Solved Examples & Key Concepts

Example 1: The “Handshake” Problem (Complete Graphs)

Question: If 5 people meet in a room and everyone shakes hands with everyone else exactly once, how many handshakes happen?

Solution: This is identical to the “Complete Graph” problem with 5 points ($K_5$).

1. Person A shakes hands with 4 others.
2. Person B shakes hands with 3 others (excluding A, already done).
3. Person C shakes hands with 2 others.
4. Person D shakes hands with 1 other.
5. Person E has no new people to shake hands with.

Total = $4 + 3 + 2 + 1 = 10$. This is the 4th Triangular Number.

Example 2: Hexagonal Numbers Logic

Question: What is the rule for forming Hexagonal Numbers: 1, 7, 19, 37…?

Solution: Let’s look at the difference between consecutive terms:

• $7 - 1 = 6$
• $19 - 7 = 12$
• $37 - 19 = 18$

The differences are multiples of 6 ($6 \times 1, 6 \times 2, 6 \times 3$). To find the next number:

1. Next difference will be $6 \times 4 = 24$.
2. Add this to the last number: $37 + 24 = 61$.

Example 3: Powers of 2

Question: Why does adding the sequence of Powers of 2 (starting from 1) almost give the next power of 2?

Concept: The sum of powers of 2 up to $2^n$ is always one less than the next power ($2^{n+1} - 1$).