Quod Erat Demonstrandum

2009/04/20

Just reply to my so-called ETV

Filed under: Fun,Junior Form Mathematics — johnmayhk @ 12:45 上午

Here is just a reply to my primary school mathematics ETV (magic square).

When I was a primary school boy, my mathematics teacher taught us how to construct magic square, today, I still remember what she had taught me. For your reference, here is an easy way to construct magic square of odd order:

http://en.wikipedia.org/wiki/Magic_square#A_method_for_constructing_a_magic_square_of_odd_order

Students, try to construct magic squares of order 5 and 7 by using the method mentioned above.

(Sudden question: will the magic squares of order 5 and 7 etc. have the same property as described in the video?)

To see (not prove) the reason why there is a property as described in the video, let’s look at the magic square of order 3 again and do some junior form mathematics.

– – – – – – M1

It is actually ‘equivalent’ to the following magic square:

– – – – – – M2

M2 is formed by subtracting 5 from each entry in M1.

Look at M2, it may be easier to see the sums of entries of the same row, same column and same diagonal are zero. That is actually the property of magic square. Right?

For simplicity, let a = 100, b = 10.

Look at M1, the number formed from the first row, i.e. 816, can be written as 8a + b + 6.

To have the linkage with M2, I write

816^2
= (8a + b + 6)^2
= (3a - 4b + 1 + 555)^2
= (3a - 4b + 1)^2 + c(3a - 4b + 1) + d – – – – – – (*)

(where c = 2 \times 555 = 1110, d = 555^2 = 308025)

See, 3, -4, 1 are numbers in the first row in M2.

Look at M1 again, the number formed from the third row, i.e. 492, and the ‘reverse’ of this number is 294.

In fact, 294 is quite ‘related’ to the number 816, consider

294^2
= (2a + 9b + 4)^2
= (-3a + 4b - 1 + 555)^2
= (-3a + 4b - 1)^2 + c(-3a + 4b - 1) + d
= (3a - 4b + 1)^2 - c(3a - 4b + 1) + d

Compare with (*), their middle terms are different from a negative sign only.

This is not by accident, it is all about the construction of magic square.

The ‘proof’ is done by the ‘zero sum’ in M2. What? Okay, let me write the whole story.

816^2 + 357^2 + 492^2
= (3a - 4b + 1)^2 + c(3a - 4b + 1) + d
+ (-2a + 2)^2 + c(-2a + 2) + d
+ (-a + 4b -3)^2 + c(-a + 4b -3) + d
= (3a - 4b + 1)^2 + (-2a + 2)^2 + (-a + 4b -3)^2 + 3d – – – – – – (#)

The construction of magic square ensures the terms involving c being cancelled.

Now, the ‘reversed sum’

618^2 + 753^2 + 294^2
= 294^2 + 753^2 + 618^2
= (3a - 4b + 1)^2 - c(3a - 4b + 1) + d
+ (-2a + 2)^2 - c(-2a + 2) + d
+ (-a + 4b -3)^2 - c(-a + 4b -3) + d
= (3a - 4b + 1)^2 + (-2a + 2)^2 + (-a + 4b -3)^2 + 3d

Also, the terms involving c are cancelled, and the result is just the same as (#) and that is why

816^2 + 357^2 + 492^2 = 618^2 + 753^2 + 294^2

Instead of considering rows, as shown in the video, the property remains when we consider numbers formed from columns. And the so-called proof is exactly the same as before.

Urm, we may ask naturally that will the following be true for some positive integer n (other than n = 1, 2)?

816^n + 357^n + 492^n = 618^n + 753^n + 294^n

Students, explore it if you are interested.

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