Transcript Math Olympiad Strategies

Math Olympiad Strategies
Gauss: Patterns and
Sums
A Child Prodigy
 Karl Friedrich Gauss (1777-1855) was one of the world’s greatest
mathematicians. There is a story that when Gauss was a nine-year old
schoolboy, the teacher of his class assigned the rather tedious task of
finding the sum of all the counting numbers from 1-100 inclusive. To
the teacher’s surprise, Gauss arrived at the correct answer within
seconds. What is the sum?
1 + 2 + 3 + 4 + …+ 97 + 98 + 99 + 100
How did Gauss get 5050?
 He may have added the ends together:
 1 + 2 + 3 + 4 + …+ 97 + 98 + 99 + 100
101 +101 +101 +101 ….50 pairs…..so 101 x 50 = 5,050
Gauss may have added two lists of 1 to
100 together:
 1 + 2 + 3 + 4 + …+ 97 + 98 + 99 + 100
100+99+98+97+…. 4 + 3 + 2 + 1
101+101+101 + 101+
……101 +101 + 101+ 101
So…101 x100 = 10, 100
But that 2 sets of 1-100 so we must divide by 2
10,100/2= 5050!
Gauss’ Formula for sum of the first n
counting numbers:
n(n+1)/2
Use the formula to find the even and odd sums from 1 to n:
n=10
10 x (10 +1)/2
10/2 X 11
5 x 11
55
N(N+1)/2
 Find the sum of the numbers from 1 to 17
 N= 17
 17 x (17 +1)/2
 17 x 18/2
 17 X 9
 153
 Find the sum from 1 to 24
 N=24


24 x (24 + 1)/2
24/2 x 25 =
12 x 25 = 300
The Handshake Problem
 Using a table, Let’s see if we can find a formula for the
famous handshake problem. Is it the same as the Gauss
formula?
Number of people:
1
2
3
4
…
Number of handshakes
0
1
3
6
…
n
H=
Diagonal Table
 A polygon is a closed figure formed from line segments that
meet only at their endpoints.
 A diagonal is a line segment that joins 2 vertices of a
polygon but is not a side of the polygon.
Number of sides of a
polygon
3
4
5
6
…
Number of
diagonals:
0
2
5
9
…
n
D=
Formulas
 Gauss: n(n+1)/2
n= last counting number
 Handshake: n(n-1)/2
n= the number of people
 Diagonals: n(n-3)/2
n= the number of sides of a polygon
Gauss and Sums Homework
NAME:______________________________
Due:____________________________
1.
What is the sum of each of the following series of numbers?
1-15
b. 1-40
c. 1-75
2.
Ten people enter a room one at a time. Each new entrant shakes hands with each person already in the room.
How many handshakes occur?
3.
January 1, 1941 fell on a Wednesday. On which day of the week was January 1, 1942?
4.
A diagonal of a polygon is a line drawn from a vertex to another non-adjacent vertex. How many diagonal does
each of the following polygons have?
Bonus- Can you find the formula for
the number of diagonals for an n-gon
Polygon
Pentagon
Hexagon
Heptagon
Octagon
# of diagonals
Gauss’s Challenge from Historical
Connections Volume 1: Day 2 in Class