Dear fellow magicians,

I'm kind of stuck in training my skills in creating instant 4x4 magic squares. The number which killed me is 67.
I've tried Sam Dalal's method from Patterns of Perfection revisited and also Michael Daniels' version presented in Perfectly Possible.

My problem is the sum of the outer/middle 2x2 squares - two of those squares are off by one (reminds me of Kenton Knepper ).
Perhaps it's my fault, but I get those aberrations with different methods - and Michaels training-script accepts my solution as valid.

Unfortunately I haven't found any explanation regarding this topic, but it may be due to an insufficient search not really knowing what to search for...

Can someone please enlighten me about this.

Thanks in advance an best season greetings

Chris

P.S.: When naming the cells from A to P the aberring squares are BCFG and JKNO or EFIJ and GHKL

Here's a vey simple approach:
http://www.mathsbusking.com/shows-magic-squares.html

Hi Larry,

thank you for the link.
After finishing Michaels Perfectly Possible I realized that he elaborates on the theme that squares of odd sums aren't that perfect as even ones. They seem to lack quite a few of the summation patterns.

While not being pleasant this is important for my presentation, since with even numbers you can go rampage when presenting the possibilities to sum up the cells, with uneven results one should be more careful in ones presentation...

Thank you for your help.

Cheers Chris

Hi Chris,

Yes, there is nothing unique about the number 67. The issue you describe applies to ALL odd-total 4x4 squares.

Odd-total "mostly perfect" squares have 36 symmetrical summation patterns, which are a subset of the 52 patterns achieved by even-total "most perfect" squares.

Of the missing 16 patterns, the ones you mention are the most notable - which of the two combinations you list will occur depends on the way the square is rotated. With my Perfectly Possible method it will always be EFIJ and GHKL.

If odd-total squares are a problem for your presentation, you could always force an even total by getting the spectator to double the number they first thought of - "to make it even more difficult".

Mike

Hi Mike,

thanks for your further information.
I knew the "normal" development as in Larry's link - having a fixed pattern and adjusting four "key-cells" to reach the needed number/sum. Or as John Archer teaches in his lectures using different cribs for different ranges to minimize the offset between the fixed numbers and the newly created ones, so that the squares don't look that "pre-made".
And for some reason the audience wanted even numbers or was that fascinated with the result when using odd sums, that I never really looked deeper in the mechanics of the magic squares until now.

Some weeks ago I bought and read Sam Dalal's PoP revisited and started to develop the squares "on the fly" - and allthough it should have dawned on me much earlier I just stuck with the creation of a square for 67 (I even never looked after other odd ones afterwards since I seemingly had something misunderstood in the creation).

Yesterday, after further looking into this theme I found your script/book and bought it. I really like your approach, but after understanding the method I didn't read further (my fault!), thus missing your explanation of what patterns have to fail with odd sums. This lead to my initial post above

Unfortunately one doesn't find easy information about those specialties of "odd-squares" (or I didn't use the right search-terms).

But thanks again for all the help, it's really appreciated.

Best season greetings

Chris

Just to add a bit of detail, there are four levels of success that a four-by-four magic square can achieve. Of course, the 10 basic patters--the rows, columns, and diagonals--must add to the magic sum in order to qualify, but there are 4 other patterns that will automatically match. So even the poorest square will have 14 successful patterns. Most methods of construction, including the 'instant' method preferred by many magicians, will achieve 10 more. Improved methods that have appeared more recently, including those of Michael Daniels, will achieve 36 predictable patterns when the magic sum is odd (the maximum possible for odd magic sums), and 52 when the magic sum is even (the theoretical limit, or 'perfection'). Those of us who realize how much success is possible find the older methods to be less than satisfying.

If you want more details, see the postings in this forum under "Magic Squares Solution Chart."

Unable to find this post "Magic Squares Solution Chart."

This thread: https://www.themagiccafe.com/forums/view......&start=0

The easily-learned methodology that I teach (in my Calculated Thoughts book) always yields the 52 "interesting" patterns for perfect squares, and 36 for others. It's worth noting that there are additional symmetric patterns produced, but most authors ignore these, as they are not very appealing visually.