The Mystery and Mystique of Magic Squares

Magic squares are square arrays of numbers that have the same sum horizontally, vertically, and diagonally. It is the arrangement of n^2 numbers in an n \times n square. They were known to mathematicians since antiquity and they were believed to have mystical powers.  The most popular magic square is the 3 \times 3 Lo Shu square dated about 650 BC as shown in the first figure.

Magic Square

Magic squares are of different sizes. The magic squares with 16, 25, and 36 numbers are shown below. Of course there are still larger magic squares. In fact, in 1663, Muramatsu, a Japanese mathematician constructed a 19 \times 19 magic square. Some mathematicians even found ways to generate magic squares. The 3 \times 3 square above can be generated using elementary school algebra. » Read more

Latitudes, Longitudes and the Polar Coordinate System

In the early times, mathematicians and astronomers observed that the stars rotate about a point near the star now known as Polaris.  This is one of the bases why some of them, particularly Copernicus, believed that the Earth spins on its axis and revolves around the sun.

north-pole

The longitude, the “vertical divisions” of Earth, meet at the North and the South Poles. With the addition of latitudes, the Earth’s “horizontal divisions,” a view of these lines from one of the poles is shown above. The longitudes appear as lines radiating from the pole (the center) and the latitudes as concentric circles. The largest circle is the equator. » Read more

The Mathematics of Tenure, Car Lights, and Logic Gates

In a state office, an employee maybe tenured if (1) he has passed the Career Service Examination, or (2) he has worked as a probationary employee for more than 10 years.

Notice that the employee maybe tenured if either one of the conditions above is satisfied. He may also be tenured if both conditions are satisfied. If we create a table and rephrase the statements above to question form, we can see that the only reason for an employee not to be tenured is if the answer to both questions is NO.

tenure-table

The first table is also the same as the table shown below.  In the second table,  the situation describes the relationship between a car’s ceiling light and its two doors. Unless the ceiling light is turned on or off manually, it will always be ON if both doors or one of the two doors is open. The only time that the light will be OFF is if the two doors are shut. » Read more

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