### PERFECT MAGIC HYPERCUBES

Perfection

Once one comes up with something, someone tries to do better. For a magic square, this was achieved with the concept of a pandiagonal magic square. Not only do the rows, columns and two diagonals sum the magic sum, (which is the basic requirement,) but also so do all the broken diagonals, as well.

 01 15 04 14 12 06 09 07 13 03 16 02 08 10 05 11 Figure 1

The red numbers in Figure 1 indicate a broken diagonal. If another copy is moved in beside it then the broken diagonal can become continuous. Computer people think in terms of the "wrap around." One way to show it effectively is to put the square on the surface of a donut. Others will fill the plane with identical squares. Some talk in terms of modular space, there are 4 rows, 4 columns, 4 diagonals one way and 4 diagonals the other way. That is 16 straight-line sums. For a square of order m, there would be 4m sums.

When it comes to the magic cube, there is not only a new z-direction put in, but there are two kinds of diagonals. The term triagonal was coined to differentiate the three-dimensional characteristic of the true diagonals of the cube from the regularly established planar diagonals of a magic square.

The perfect magic cube begins at order 8. All rows, columns and pillars must sum the magic constant and the four continuous triagonals to meet the requirements generally accepted by all. Instead of 4 ways to move suddenly there are 13 ways in a cube, unless you want to consider the reverse directions and make it 26. To be perfect means all diagonals and all triagonals, including the broken ones also do.

Many false claims of a perfect cube have been noticed with calling a pandiagonal cube perfect when the triagonals do not all sum the magic sum. The late David M. Collison showed a pandiagonal magic cube of order 7. But, it is not perfect.

### The Perfect Magic Tesseract.

The lowest order where it occurs is at order 16. There are 65,536 numbers to put in. The magic sum will be S=524,296 by formula. There are 40 different directions through each cell which sum S. There are 32,768 “quadragonals”  besides the 65,536  triagonals, and the 49,152 diagonals to sum S. Oh yes! There are 1536 pandiagonal magic squares included, as well as 64 perfect magic cubes embedded. With the tiniest type, you could show it on 1536 pieces of paper.

The alternative is to generate selected parts of it as needed by a program.

 N= 1 Path #1 9 Path #2 1 by John R. Hendricks 10-Sept. -01 Order 16 Y = Z 61733 27348 20003 9682 289 39641 48682 54747 61740 27357 20014 9695 304 39640 48679 54742 524296 592 39352 48455 54966 62021 27060 19779 9906 577 39353 48458 54971 62028 27069 19790 9919 524296 62316 26781 19566 10143 880 39064 48231 55190 62309 26772 19555 10130 865 39065 48234 55195 524296 1265 36873 48122 57099 62716 24589 19454 12047 1280 36872 48119 57094 62709 24580 19443 12034 524296 62933 24868 19155 11810 1489 37161 47834 56875 62940 24877 19166 11823 1504 37160 47831 56870 524296 1728 37448 47543 56646 63157 25156 18867 11586 1713 37449 47546 56651 63164 25165 18878 11599 524296 63388 25453 18590 11375 1952 37736 47255 56422 63381 25444 18579 11362 1937 37737 47258 56427 524296 3841 38137 45066 56315 65292 25853 16398 11263 3856 38136 45063 56310 65285 25844 16387 11250 524296 2nd Direction 65061 26068 16675 10962 3617 38361 45354 56027 65068 26077 16686 10975 3632 38360 45351 56022 524296 3408 38584 45639 55734 64837 26292 16963 10674 3393 38585 45642 55739 64844 26301 16974 10687 524296 64620 26525 17262 10399 3184 38808 45927 55446 64613 26516 17251 10386 3169 38809 45930 55451 524296 3057 40713 46330 53259 64508 28429 17662 8207 3072 40712 46327 53254 64501 28420 17651 8194 524296 64213 28196 17875 8482 2769 40489 46554 53547 64220 28205 17886 8495 2784 40488 46551 53542 524296 2496 40264 46775 53830 63925 27972 18099 8770 2481 40265 46778 53835 63932 27981 18110 8783 524296 63644 27757 18334 9071 2208 40040 46999 54118 63637 27748 18323 9058 2193 40041 47002 54123 524296 1 39929 48906 54523 61452 27645 20238 9471 16 39928 48903 54518 61445 27636 20227 9458 524296 First Direction X =X

Figure. One page of the perfect magic tesseract of order 16.

A perfect magic tesseract means that all squares contained in it are pandiagonal magic squares. It also means that all magic cubes contained within it are perfect too. It must also be panquadragonal. It is totally and utterly wrapped around no matter how you look at it.  Unfortunately, the smallest perfect magic tesseract  is of order 16. The best one can do is to show it as a distorted projection on many pieces of paper taped together.

The program has been made so that all coordinates are  between zero and 15 inclusive. If a negative route is chosen, then it shows the set of 16 numbers in reverse order, Only change the values in the green shaded zones and the rest is given.  You are given a line at a time to do with what you want,

There are the numbers 1 to 65,536 used. The magic sum is 524,296. There are 32,768 four-dimensional diagonals (quadragonals); 65,536 triagonals; 49,152 diagonals to check out. Contained are 1536 pandiagonal magic squares of order 16 and 64 perfect magic cubes of order 16.  And, oh yes, there are 16,384 rows, columns, pillars and files to check.

### The program

The easiest way is to use a coordinate system and show any row, column, or whatever you want. Pick any starting position (w, x, y, z) and any route from 1 to 40 and enter the values into the computer. Out come the numbers and the sum. If you want the reverse direction enter –5 instead of 5 and the numbers will all be reversed. For coordinates, 0 is the same as 16. For routes zero is not a route.

Some of the routes:

 Route #1 is a pillar. Z-changes. Route #3 is a column. Only y changes. Route #9 is a row. Only x changes. Route #27 is a file. Only w changes. Route #40 is the main quadragonal.

You figure out what the rest of them are, and I think they are all diagonals, triagonals, and quadragonals of one type, or another. .

If you would like a copy of the program, please request one by email.