- For example, a person chooses the
number 25 and keeps that number to
him- or herself. Ask the person to com-
pute 3 n. (3 25 75) - Ask if the number is odd or even. (odd)
- If the number is even, tell him or her
to divide the number by 2; if the num-
ber is odd, tell him or her to add 1 to
the number, then divide that number
by 2. (75 1 76/2 38) - Tell him or her to take that number
and multiply by 3, then divide by 9. (38
3 114/9 12.666 ...) - Take that number and multiply by 2
(12.666 ... 2 approximately 25).
The answer should be the original
number—or close to it.
What is a magic square?
A magic square is an array of numbers in
an nby nsquare that contains positive integers—from 1 to n^2 —with each number
occurring only once. The numbers in the squares are arranged so that the sum in any
horizontal, vertical, or main diagonal direction is always the same. This is shown in
the following formula: n(n^2 1)/2.
Magic squares are often divided into orders; for example, a three-order magic
square means three boxes per row and three boxes per column. In reality, “magic”
squares are actually matrices (for more information about a matrix, see “Algebra”);
they can be odd-order (such as a 3-by-3, or 5-by-5 matrix) or even-order (such as a 4-
by-4 or 6-by-6 matrix) magic squares. Perhaps the simplest magic square is the 1-by-1
square, whose only entry is the number 1.
Such magic squares have been known for centuries. For example, the Chinese
knew about the three unique normal squares of order three. In Chinese literature dat-
ing from as early as 2800 BCE, a magic square known as the Loh-Shu,or “scroll of the
river Loh,” was invented by Fuh-Hi, who is thought of as the mythical founder of the
Chinese civilization.What is Pascal’s triangle?
Pascal’s triangle, as the name implies, is a collection of numbers in the shape of a trian-
438 gle. Each number in the triangle is the sum of the two directly above. For example, in the
In these 3-by-3 Chinese magic squares, the numbers
are arranged so that when added vertically, horizon-
tally, or diagonally, they are always equal to 3 times
the center number.