Bao Qi-shou lived in the latter part of the
nineteenth century. He was an unsuccessful businessman, a soldier, a
satirical poem writer, and finally a student of astronomy and
mathematics and a teacher of mathematics to gifted students. He
self-published a book in about 1880 called Bi-Nai Mountain Hut
Records which included an article Zen Bu Suan Fa Hun Yuan Tu
(Augmented Solid and Spherical Mathematical Diagrams)
It is that article (which appeared as volume 2 of
Bao’s book) that is the subject of this web page. The 24 pages of
that volume are an appendix to Lih (1986 ) and are shown on this
I first became aware of Bao’s work when I received
an email from Tung Hsu in August of 2010. He mentioned that his
mothers Grandfather (Bao Qi-Shou) had found magic perimeter
solutions for the five platonic solids in the late 1800’s. He later
sent me some supporting documents , , and several photographs.
I contacted Dr. Ko-Wei Lih and received helpful suggestions and
material from him.
- Lih, Ko-Wei, Bao Qi-shou and his Polyhedral Hun Yaun Yu. 1993,
Cheng-hung Lin and Daiwie Fu (Eds.), Philosophy and Conceptual History
of Science in Taiwan, pps 209-220.
- Lih, Ko-Wei, 論保其壽的渾圓圖, 1986, 22 pages, All text in this paper is
Chinese, but it contains images of all pages from Vol. 2 of Bao.
I am indebted to Dr. Ko-Wei Lih for providing me with both of these
papers. Without them this web page could not have been written.
- Contains short English language descriptions of most of the pages in
Bao’s volume 2 along with translations of many diagrams.
- Contains images of all 24 pages of Bao’s volume 2.
More elaborate descriptions of these two papers, as well as other
references, are located at the end of this page.
And , of course, I greatly appreciate Tung Hsu bringing this subject to
my attention. Tung is interested in solving the problem of assigning
numbers 1 to 60 to the vertices of the buckyball,
so faces are magic. Is this possible? Suggestions
Above; Bao's two volumes
Below; Cover of volume 2
My text relies heavily on the brief notes in Lih (1993)
, as I found that the Web
translation services did not produce useful results. I found interesting
the fact that Bao used different methods (3-dimensional line drawings,
planar graphs, and diagrammatic illustrations) to illustrate his figures!
|These are the only pages in Bao's volume 2 that do
not contain illustrations.
They are pages 1, 2, and 24.
I include them for the benefit of any
possible Chinese language readers.
[a] Lih (1993, ) quoting
an earlier research paper of 1935.
Types of perimeter magic:
Vertex V-magic The numbers at the vertices of each surface
(face) sum to a constant. If V-magic, then F-magic (if no numbers are on
Edge E-magic the numbers along each edge of the graph sum
to a constant. If E-magic then F-magic, but F-magic does not require
E-magic. i.e. all edges may not sum the same
Face F-magic the sum of all the lines around the perimeter
of the face equals a constant.
These designations were first proposed by Charles Trigg in
numerous articles on the subject written in the 1970’s.
 Although Lih uses a different
terminology in his papers, I have chosen to use Trigg’s, so as to be
consistent with my other pages.
The Platonic Solids:
V=8, E=12, F=6
V=4, E=6, F=4
V=6, E=12, F=8
V=12, E=30, F=20
V=20, E=30, F=12
Two other shapes (not platonic) will be illustrated and
The icosadodecahedron consists of 12
pentagonal faces and 20 triangular faces (the six-banded sphere).
The Buckyball is a round structure consisting
of 12 pentagons and 20 hexagons with a total of 60 vertices (carbon
Page 4 --This 3 dimensional figure shows numbers 1 to 8 arranged
in a vertex face magic configuration such that each face sums to 18.
Page 5 --This figure shows the edges labelled to form a face
magic cube. Numbers used are 1 to 12. Each face sums to 26.
Page 6 -- This fully perimeter magic cube is obtained by
combining the previous edge and vertex magic figures. In this case the
range of numbers from 1 to 12 have been increased by the constant 8 to
provide a range of unique numbers from 1 to 20. Each face sums to 76.
The lower range of numbers could have been increased in value instead (by
the constant 12). The face constant then would have been a larger number.
Note that in Boa’s original diagram, the label 2 on the top back edge
should have been a 20.
Page 7 --This drawing illustrates the principle that
complementary pairs may be inserted on the edges of a vertex face magic
cube to produce an order-4 perimeter magic cube. Here the vertices are the
series from 1 to 8. The edges use the numbers from 9 to 32 in complement
pairs each summing to 41. Each face sums to the constant 182.
See my previously posted reference to this figure here. There C. Pickover
improved on Bao’s figure by rearranging the interior edge numbers so each
edge sums to 50 (with the face constant still 182).
|Page 8 --Uses the numbers 1 to 12 in complement
pairs to label the edges of the tetrahedron, so the faces all sum to 39.
Page 9 --This planar net figure uses numbers 1 to 4 placed on the
vertices and the faces labelled with numbers 5 to 8. The result is face
magic with S=14. Strictly speaking, this is not perimeter magic.
Page 10 --This 3-dimensional drawing of a tetrahedron shows the
same arrangement of the numbers 1 to 8.
Page 11 --This planar net figure uses numbers 1 to 16 and is
face magic with S=72. My translation is a 3-dimensional drawing.
Page 12 --shows the figures of pages 9 and 11 but labelled with
the complements of those numbers. i.e. 8 replaces 1, 7 replaces 2, etc. in
the first figure; 16 replaces 1, 15 replaces 2, etc. in the second figure.
|In the text on these pages Bao explains how, by trial and
error, it is easy to see that there is no way to produce constant face
sums with consecutively labelled edges. [1,
To the right is the translation of Bao's page 12 images.
Page 3 --Two representations based on the same consecutive
vertex labelling. [1, pp 214-5]
I decided not to attempt a translation of these figures.
Page 13 --Bao specifically discusses the method of
switching to complements. i.e. each number switched to the difference
between it and 19. This results in all faces summing to 66. Lih
[1 p.314] comments that Bao also
mentioned substituting numbers 1 to 12 with 7 to 18, and 13 to 18 changed
to 1 to 6. But this didn't wok for me! It results in an octahedron with 5
faces summing to 48, two summing to 66, and one equal to 84!
As shown, this is an order-3 perimeter E-magic octahedron using numbers 1
to 18 with S = 48.
Page 14 --Edge labelling of complement pairs. This
figure uses complement pairs of the numbers 19 to 42. So each edge sums to
61 making the figure edge magic with each face sum 183.As each pair sums
to the constant 61, the pairs may be placed on any edge of the previous
figure to produce an order-5 perimeter magic octahedron.
Page 15 --The consecutive numbers 1 to 72 are placed so that the
faces of all 20 triangles sum to 279. Bao’s page 15 has 2 numbers
missing. 22 and 50 should appear between 9 and 15. This is corrected
in Lih’s translation drawing.
Lih's translation of pae 15 [2, fig.
Page 16 --Lih [1, p 215]
mentions only that Bao has one mistake on this page. The number 24
between 70 and 2 should be 80.
He mentions nothing else about what this diagram represented.
|I find it interesting that Bao used a variety of
illustrations to demonstrate his perimeter magic figures.
For the dodecahedron he uses a planar net drawing on page 17, then on
pages 18 and 19 he uses 3 dimensional illustrations rather then line
drawings as on some previous pages.
Page 17 -- A planar graph
representation with vertices consecutively labelled from 1 to 20, Faces
are anti-magic. The 5 vertex numbers around each pentagon face sum to
consecutive numbers from 47 to 58. So this figure is V-antimagic.
Page 18 -- Presumably Bao used this illustration to emphasize that
it is impossible to have constant face sums using consecutive vertex
Page 19 -- My version of Bao's 3-dimensional illustration shows
the numbers on the back side upside down, as does Bao's.
This figure is Face magic: S = 230. It uses consecutive numbers from 1 to
The vertices use consecutive numbers from 1 to 20 (same as bao-17).
When the vertex numbers (1 to 20) are removed and the edge numbers reduced
by 20, the figure is E-antimagic, with sums equal to the consecutive
numbers from 72 to 83.
Page 20 -- Lih 
p.218 says there is no indication of how Bao arrived at this
complicated configuration. Numbers 1 through 90 are assigned so that six
numbers around each of the 20 triangles add up to 228 and the 10 numbers
around each of the 12 pentagons add up to 380. The vertices are labelled
with the consecutive numbers 1 to 30. (I show a translation below.)
Page 21 -- This figure is obtained by adding 60 to
each vertex and subtracting 30 from each edge of the page 20 figure. The
result is then reflected vertically. 
Page 22 -- This is the complementary numbering of
page 20.  p.218
Page 23 -- This image is of a ball wrapped with 6
intertwined paper bands. If these are regarded as edges, we have a
combined Icosahedron and Dodecahedron. Thus it has 20 triangle faces and
12 pentagon faces. Below is shown Tung Hsu’s model of this figure with the
numbers from the planar diagram of page 20.
Page 24 -- (Shown at the top of this web page) is
translated by Lih  as follows:
Further diagrams can be constructed by the
previous methods. It becomes easier to make variations when more and more
numbers are used. However, it is pointless to give these illustrations
when all methods have been exhaustively demonstrated.
|The Buckyball is a round structure consisting of 12
pentagons and 20 hexagons with a total of 60 vertices (carbon
atoms). The C60 carbon molecule, which is harder then diamond, has a
carbon atom at each vertex of this structure.
If this structure
looks familiar, it probably is. The most popular design of soccer
ball is the 32-panel.
It uses this same 12 pentagon and 20 hexagon pattern.
Tung Hsu, Bao's Great Granson, is interested in the
problem of assigning a number at each vertex of this structure.
Is it possible to assign the
consecutive numbers from 1 to 60 in such a manner that all 12 pentagon
faces sum to a constant, and all 20 hexagon faces sum to a constant?
The figure on the right is a 2-D version. If working on
this problem, it may be helpful to print out an enlarged version of this
image to help in assigning numbers.
Lih, Ko-Wei, Bao Qi-shou and his Polyhedral Hun Yaun Yu.
1993, Cheng-hung Lin and Daiwie Fu (Eds.), Philosophy and Conceptual
History of Science in Taiwan, pps 209-220. 12 page English paper on Bao
It contains a short biography of Bao Qi-Shou and a very brief history of
magic squares in China. The author then presents a brief description of
most of the pages of Bao’s book along with some translated diagrams . He
also refers to an earlier translation of Bao’s work by Li Yan in his 1935
article A Study of Chinese Mathematicians’ Magic Squares.
It is available for downloading from this site as
Lih, Ko-Wei, 論保其壽的渾圓圖, 1986, 22 pages, All text in this
paper is Chinese, but it contains images of all pages from Vol. 2 of Bao.
It is available at
http://episte.math.ntu.edu.tw/articles/ar/ar_li031207_1/page2.html . I
have used these images (with permission) on my page.
Lih, Ko-Wei, On Magic and Consecutive Labelling of
plane graphs, 1980, 33 pages, , Utilitas Mathematica, 24(1983),
p.165-p.197. No mention of Bao, but lots of magic labelled graphs.
I have not used material from this paper, but include the reference for
It is available for downloading from this site as
Tung Hsu, "An introduction to Bao Qi-shou and his 'Suan Fa
Hun Yuan'" (in Chinese), Bulletin of the Association for the History of
Science, No. 13, 2009, pp. 85 - 90.
It contains an image of the two volumes of Bao’s essay.
Charles W. Trigg, Second Order Perimeter-magic and
Perimeter Anti-magic Cubes, Mathematics Magazine, 47(3), 1974, pp
See more information and other references on my other 3
Many thanks again to Professor Tung Hsu and Dr. Ko-Wei Lih!
Tung Hsu is a recently retired physics professor now living in the United
Dr. Ko-Wei Lih, is a mathematician with Institute of Mathematics, Academia