problem solving notes

imperfect_notes/Intro_to_Problem_Solving.md

@@ -65,23 +65,112 @@
         
 - Ask if it's now clear that DSA is the basic building block
         
- ## How Problem Solving works?
+## How Problem Solving works?
+- Dicuss about the computer and decision making
+- How the computer doesn't have mind of its own
+- How every step needed to be told to computer based on several decisions
  
- - Dicuss about the computer and decision making
- - How the computer doesn't have mind of its own
- - How every step needed to be told to computer based on several decisions
- 
- - Example 1 : **ATM Machine**
+- Example 1 : **ATM Machine**
     - Discuss the process outline
     - Discuss the process in detail using decision tree (Don't introduce the name yet)
     - after drawing a basic decision tree - introduce with the name "Decision Tree"
     - Keep on focusing on the importance of decisions in problem solving
- 
- 
- 
- 
- 
- 
+
+- Example 2 : **Super Mario**
+    - Engage students by asking about the game
+    - Discuss what decision making happend when you press ```->``` arrow
+
+- Example 3 : **Pubg Game**
+    - Engage students by asking about the game
+    - Discuss what decision making happens while a person shoots in pubg
+    - How does energy/health reduction happens , what are different variables involved to make one decision
+        - Aim
+        - Distance
+        - Type of Gun
+        - Body Armor
+
+## "Observation" - The key to solve Problems
+
+- Discuss about Guass
+    - Ask students if they know about him, what they know
+    - state the problem - tell them his teacher gave him the problem
+    - Problem : Find the sum of numbers from 1 -> 100
+    - discuss that He didn't jump on problem directly, **made observation** (focus on observation)
+ 
+ - **Prime Numbers**
+    - What are prime Numbers
+    - why 1 and 0 are not prime
+    - Given a number check if it's prime or not - ask students for the solution
+    
+        - Approach 1 : Iterate from 1 -> N and check if it is divisble or not
+        
+        - Approach 2 : Sqaure root approach - traverse only till square root 
+            - Intution - factors always occurs in pairs - focus on presenting it as an observation
+            - give proof by examples
+            - Time analysis between O(N) and O(squareRoot(N))
+            
+                - Considering one % (mod operation - ask if they know) takes 1 msec
+                
+                    N | Approach 1 - N steps | Approach 2 - sqaureRoot(N) steps
+                    -- | -------------------- | --------------------------------
+                    11 | 11 msec | 3 msec
+                    101 | 101 msec | 10 msec
+                    100000 | 16.6 min | 1 sec
+                    10^18 | 116 days | 1.66 min
+             - Ask them if they appreciate the improvement in timings - put more focus
+
+- **Find sqaure root of a given Number**
+    - Explain the problem - have to find the integer part
+    
+    - Approach - 1 :
+        - iterate i from 1 to N and check if square of i is <= N
+        - if square <= N - it is current ans
+        - else stop the process - **WHY?** - Explain
+    - Ask if the Approach 1 is fast enough to increase the curiosity
+    
+    - Approach - 2 :
+        - start by stating an observation
+        - pick some random element and check if it's square is <= or > than N
+        - based on previous checks decide to ignore one part - properly explain this with example
+        - now rather than picking random , pick middle element
+        - reason for picking middle element - it always removes atleast half of the prospect solution
+    
+    - Discuss the time Analysis
+    
+## How Data Structures helps in Problem Solving?
+
+- Discuss how just the way of organising the data can optimise the solution
+- Give example of toothBrush , how toothbrush stored in kitchen would result in ineffeciency
+
+- **Dictionary Word Search**
+    - Discuss the process of searching a word in dictionary
+    - Relation with the sqaure root problem
+    - Name of the Algorithm - Binary Search
+    
+- How are we able to apply the Binary Search
+    - Order matters (Main situation)
+    - Arrangement of Data Matters / Organisation of Data Matters
+    
+- **HC Verma** Example - searching a word in HC verma
+    - Why Binary search won't work 
+    - No particular organisation of data
+    - Can sort all the words - But is it effecient ? NO!
+    
+    - Discuss Index / Appendix
+    - How data is stored in a different way which improves efficency
+    
+    word | page nos.
+    ---- | ---------
+    velocity | 100 101 167
+    
+    - Data Structure used - **Inverted Index**
+    - All search engines use this to optumise the search
+    
+## Conclusion
+    - Importance of Observation
+    - Importance of Data structures and Algorithms
+    - Importance of 3 essential qualities
+    - Further work