[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

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[R] Reasoning Tricks - 2 [Cubes/Dices]

In this article I will take up CUBES/DICES. They may pose a problem to you if you are not good at imagining things in 3D. So let's begin.
Firstly you should know that there are 2 types of dice - Standard and Ordinary.
In a standard dice, the sum of opposite faces is 7 [Remember that], so if they ask which face is opposite to 2, you can say it is 7-2 = 5. But mostly the questions are from ordinary dice, where the sum is not 7.

There are various rules to solve the problems on dice -

1. If two sides of Dices are common
If two sides of cubes are common( has same numbers or symbols), then the remaining two will be opposite to each other.

2. If one side of Dices is common
If one side of given dices is common then list these sides (numbers on them) either in clock-wise or anti-clockwise. Comparing the numbers obtained from both dices will give you the opposite numbers.

3. If one side is common and it's place is same in both dices.

If one side is common in both cubes and its place is also same, then the remaining two sides of respective dices will be the opposite to each other.

Now let's move to a typical CGL question - 

In this question the face opposite to 2 is asked, so just look at each cube and list the neighbours of 2
From 1st cube :
Neighbours of 2 = 3,4                  ...(1)
From 2nd cube :
Neighbours of 2 = 1,3                  ...(2)
In the third cube '2' is not present, so we will skip this cube
From 4th cube :
Neighbours of 2 = 4,6                   ...(3)
Now merge all the three results and prepare a consolidated list of the neighbours
Neighbours of 2 = 1, 3, 4, 6
Note the number which is missing from the list. Here it is 5
So 5 is opposite to 2.

Answer : 5

Note : For each face, there are 4 neighbours and 1 opposite face. So make sure when you make the final consolidated list for a face, it has 4 neighbours [like our list contained 4 neighbours]

Next question

From the above figure, neighbours of 6 = 1,3,5
Note that there are only 3 neighbours in the list and hence we can't be sure of the face that is opposite to 6. Either it can be 2 or 4 (both are missing from the list)
When you encounter such situations, pick the face which is featured in most of the cubes. We can see that 3 is present in three of the cubes, so let's pick it.
Neighbours of 3 = 1, 4, 5, 6
From above data we can deduce that the face opposite to 3 is 2. Now since 2 is opposite to 3, it cant be opposite to 6. Hence the face opposite to 6 is 4
Answer : 4

Unfolded Cubes

Remember the following things when you get an unfolded cube

In short, no matter which figure of an unfolded cube is given in the question, the alternate faces are opposite to each other.

Now a SSC question - 

From the unfolded cube it is clear that-

Circle is opposite to Triangle
Square is opposite to Heart
Diamond(♢) is opposite to Clubs(♧)

(A) In option A we can see that triangle is shown adjacent to circle, but we know that they are opposite. Hence option A is wrong

(C) In option C, square is adjacent to heart, but they are opposite. Hence C is wrong
(D) In option D, diamond is adjacent to clubs, but they are opposite. Hence D is wrong.
Answer : (B)

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[R] Reasoning Tricks - 1

The moment you receive your question paper, just write A=1, B=2, C=3...Z=26 at the back of it. This is better than memorizing the values because exam pressure is a bitch :) you may end up recalling wrong values. Moreover, it is easier to decipher the pattern if you have written down the alphabets on a paper.

Since the reasoning questions are pre-defined, I will try to give tricks for few of them.

1. 2-3 questions in reasoning section are vocabulary based. Many students leave these questions because they are confusing/difficult. But these questions are the easiest.

            (1)

 (2)

These questions may pose difficult if you don't know the meaning of the words giddiness, expunge, etc. But don't worry SSC itself solves these questions for you. How? Have a look at the hindi translation of these questions(which is usually given on the adjacent page)...

    (1)

(2)

Looking at this, you can easily decipher that the answers for the above questions are (A) and (C)

Always refer the hindi translation when ever you face any difficulty in solving vocabulary based questions of Reasoning section.

2. Then there are questions, where you have to arrange some words alphabetically. Don't commit the mistake of forming the entire sequence. Learn to solve such questions with options.

Like in the above question it is clear that the first 2 words will be Recollect and Remember. So the sequence will start with 1, 2, ...Stop here only and look at the options. Only option D starts with 1, 2...Don't go on arranging the rest of the words because it is not required and will only waste your time. Similarly for the below question

Smallest organism among these is the mosquito. Hence the sequence will start with 3. Option C is the only option starting with C. 

3. Now let us see a very famous question of CGL - Paper cutting. There are two rules to solve such questions - 

a) Note the position at which the cut is made(most of the questions can be solved with this rule only)

b) Count the layers

This will make sense if we take any example - 

To solve this question, just look at the position at which the cut is made (let's call it position X)

Now match the pattern of X with all the 4 options and see which option has exactly the same pattern.

Answer : C

Again you can see that option A matches the pattern

Answer : A

Both the questions above were solved with rule (a). Now let's see some questions where rule (b) is helpful - 

First of all apply rule (a)

Rule (a) only eliminates Option (2).

So here we will apply rule (b), i.e. counting the number of layers on which the cut is made.

When you are holding a piece of paper, you are actually holding 1 layer. When you fold it once, you will get 2 layers. Making a third fold will result in 4 layers and fourth fold will result in 8 layers.

In this question, the paper is folded twice and hence total no. of layers formed are 4.

There are total 4 layers and two holes are made on them.

So total holes in the final figure  = No of layers * No of holes made = 4*2 = 8

This means in the answer, there will be 8 holes in total

Options (1) and (2) have 8 holes, but option (2) was eliminated with rule (a). Hence 

Answer : (1)

Bonus Trick :

In SSC CGL, there is a very common question of nested square roots and there are total 3 cases for it (addition, subtraction and multiplication)

Case 1

In such questions, you should avoid forming quadratic equations because that will be time consuming. Here is the trick : Break the number written inside the square root into n*(n + 1) form. So 12 = 3*4

Answer is (n+1), i.e., 4

Similarly, for 72 (written as 8*9), the the answer will be 9; for 20(written as 4*5) the answer will be 5, and so on. In CGL questions, the number will always break into n*(n + 1) form.

Case 2

If instead of '+' you have '-' (above image), then

72 = 8*9

The answer is n, instead of n+1. So the above expression evaluates to 8.

If instead of + you have * (multiplication), then the answer is the number itself. E.g. - 

Case 3

Simple, isn't it? :)

I will cover more Reasoning topics in Part-2.

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