Challenge Problem : Handshake problem: If you are in the room with one other person and you shake hands. You shake hands once and the total number of handshakes is one. If there are two other persons in the room, you shake hands twice, but the total number of handshakes is three (because the other two people also shake hands).
Have you tried filling this out yet?
# People

1

2

3

4

5

… 
n 
... 
100 
MY # handshakes

0

1

2

# People

1

2

3

4

5

… 
n 
... 
100 
Total # handshakes in group

0

1

3

Don't cheat!!
# People

1

2

3

4

5

… 
n 
... 
100 
MY # handshakes

0

1

2

3 
4 
n1 
99 
# People

1

2

3

4

5

… 
n 
... 
100 
Total # handshakes in group

0

1

3

6 
10 
??? 
???
Back up a step. If you were in a group with 5 people and shook hands with 4 people. Shouldn't every person shake hands with 4 people? YES!! But then why is it 10 total handshakes and not 20 total handshakes?
Because you'd be counting all the handshakes TWICE. If you shake hands with Fred, then you cannot count his handshake with you as a new one. So if I know the number of handshake that I MAKE and I know the number of people in the room, then I multiply these numbers AND DIVIDE BY 2!! This is very common in networks.
# People

1

2

3

4

5

… 
n 
... 
100 
MY # handshakes

0

1

2

3 
4 
n1 
99 
# People

1

2

3

4

5

… 
n 
... 
100 
Total # handshakes in group

0

1

3

6 
10 
n(n1)/2 
100(99)/2 
If you at this again, you'll notice this would also be the answer for the question of how many total sides and diagonals in a figure with 100 sides (each side or diagonal is the same as a handshake).
SIMILAR PROBLEM:
How many diagonals can you draw from ONE vertex of a 100 sided polygon (a 100gon)?
How many total diagonals are there in a 100gon?
# Sides

3

4

5

6

7

… 
n 
... 
100 
# diagonals from a single vertex

0

1

2

# Sides

3

4

5

6

7

… 
n 
... 
100 
Total # diagonals in a polygon

0

2

5


Don't cheat!!
# People

1

2

3

4

5

… 
n 
... 
100 
MY # handshakes

0

1

2

3 
4 
n1 
99 
# People

1

2

3

4

5

… 
n 
... 
100 
Total # handshakes in group

0

1

3

6 
10 
n(n1)/2 
100(99)/2 