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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