1. Logical Problems
In this article, we will look into logical problems, how to solve them, and a few applications of them. There is no one specific formula to solve logical problems. You have to use different formulas and logic to solve these questions.
1.1 Magic Square
In a magic square, numbers are arranged in such as way that the sum of each row, column, and diagonal is one constant number.
Example:
Here the sum of numbers in each row, column, and diagonal is 15.
Steps to solve magic square questions:
- Try to find one missing number in a diagonal, row or column by subtracting the sum of two numbers from one constant magic number given.
- Using these numbers, the rest of the missing numbers can be found.
Example: The following magic square has all rows, columns and diagonals adding upto 18. Several numbers have been missed out.
Which number replaces A?
Solution:
Sum of all rows = 18
Find the missing number in the last row containing 10 and 3.
Missing number = 18 - (10 + 3)
= 5
Sum of all diagonal numbers = 18
Find the missing number A in diagonal elements 7, A and 5
A = 18 - (7 + 5)
A = 6
1.2 Alphabet codes
There are different logical questions involving alphabet codes. In alphabet codes, different letters are represented by different numbers. Find out the value of words by either adding or multiplying numbers as mentioned in the question.
Example: The letters of the alphabet are assigned number values: A = 2, B = 4, C = 6 and so on. The value of a word is found by adding the value of the letters. What is the value of the word ‘MATH’?
Solution: A is the first alphabet, its value is 2 (2 × 1).
B is the second alphabet, its value is 4 (2 × 2).
C is the third alphabet, its value is 6 (2 × 3).
M is the 13th alphabet, its value is 2 × 13 = 26
T is the 20th alphabet, its value is 2 × 20 = 40
H is the 8th alphabet, its value is 2 × 8 = 16
Value of word MATH = 26 + 2 + 40 + 16 = 94
1.3 Palindromic numbers
A number that can be read the same way forwards and backwards is called a palindrome.
Examples:
- All single digit numbers are palindromes.
- Two digit numbers - 11, 22, 33, 44, 55, 66, 77, 88, 99
- Three digit numbers - 101, 202, 252, 787, etc
- Four digit numbers - 1111, 1221, 1771. etc
- Five digit numbers - 11111, 12121, 14741, etc
Example: How many palindromes are there between 100 and 200?
Solution: Palindromes between 100 and 200 are:
101, 111, 121, 131, 141, 151, 161, 171, 181, 191
Total number of palindromes = 10
There are 10 palindromes between 100 and 200.
2. Quadratic Equations
Equations containing terms up to x² are called quadratic equations. Let us look at a couple of examples to understand how to solve such equations.
Example:
a) a² = 49
Solution:
a × a=49
Find a number which multiplied by itself gives 49.
7 × 7 = 49 or -7 × -7 = 49
Hence, x=+7 or x=-7
b) x² + x = 90
x(x+1)=90
Find the factors of 90 which have a difference of 1.
9 × 10 = 90 or -9 × -10 = 90
Comparing the factors with x(x+1)
We get, x=9 or x=-10
