MATHEMATICS IN MAGIC C Deanna Mele Background Magic

MATHEMATICS IN MAGIC C Deanna Mele

Background • Magic is a form of theater that depicts impossible events as though they were really happening • Doing tricks magicians do requires the audience to be able to experience the magic as real and unreal at the same time • In other words, we know that magic is not real, but the illusions behind it are and so are the concepts behind every “magic trick” • The goal is to make the audience question how a “trick” was possible

C 4 X 4 GRID

4 x 4 Grid You should write down the following numbers you choose: All 4 numbers should be different! 1. Pick ANY number from the grid 2. Pick another number, but a different row AND column 3. Pick a third number, yet again a different row AND column 4. Do this one more time; pick a fourth number not using the same row AND column 5. Add up all 4 numbers!

4 x 4 Grid Without knowing what numbers you chose, I know the sum will ALWAYS be 34!

Why? • The trace of this 4 x 4 matrix is 34 so the sum of four numbers, no two in the same row or column, will be 34 • If you picked a number in the 1 st row, it will be 1, 2, or 3 spots from the diagonal. • This will add +1, +2, or +3 to the sum of the numbers. • Now choosing another row, you’ll have to move that number 1, 2, or 3 spots to the left of the diagonal • This would mean -1, -2, or -3 to the sum of the numbers.

Why? • When you transpose the columns of two numbers in different rows, you add and subtract the same distance to the diagonal. • The total sum remains the same as the sum of the diagonal.

Lets think of this another way… • Write “abcd” to mean 1234 • • 1 corresponds to row 1 column a 2 corresponds to row 2 column b 3 corresponds to row 3 column c • For any other permutation, “abcd”, the sum of the numbers will be 34 plus the adjustment for changed positions, either left or right, and can be written as follows: 4 corresponds to row 4 column d • Every combination of 4 numbers following the “no same row and column rule”, will correspond to the permutation 1234 which results in the diagonal sum equaling 34 • Each permutation of 1234 has the same sum of 10 = a + b + c + d. 34 + (a – 1) + (b – 2) + (c – 3) + (d – 4)

Lets think of this another way… 34 + (a – 1) + (b – 2) + (c – 3) + (d – 4) Knowing that permutation 1234 has the sum 10=a+b+c+d, we can plug in and solve =34 -10+(a+b+c+d) =34 -10+10 =34

4 x 4 Grid Trick Conclusion Regardless of the number chosen first, second, third, and fourth, the sum will always be 34!

C MAGIC SQUARE

Magic Square • The sum of all columns equal 15

Magic Square • The sum of all rows equals 15

Magic Square • The sum of both diagonals equals 15

Magic Square • 15 is the magic sum of the square!

Magic Square It is well known that the magic sum of the square is 15, but what other interesting facts are there that are less well known?

Magic Product • If you multiply the three numbers in each row and add the products, you will get the same solution as if you added the three numbers in each column and added the products! (8 x 1 x 6)+(3 x 5 x 7)+(4 x 9 x 2)=225 (8 x 3 x 4)+(1 x 5 x 9)+(6 x 7 x 2)=225 • 225 is the magic product

Magic Pairwise Product • If you multiply the pairwise products of each row and add them, you will get the same solution as if you took the pairwise products of each column and added them together (8 x 1+1 x 6+6 x 8)+(3 x 5+5 x 7+7 x 3)+(4 x 9+9 x 2+4 x 6)=195 (8 x 3+3 x 4+4 x 8)+(1 x 5+5 x 9+9 x 1)+(6 x 7+7 x 2+2 x 6)=195 • 195 is the magic pairwise product of the square

How to find all 3 x 3 Magic Squares! • Let the number in the middle = x • Let the magic sum of each row, column, and diagonal be = R

How to find all 3 x 3 Magic Squares! • Add the middle column and middle row and both diagonals. • Now we have 4 R • This includes the middle number four times and all the other numbers once so it must add up to the total of all the numbers (3 R) plus three times the middle number • So now we have: 4 R = 3 R+ 3 x -3 R=-3 R+ 3 x R= 3 x • This also tells us the total of all the numbers in the square is 9 x

Now you can make your own magic squares! 1. You just choose the middle number 2. Pick two numbers in other positions 3. Complete the whole magic square so each line adds up to three times the middle number