calculus
posted by Jacob on .
A magician writes a list of numbers, 1 through 10, on the board then turns his back to the board. Two audience members are asked to write different numbers from 110 in the first two spaces. One of the spectators adds the two numbers and places the result in position 3. Then they add the second and third number and write the result in position 4, add the third and fourth number and place result in 5th space, etc., until all 10 spaces are filled. The magician glances quickly at the board to make sure they have filled all 10 spaces, then turns away from the board. The spectators are then asked to add the column of 10 numbers and place the grand total below at the bottom of the column. Before they can add the numbers, the magician announces the total!
Discuss the basic mathematics behind the above magic effect. Discuss how the magician can quickly tell the sum of all the numbers before the participants are able to add them up. (

let the first two numbers written down be
x and y
the numbers would appear this way
x
y
x+y
x+2y
2x+3y
3x+5y
5x+8y
8x+13y
13x+21y
21x+34y , notice the "Fibonacci Number" pattern
total = 55x + 88y
= 11(5x + 8y)
The magician should be reasonably capable of doing some quick mental arithmetic
.... multiply the first number by 5 and the 2nd by 8, add those up, and multiply the result by 11
suppose the first 2 numbers written down are
4 and 7
5 times 4= 20, and 8 times 7 = 56
20+56 = 76
multiplying a two digit number by 11 is easy ...
11x76 = 836