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 1-10 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
The basic mathematics behind the magic effect described is based on a concept called arithmetic progressions. An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. In this case, the difference between each pair of consecutive numbers is always the same.
Let's break down the steps to understand how the magician can quickly determine the sum of all the numbers:
Step 1: The magician writes the numbers 1 through 10 on the board.
Step 2: The first participant selects a number, let's say x, and writes it in the first space. The second participant selects a different number, let's say y, and writes it in the second space.
Step 3: The magician quickly glances at the board to ensure all spaces are filled.
Step 4: The participants begin to fill in the remaining spaces by adding the previous two numbers.
Now, let's consider the numbers in the list:
1, 2, x, y, (x + y), (x + 2y), (2x + 3y), (3x + 5y), (5x + 8y), (8x + 13y)
Here's where the magic lies: The magician is able to determine the final sum without seeing the specific values of x and y.
The key lies in noticing that each number in the list can be represented as a combination of x and y, using Fibonacci-like patterns. In this case, the pattern is similar to the Fibonacci sequence, where each term is the sum of the previous two terms.
For example, consider the fifth number in the pattern: (x + y). This is the sum of the second and third numbers. Similarly, the sixth number, (x + 2y), is the sum of the third and fourth numbers. This pattern continues throughout the list.
Now, if we look at the final number in the pattern (8x + 13y), we can observe that the coefficient in front of x is the Fibonacci number in position 7 (8), and the coefficient in front of y is the Fibonacci number in position 8 (13).
The magician has memorized the Fibonacci sequence up to a certain position and is able to quickly calculate the coefficients of x and y based on the 7th and 8th Fibonacci numbers. With this information, the magician can tell the sum of all the numbers without explicitly knowing the individual values of x and y.
By mentally multiplying the 7th Fibonacci number (8) by x and the 8th Fibonacci number (13) by y, the magician can determine the sum of all the numbers on the board.
This magical effect relies on the mathematical observation of an arithmetic progression and the application of Fibonacci numbers. By understanding these concepts, one can recognize the pattern and quickly calculate the sum before the participants have a chance to add them up.