triangles easily.

    Sugar Technology

    ReplyDelete
  19. Plz sir how to answer properly syllogism...my what's app no 7894888329
    Plz plz sir

    ReplyDelete