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In Memoriam 2015

January 1: Donna Douglas: Played daughter Elly May Clampett in The Beverly Hillbillies. (Age 82). 1: Mario Cuomo: Governor of New York (1983 to 1994) (Age 82). 2: James Cecil Dickens: Known as Little Jimmy Dickens, best known for his song May the Bird of Paradise Fly Up ...

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The disappearance of misc.activism.progressive and the emergence of Thought Crime Radio

Almost four years ago, the articles in the USENET newsgroup misc.activism.progressive ground to a halt, and moderator Rich Winkel has all but disappeared from the USENET, whom I learn resided in Harrisburg (up until 2010, at least), a half hour or so drive from his ...

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Sounding off on the end of CanCon and the CRTC

I guess with the recent decision to axe all cancon requirements for daytime programming in Canada, the CRTC is crawling toward its own irrelevance. Let's not be naive, Canadian culture is that much more weakened without the protection it partially enjoyed from American influence. With ...

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Eldred, Saskatchewan on the map … barely

Eldred, Saskatchewan on the map ... barely

I've written about obscure Saskatchewan communities before. Here is another community far to the north of Unity. My ancestors from France settled here. Many of my ancestors were pioneers that broke new farming ground nearest to a community called Eldred, Saskatchewan. Eldred was about 10 km ...

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Zero

Once upon a time, around the year 525 during the reign of Pope John I, a monk named Dionysius invented the idea of Anno Domini by producing a calendar which marked the time since the birth of Christ. The numbering of the years was adopted ...

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Fortune Cookies for Human Rights

Fortune Cookies for Human Rights

You know, I was minding my own business in this classy Chinese restaurant, engorging myself on their copious buffet, had my fill, and was handed the bill with an accompanying fortune cookie. This fortune cookie (the one to the left) really existed, and I never saw ...

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Getting f(x) notation to work in Maple

Getting f(x) notation to work in Maple

Maple is a robust math environment which can graph, solve equations, and solve for the unknown with the aid of its computer algebra solver (CAS), which is capable of computing exact roots of cubic functions, for example. I wanted to demonstrate for myself that Maple could ...

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Kudos to the 1050 CHUM Memorial Blog

Kudos to the 1050 CHUM Memorial Blog

Recently, I've been hit (my website that is) by someone possibly checking his plethora of links from his/her website, and when I back-traced it, I find this cool blog which acts as a convincing historical shrine to the late great 1050 CHUM Radio in Toronto. ...

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The Obfuscation of Electronics: The Behringer Xenyx 502

The Obfuscation of Electronics: The Behringer Xenyx 502

This is more like a meta-review. I have gone to Canada Computes where nearly the entire Behringer line is sold, and was impressed by the specs. But does it do what I want, the way I want it? I face a number of obstacles, being a ...

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Recreational Math I: Magic Squares: the “really good” kind – Part 2

Last time I introduced the idea of magic squares. I promised I would show you how to make one. In this post, I will begin by discussing “trivial” squares, or squares made by simple rules of following diagonals and wrapping.

When I say a square is “magic”, I mean that all rows, columns, and diagonals add up to the same number. While other sources, such as Wolfram’s Mathematica, say that only the main diagonal of the matrix need be magic, I will take the more strict requirement that both leftward and rightward diagonals have to be magic.

There are trivial magic squares that begins by following a rule where you start with “1” in the top middle square, then move up and to the right one square, and place a “2” there.

_  1  _
_  _  _
_  _  _

But you may have noticed that if you start at the top, how can you move “up and to the right”? You get around this by “wrapping” to the bottom, treating the bottom of the rightward column as though it is above.

_  1  _
_  _  _
_  _  2

OK, you say, but now there’s no “right” after the last column. Now what? Now you can wrap so that the leftmost column is treated as “right of” the rightmost column:

_  1  _
3  _  _
_  _  2

Now another problem: up and to the right of “3”, there is a “1” in the way. If this happens, you are allowed to place the fourth number below the “3”:

_  1  _
3  _  _
4  _  2

Now, following these rules and exceptions, we can keep going:

8  1  6
3  5  7
4  9  2

The result is a magic square whose rows, columns and diagonals add up to 15.

I found that if I moved the 1 elsewhere and followed these rules in the same manner, some or all of the “magic” is lost. There seems to be only one magic square that can be made using these rules, at least one that adds up to 15 in all of its rows, columns and diagonals. The following 3×3 magic squares were the closest I could come to any credible “magic” by placing the “1” in a different position:

6  8  1                4  9  2
7  3  5 and similarly: 8  1  6
2  4  9                3  5  7

But notice in both cases, neither of the diagonals add up to 15. In the next post, I will discuss a way to break this limitation, making it possible to construct up to 36 3×3 magic squares.

Meanwhile, let’s expand the idea to 5×5, using the same, identical rules. This one seems easier in a way, since there aren’t as many blockages early on:

17  24   1   8  15
23   5   7  14  16
 4   6  13  20  22
10  12  19  21   3
11  18  25   2   9

I particularly like 5×5 squares. But my experience with placing the “1” elsewhere than the exact middle position of the top row has resulted in a loss of “magic”. However, I was lucky on my first attempt with moving the “1” around. The following magic square has a “Mathematica” level of magic:

11  18  25   2   9
17  24   1   8  15
23   5   7  14  16
 4   6  13  20  22
10  12  19  21   3

Later in this series, we can break this limitation, too. But next, we shall discuss some 4×4 magic squares, including one that made history.

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