During the last four or five years there
have been tremendous advances made in the field of magic cubes. Indications are
that this trend is going to continue. For example, this page contains material
that I obtained during the few weeks after I ‘finalized’ my magic cube site at
the end of 2003.
At that time I was prepared to edit and
refine pages as a result of correspondence with readers. However two weeks
later, I now realize that there is going to be an abundance of new material
begging to be published. I think the best approach to handling this material is
to put the new items onto an update page, more or less in the order that they
come to me.
This page, therefore, will contain a
variety of cube subject matter, unlike the previously posted pages with their
specialized subjects. I will edit my previous pages only to the extent of
inserting links between each entry and previous pages on that subject.
Additional Update pages will be posted as necessary.
In 2003, Peter Loly of the University of
Manitoba published an order 4 square [1] that was not magic because the rows
and columns all summed to different values. However, both main diagonals and
all 6 broken diagonals summed to the required 34.
When I challenged readers to construct a cube with these
same characteristics, Aale de Winkel supplied me with an order 4 cube
(Dec.18, 2002) that came quite close. All pantriagonals are correct.
Unfortunately, no planar diagonals are correct. Also, all pillars are
correct (which they should not be).
Aale de Winkel's cube is shown on my Unusual Cubes Page.
On Jan. 4, 2004, I received this cube from Guenter
Stertenbrink.
This order 4 cube has NO monagonals correct. ALL
diagonals and ALL triagonals are correct. Because rows, columns, and pillars
do not sum correctly, this cube may not be called magic. I think, though,
that it is definitely magic of a different kind!
I am calling this type of cube ‘purely pandiagonal” in
reference to Peter Loly’s square of this type.
[1] Peter D. Loly, A Purely Pandiagonal 4*4 Square..., Journal of
Recreational Mathematics, Vol. 31, No. 1, 20022003, pp 2931 

1 29 49 45 8 28 56 44 13 17 61 33 12 24 60 40
50 46 2 30 55 43 7 27 62 34 14 18 59 39 11 23
4 32 52 48 5 25 53 41 16 20 64 36 9 21 57 37
51 47 3 31 54 42 6 26 63 35 15 19 58 38 10 22
Heterosquare: similar to a magic square except all
rows, columns and main diagonals have different sums.
Antimagic Square: a subset of heterosquares where all rows, columns and
main diagonals have consecutive sums.
See Joseph S. Madachy, Mathemaics On
Vacation, 1966, pp 101110.
Or the same material in Joseph S. Madachy, Madachy’s Mathematical
Recreations, 1979, pp 101110.
My heterosquare and antimagic square page includes
samples of Peter Bartsch heterosquares sent to me between November 2002 and
March 2003. For convenience I show one of his heterosquares here. As part of
his research was to find heterosquares with the smallest possible 'sum of
sums' , this figure shows 111 as the sum of these line totals. 
A heteromagic square 
The rest of this
section will consist of heteromagic cubes provided by Bartsch in early
January of 2004
I received this cube (Fig. a) from Peter
Bartsch on January 5, 2004
This cube uses the consecutive numbers from 1 to 27.
The totals of the rows, columns pillars, and triagonals are all
different so this qualifies as a heteromagic cube.
For convenience, I show the totals listed in
order of magnitude. A quick inspection reveals that they are all
different.
In this cube, the sum is 1300.
This is also the smallest possible order 3
heterocube, based on the fact that it uses the lowest set of 27
consecutive numbers.
But is it possible to rearrange these numbers
so that the total of the sums is lower?
Because some of the totals are the same as
numbers that are in the cube, we will call it a ‘regular’
heteromagic cube.
If all 31 totals were different then the numbers within the cube, we
would call it a ‘real’ heterocube.
I received the cube below (Fig. b) from Peter
Bartsch on January 9, 2004
This cube also uses the consecutive numbers from 1 to 27
This is a 'real' heteromagic cube because all
totals are different and none of them appear within the cube! Bold
totals are in consecutive order.
The 31 totals contain a consecutive series of
29 numbers.
If ALL 31 sums were consecutive, this would be an antimagic cube.
The total would then be 1333, the smallest possible for a real order
3 heterocube.

Fig. a 
Two Prime Number Heteromagic Cubes.
I received the Fig. c cube on January 12, 2004.
The 27 numbers in the cube and the 31 line sums are all prime numbers.
The total sum is obviously not a prime number.
Totals shown in bold are a consecutive series of prime numbers.
I received the Fig. d prime heteromagic cube on Jan.
13/04
In this case, the total sum is also a prime number.
In both cases, all sums of rows, columns, pillars
and triagonals are unique prime numbers but some appear also in the cube,
so these are ‘regular’ heteromagic cubes.
These cubes are the smallest possible based on the
fact that they use the lowest set of 27 consecutive prime numbers (3 to
107). But is it possible to rearrange these numbers so that the total of
the sums is lower?
Magic ratios
In early January of 2004, Walter Trump suggested a
new definition useful for cubes that were not quite magic, as a means of
measuring the richness of their features. Aale de Winkel suggested the
name 'magic ratio.
Magic ratio: correct lines divided by total rows,
columns, pillars, and 4 triagonals.
So correct lines/3m^{2} + 6m + 4. That is 3m^{2}
monagonals, 6m planar diagonals, and 4 main triagonals.
If all these lines are correct, the cube under discussion is
‘perfect’ by the Boyer/Trump definition, ‘diagonal’ by the
Hendricks/Heinz definition. 
As an example I will use an order 4 cube I received
from Guenter Stertenbrink on Oct. 29, 2003.
It is not magic, but is interesting in it’s own right because the
consecutive numbers form a closed knight tour.
All 3m2 monagonals are correct but no diagonals and only 2 triagonals.
So the magic ratio is 3m^{2} + 2/3m^{2} + 6m
+ 4 = 50/76 = 65.8%
1 46 23 60 42 5 64 19 55 28 33 14 32 51 10 37
56 27 2 45 31 52 41 6 34 13 24 59 9 38 63 20
47 4 57 22 8 43 18 61 25 54 15 36 50 29 40 11
26 53 48 3 49 30 7 44 16 35 58 21 39 12 17 62
To rate a cube such as Guenter Stertenbrink’s purely pandiagonal cube we
have defined another ratio.
Panmagic ratio: divide the correct lines in the cube by
the total lines as used in magic ratio plus total possible broken
diagonals and broken triagonals.
That is 3m^{2} monagonals, 6m^{2} pandiagonals, 4m^{2}
pantriagonals. So correct lines/13m^{2}.
If the ratio is 100% (i.e. 1:1) then this cube is ‘perfect' (nasik)
by the Hendricks/Heinz definition. 
The cube in the above example has no
correct broken diagonals but 16 correct pantriagonals.
The panmagic ratio is therefore (48+0+16)/48+96+64) = 64/208 = 30.8%
For the purely pandiagonal cube at the
top of this page is:
ALL pandiagonals and pantriagonals are
correct but NO monagonals are correct.
Magic ratio = 36.8 %. Possible correct lines are 3m^{2} + 6m + 4.
Correct lines are 24 + 4 = 28 out of the 76 possible.
Panmagic ratio = 76.9 %. Possible correct lines are 3m^{2} + 6m^{2}
+ 4m^{2}.
Correct lines are 96 + 64 = 160 out of the 208 possible.
A final example; I obtained this cube
from Guenter Stertenbrink on Jan 18, 2004.
1 26 51 101 76 125 100 25 50 75 32 57 107 82 7 94 19 44 69 119 63 113 88 13 38
17 42 67 117 92 114 89 14 39 64 28 53 103 78 3 96 21 46 71 121 60 110 85 10 35
23 48 73 123 98 106 81 6 31 56 45 70 120 95 20 87 12 37 62 112 54 104 79 4 29
15 40 65 115 90 102 77 2 27 52 49 74 124 99 24 83 8 33 58 108 66 116 91 16 41
9 34 59 109 84 118 93 18 43 68 36 61 111 86 11 80 5 30 55 105 72 122 97 22 47
This cube has 31
correct monagonals (out of 75) so is not magic. All pandiagonals and
pantriagonals are correct.
The magic ratio is therefore (31+30+4)/75+30+4) = 65/109 = 59.6%
The panmagic ratio is therefore (31+150+100)/75+150+100) = 281/325 = 86.5%
To summarize:
Magic ratio equals correct lines/3m^{2} + 6m + 4 A
Diagonal (Boyer/Trump perfect) cube would be 100%
Panmagic ratio equals correct lines/13m^{2 } A Perfect
(Boyer/Trump perfect enhanced) cube would be 100%
These terms will probably not see much use, but are another method of measuring
the feature richness of cubes that are not necessarily perfect.
On January 24, 2004 Aale de Winkel wrote
Dear Friends
Guenter alerted me that the proof for the none existence of normal pandiagonal
squares
of doubly odd order could be generalized to prove the noneexistence of
doubly odd order normal pandiagonal hypercubes of any dimension.
http://www.magichypercubes.com/Encyclopedia/index.html
The crux of
the matter is that in the orthogonal planes, both the monagonal and both
diagonal directions needs to be summing.
In the other planes such as the oblique planes in the cube one of the monagonal
direction
need not to sum, thus {pantriagonal} of doubly odd order (hyper)cubes can exist
as
Abe's order 6 exemplifies.
On Jan. 30, 2004 Guenter Stertenbrink
wrote on :Subject: no pandiagonal cube of order 8p+4
exists.
The ensuing exchange of 10 messages (to Feb. 2) between him and Aale de Winkel
seems to indicate that no normal Pandiagonal or Perfect magic cubes of
order 12 can exist.
See
http://groups.yahoo.com/group/magiccubes/message/31 for a discussion
between Stertenbrink and de Winkel regarding order 12 cubes of this type.
This answers a question on my Cube
Summary page and allows me to fill some holes in the
summary table.
