Magic Cube Update-1

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Introduction

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.

Purely pandiagonal Cubes

Heteromagic cubes

Magic ratios

Pandiagonal impossibility proof

Purely pandiagonal Cubes

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, 2002-2003, pp 29-31

 

 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

Heteromagic cubes

Heterosquare: similar to a magic square except all rows, columns and main diagonals have different sums.
Anti-magic Square: a subset of heterosquares where all rows, columns and main diagonals have consecutive sums.

See Joseph S. Madachy, Mathemaics On Vacation, 1966, pp 101-110.
Or the same material in Joseph S. Madachy, Madachy’s Mathematical Recreations, 1979, pp 101-110.

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 anti-magic cube.
The total would then be 1333, the smallest possible for a real order 3 heterocube.
 


Fig. a

Fig. b

Fig. c

Fig. d

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/3m2 + 6m + 4. That is 3m2 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 3m2 + 2/3m2 + 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 3m2 monagonals, 6m2 pandiagonals, 4m2 pantriagonals. So correct lines/13m2.
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 3m2 + 6m + 4.
Correct lines are 24 + 4 = 28 out of the 76 possible.
Panmagic ratio = 76.9 %. Possible correct lines are 3m2 + 6m2 + 4m2.
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/3m2 + 6m + 4       A Diagonal (Boyer/Trump perfect) cube would be 100%
Panmagic ratio equals correct lines/13m2                   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.

Pandiagonal impossibility proof

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 none-existence 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.

This page was originally posted January 2004
It was last updated March 10, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz