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The Magic of Numbers

June13

MagicSqIt is not known who created this magic square, but it is shown in Durer’s famous 1514 engraving, known as Melancholia. What is quite remarkable about this square is that:

1. All the rows, columns and the two diagonals add up to the same number, just like ordinary magic squares. In this case the number is 34.

2. Add up the four corners and what do you get?

3. What is the sum of the four center squares?

4. Add up the opposite pairs (e.g. 3+2+15+14). What do you get? Now try the second opposite pair (8+12+5+9). Again, what is their sum?

5. Try adding up the slanting squares – 2+8+15+9? Then its twin sum – 3+5+12+14?

6. How about the sum of the corner squares? That is, from the top left: 16+3+5+10, then 2+13+8+11, then 7+12+1+14 and, finally, 9+6+15+4.

7. Actually, this is only the beginning of the magic to be found with numbers in this magic square. For example, the sum of the first and second columns equals the sum of the third and fourth columns; and the sum of the first and third columns equals the sum of the second and fourth columns. But, wait, we are only getting started!

8. The numbers in the diagonals add up to the numbers that are not in the diagonals and the sum of their squares, as well as the sum of their cubes are equal!

By the way, look at the bottom of the square, where it records the date of the engraving – 15 14.

No, we did not cover all the magic in this square but it does go someway to showing you that there is nothing boring in Mathematics if you look closely enough!

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10 x 9 x 8 + (7 + 6) x 5 x 4 x (3 + 2) x 1 = 2020

NCEA Level 2 Algebra Problem. Using the information given, the shaded area = 9, that is:
y(y-8) = 9 –> y.y – 8y – 9 =0
–> (y-9)(y+1) = 0, therefore y = 9 (can’t have a distance of – 1 for the other solution for y)
Using the top and bottom of the rectangle,
x = (y-8)(y+2) = (9-8)(9+2) = 11
but, the left side = (x-4) = 11-4 = 7, but rhs = y+? = 9+?, which is greater than the value of the opp. side??
[I think that the left had side was a mistake and should have read (x+4)?]

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