Competitive Programming: Essential Maths and Bit Manipulation Problems

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If you enjoy Maths and want to start competitive programming and DSA, here is a list of all Maths and Bit Manipulation problems for you.

Pattern 1: Prime Numbers & Sieve Methods

• Prime Subtraction Operation
• Count Primes
• Sum of All Subset XOR Totals
• Four Divisors
• Prime Number Maze

Pattern 2: Combinatorics & Counting

• Unique Binary Search Trees
• Distinct Subsequences II
• Number of Dice Rolls
• Different Ways to Add Parentheses

Pattern 3: Modular Arithmetic

• Count Array Pairs Divisible by K
• Number of Music Playlists
• Sum of Divisors
• Modulo 100

Pattern 4: Number Theory & Divisors

• Common Divisors
• Counting Divisors
• Sum of Divisors
• Greatest Common Divisor

Pattern 5: Geometric Algorithms

• Valid Square
• Circle and Rectangle Overlapping
• Two Circles

Pattern 6: Game Theory

• Stone Game
• Nim Game
• Another Game
• Grundy’s Game

Pattern 7: Probability Theory

• Dice Roll Simulation
• Random Pick with Weight
• Generate Random Point in Circle
• Dice Expectation

Bit Manipulation

Pattern 1: Bitwise AND Operations

• Bitwise AND of Numbers Range
• Largest Combination With Bitwise AND Greater Than Zero
• SOS Bit Problem

Pattern 2: Bitwise OR Operations

• Bitwise ORs of Subarrays

Pattern 3: Bitwise XOR Operations

• Single Number
• Maximum Xor Subarray
• All Subarray Xors
• Number of Subset Xors
• Count Pairs With XOR in a Range

Pattern 4: Finding Unique/Missing Elements Using Bits

• Single Number
• Single Number II
• Single Number III

Pattern 5: Bit Counting & Manipulation

• Number of 1 Bits
• Reverse Bits

You Should Know:

Linux Commands for Bit Manipulation & Maths

 Check if a number is prime (bash) 
is_prime() { 
local num=$1 
for ((i=2; ii<=num; i++)); do 
if ((num % i == 0)); then 
echo "$num is not prime" 
return 1 
fi 
done 
echo "$num is prime" 
}

Generate prime numbers using Sieve of Eratosthenes (Python) 
python3 -c " 
def sieve(n): 
sieve = [bash]  (n+1) 
sieve[bash] = sieve[bash] = False 
for i in range(2, int(n0.5)+1): 
if sieve[bash]: 
sieve[ii::i] = [bash]  len(sieve[ii::i]) 
return [i for i, val in enumerate(sieve) if val] 
print(sieve(100)) 
"

Bitwise operations in Python 
python3 -c " 
a = 5  0101 
b = 3  0011 
print(f'AND: {a & b}')  0001 (1) 
print(f'OR: {a | b}')  0111 (7) 
print(f'XOR: {a ^ b}')  0110 (6) 
print(f'NOT: {~a}')  Two's complement 
print(f'Left Shift: {a << 1}')  1010 (10) 
print(f'Right Shift: {a >> 1}')  0010 (2) 
" 

Windows CMD for Bitwise Checks

:: Check if a number is even or odd 
@echo off 
set /p num=Enter a number: 
set /a mod=%num% %% 2 
if %mod% equ 0 (echo Even) else (echo Odd)

:: Bitwise AND example 
set /a "result=5 & 3" 
echo Bitwise AND of 5 and 3: %result% 

What Undercode Say:

Competitive programming requires mastering mathematical concepts and bit manipulation for optimization. The provided problems cover essential patterns, and the Linux/Windows commands help verify solutions efficiently.

Prediction:

As AI and quantum computing rise, bit manipulation will become even more critical in low-level optimizations and cryptography.

Expected Output:

5 is prime 
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97] 
AND: 1 
OR: 7 
XOR: 6 
NOT: -6 
Left Shift: 10 
Right Shift: 2 

IT/Security Reporter URL:

Reported By: Shivam Shrivastava – Hackers Feeds
Extra Hub: Undercode MoN
Basic Verification: Pass ✅

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