I'm looking for information regarding the originators of the following mathematical principles used in magic:

1089 Force
Lightning Square/Cube/Fifth Root Extractions
Day of the Week Calculation
Matrix Force
Magic Square Creation
Fibonacci Lightning Addition

Day of the Week Calculation: Zeller's Congruence by Christian Zeller


1089 Force: The first appearance in any magic text was in the January and February 1928 issues of Linking Ring. C. C. Arras wrote up "A Good Book Test" in January 1928 (p. 877), and W. W. Durban commented on it the following month (p. 950). The principle itself is likely far older than that.


Magic Square Creation: This starts with Lo Shu in 2800 B.C., and the link will give you other starting points for your research.


Matrix Force: Doug Dyment's Matrix Force page has plenty of information about the origin of this concept.

Thank you. I appreciate the information and the leads.

I don't think Zeller was the first. Didn't Gauss have a formula for this?

I know of Gauss' formula for figuring the date of Easter Sunday, but not the day of the week for any date.

Scott,

You may be correct.

From the last paragraph of Chapter 8 ("Tricks of Lightning Calculators") of
Martin Gardner's Mathematical Carnival:

"Many methods for calculating the day of the week mentally were published
late in the 19th century, but I have found none earlier than a method
invented by Lewis Carroll and explained by him in Nature
(Vol. 35, March 31, 1887, page 517.)"

However...the Wikipedia article on "Zeller's Congruence" (link provided
above by Scott Cram) indicates that Zeller published even earlier.

Since the calculation for figuring the date of Easter Sunday is more
complicated than the calculation of day-for-any-date, I find it improbable
that Gauss didn't have a method.

Gotta do some more research.

I thought everyone knew that Gauss just took a peek at his PDA.

Lewis Caroll did publish a method (click here), but it's rarely used by magicians.

The one most used is Zeller's Congruence, as mentioned originally.

There's also the Doomsday method, but that is rarely used by magicians, too.

I found an old post of mine where I presented my own way to do this where you divide the year in two parts to keep the numbers low. It's fairly easy since you can do modulo 7 calculations as you go along to always keep a small number in your head at each step.

http://www.themagiccafe.com/forums/searc......=4064377

/Tomas

Scott,

It's not yet April 1...but I suspect you're pulling my leg.

I've talked to several magicians who do "day-for-any-date."
NONE of them use Zeller's Congruence.

Do you really use m=13 for Jan...and m=14 for Feb?
And then multiply (m+1) by 26...and then divide by 10 and take
the greatest integer in order to get the month key?

Does anyone have access to the latest edition of Gardner's
Mathematical Carnival? If so, has the paragraph at the
end of Chapter 8 been rewritten?

I apologize for the confusion. What I meant to say is that a modified and simplified form of Zeller's Congruence is the one most commonly used.

If anyone can properly credit the modifier of Zeller's Congruence, this would be the place to do it.

Hi all,

I'm not trying to peddle anything here, but for the record my recent book has extensive references detailing the lineage of the 1089 Force and the Matrix Force. While I've gotten good reviews in the magazines (for which I'm grateful), it's killing me that no one has mentioned yet that there are over 200 bibliographic references in it; I think that's the best part of the book.

I've written a 20 page chapter on Zeller's congruence with which I haven't done anything yet. It's geared more for a college survey text. As Scott mentions, there are lots of simplified versions of Zeller's congruence kicking around. But the original is still the best; the algorithm is the same regardless of year; there are no exceptions to take account of. Of course, any of these methods, whether the original or otherwise, fail if the country in question hadn't started using the Gregorian calendar yet. For example, what day of the week was Isaac Newton born on? (I always give that one to my students to see who's paying attention to historical influences on mathematics).




I just checked Gardner's _Mathematical Games: The Entire Collection_ CD-ROM from a couple years ago and presumably the most up-to-date version of his material. That paragraph is the same as in the first edition book (which I also have).

Thomas Henry

Scott, thanks for the link to Zeller's Congruence. I see that I just reinvented the way to divide the year in two parts.

/Tomas

...and thanks to Thomas Henry for checking the Gardner CD-ROM.

While trying to understand why Gardner doesn't mention Zeller...I thought
I had it figured out.

Here's the Gardner quote again:
"Many methods for calculating the day of the week mentally were published
late in the 19th century, but I have found none earlier than a method
invented by Lewis Carroll and explained by him in Nature..."

Of course it's possible that Gardner wasn't aware of Zeller's work when
he wrote those words. But if this were the explanation, I would think
one of Gardner's readers would have told him about Zeller's work. And
I think Gardner would have updated that paragraph in later editions of
the book.

Maybe Gardner didn't think Zeller's formula was designed for mental
calculation. But it appears to me that Zeller's formula was designed
for mental calculation. Why else would the month key be calculated using
int((m+1)*26/10)
rather than
int((m+1)*13/5) ?

To the best of my knowledge, Gardner is never "online," so we can't get
a quick answer from him. But Thomas Henry is online, so maybe he can
help me understand the following quote from his post:

Thomas Henry wrote:
"But the original is still the best; the algorithm is the same regardless of year; there are no exceptions to take account of."

I don't quite understand this comment. Which day-for-any-date algorithms have
exceptions to take account of?

The Parson is good friends with Gardner. Maybe he can get a comment from him on this.

Hi again,



What I was getting at is that in most of the simplified versions I've seen, you must adjust the final result up or down depending on the century you're in. For example, in Scott Flansburg's _Math Magic_, (New York: HarperPerennial, 1993), pp. 236-241, at a certain point, one must add 2 for the 1880's, add 3 for the 1700's, add 4 for the 1600's, subtract 1 for the 2000's, etc. And then, of course, one must subtract 1 for leap years.

With Zeller's congruence, the computation proceeds along a straight path no matter the century and no matter if in leap year or not.

In other words, it is strictly an arithmetic computation; there are no decisions to be made along the way.

What it comes down to is whether you want to permit conditionals (if/then's) or just go with straight arithmetic. Yeah, yeah, I know...there are conditionals embedded in Zeller's congruence. After all, the greatest integer function is defined in terms of a conditional. And, of course, one must remember that January and February fall in the previous year. So, now that I think of it, I guess every method has exceptional cases to keep in mind. But I'm so used to doing Zeller's that I don't even think about them as being special.

In the long run it really doesn't matter very much, for every person has different arithmetic/mental weaknesses or prejudices. What matters is whether the algorithm you use is mathematically sound and you can do it mentally without losing track of where you are. For what it's worth, despite my degrees in mathematics, I'm horrible at mental arithmetic and use every trick in the books to keep things straight. Fortunately, there are vast areas of mathematics unconcerned with computation or I would have never gotten through college!

Nonplussed! by Julian Havil. (Too bad it's not Julian Gregorian.)
Haven't seen the book, but from the Amazon website you
can do a "search inside book" for "Gauss calendar formula".