Computing, Information Systems and Mathematics 87 Rodenhurst Road
South Bank University London, SW4 8AF, England
London, SE1 0AA, England Tel/fax: 0181-674 3676
Tel: 0171-815 7411
Fax: 0171-815 7499
QUERIES ON RUSSIAN SOURCES IN RECREATIONAL MATHEMATICS
by David Singmaster
Last amended on 4 agosto 1996.
I am working on a history of recreational mathematics. I have found a
few topics which have Russian connections which I am gathering here for
convenience in correspondence. Separate letters deal with Oriental
questions and with Middle Eastern questions, i.e. Egyptian, Babylonian,
Indian, Arabic, Persian and Turkish.
There is a general question about some problems which are known from
China and Europe, but for which there are no Indian and Arabic sources known
to me; that is, the apparent transmission from the Orient has a gap in it.
For some topics, the usual transmission seems inadequate to explain the
For example, the cistern problem appears almost simultaneously in China
and Alexandria. Heron's work gives two problems, both incorrectly solved,
while the Chiu Chang Suan Ching (Jiu Zhang Suan Shu) gives a clear example
with 5 pipes and several related problems.
As another example, after 5C to 7C China, the Hundred Fowls problem is
first known to appear in Europe, Egypt and India almost simultaneously in
the late 9C. This is faster than any other example of transmission that we
know of. Further, the problem is well developed in all three places,
especially in Egypt where Abu Kamil gives a problem with five varieties of
bird and says there are 2676 solutions.
Tait's Counter Puzzle and the Chinese Rings are further examples where
there is no sign of the usual transmission through India and the Arabs.
Anatoli Kalinin says that the Chinese Rings are a old folk puzzle called
[Meleda], especially popular among the Kalmyks near the Caspian Sea, where
it is called - [Naran-shina] (stirrup ring toy). The name is derived
from a verb which is no longer in Russian.
Tangrams and the Josephus Problem are examples where there is no sign
of the usual transmission, and the transmission may well have gone the other
way. Kalinin informs me that Tangrams were unknown in Russia before the
Richter puzzles of the late 19C.
I wonder if there was some transmission over the Silk Road or other
central Asian trade routes which could have carried some information
directly between the Orient and Europe, bypassing the Indians and Arabs. If
so, there may be some evidence for this in the folk cultures of the central
Asians and Russians. I can find nothing about this and would be delighted
to hear from anyone who does know about this. Are there any collections of
early folk mathematics in the USSR?
The recreational questions are discussed more fully in my Sources or
the Queries thereto. I am currently working on the seventh preliminary
edition of this.
I am particularly interested in the works of Yakov (Jacob) Isidorovich
Perelman (. . epea). All of the copies of his works that I have seen are
quite recent editions (1950s onward), but he began publishing in 1913 and
died in 1942. A number of the problems in his books are quite interesting
as they seem to originate in this century. If Perelman published them in
his early books, he might be the originator or first publisher of them. As
will be seen below, several of these problems lead to questions of priority
between Perelman and H. E. Dudeney in the period 1915-1930. Consequently, I
am very keen to obtain information about Perelman's books (especially the
original dates) or even to obtain copies of them. I can struggle through
Russian, but early translations into English, German, French or Italian (or
even Spanish) might be more useful.
An especially interesting point arises in Fun with Maths and Physics
[FMP], MIR, Moscow, 1984. This is a collection compiled from Perelman's
works by I. I. Prusakov. On p. 194, we find the following.
Many experts in Russian literature don't suspect that the poet V.
G. Benediktov (1807-1873) was also the author of the first collection
of mathematical brain-twisters in the language. The collection wasn't
printed and remained in a manuscript form to be found only in 1924. I
had the opportunity to get acquainted with the manuscript and even
established, based on one of the problems, the year it was compiled,
namely 1869 (the manuscript wasn't dated).
Perelman then gives one of the problems, a version of 'selling different
amounts at the same prices but making the same'. This problem derives from
India, c850, and the exact same numbers already occur in 14C Europe. It is
not clear how useful this collection would be - it might contain some
problems which Benediktov attributes to earlier authors. I would like to
know if this collection has ever been printed. If not, where might the
manuscript be? Prof. Boltyanski suggested the Lenin(?) Library in
Petersburg. Would it be possible to get a copy of the manuscript? Would
any Russian speaker be interested in producing a rough translation or
summary of it in English? Kalinin has made inquiries in Moscow and St.
Petersburg but has not located this manuscript.
I have three versions of Figures for Fun [FF] (a aeaa), all
translated by G. Ivanov-Mumjiev:
Foreign Languages Publishing House, Moscow, 1957;
3rd ed., MIR, 1979;
as the first part of Mathematics can be Fun, MIR, 1985, apparently
based on the 2nd Russian ed. of 1970, translated 1973.
(Schaaf's Bibliography of Recreational Mathematics, vol. 1, p. 9,
refers to Recreational Arithmetic, 6th ed., Leningrad, 1935. Is this the
same book as translated as Figures for Fun??)
The 2nd and 3rd editions seem to be almost identical, with minor
changes to the English wording. The version in Mathematics can be Fun is on
larger pages, but with more diagrams, so the pagination is almost the same
as in the 3rd ed. - the 3rd ed. ends on p. 183, while the other ends on p.
186. The 1957 edition has 120 problems, while the 2nd and 3rd editions
omit 3 problems and add 6 others giving 123 problems. In reexamining
this, I was surprised to find that I had forgotten that the reference to
Benediktov and the above problem appear as Problem 120, pp. 141-142 & 150-
120. The Benediktov Problem. -- A great many lovers of Russian
literature probably do not even suspect that the poet Benediktov (1807-
1873) collected and compiled a whole volume of mathematical conundrums.
Had it been published, it would have been the first Russian book of
this type. But it never was and the manuscript was only found in 1924.
I had the good fortune to study the manuscript and even established -
by solving one of the brainteasers contained therein - that the
collection was completed in 1869 (the manuscript itself was not dated).
Perelman then gives the same problem as mentioned before, but in much more
detail. My memory was that Benediktov was a collector rather than an author
and probably I was recalling the phrasing here.
QUERIES ON PERELMAN PROBLEMS.
I will cite the above mentioned works as FMP with page numbers and FF
1957 and FF 1979 with problem numbers. I would be particularly grateful for
the original dates of these items.
PIGEONHOLE RECREATIONS. FMP, p. 277: Socks and gloves. This is my
earliest example of a pigeonhole problem with handed objects. I have
another example from 1943 with shoes and socks.
SPIDER & FLY PROBLEMS. Dudeney and Loyd give several versions of the
problem in a rectangular room. In 1926 Dudeney gives a version on a
cylindrical glass with the source and the target on opposite sides. FF
1957, prob. 68 = FF 1979, prob. 73 is a cylindrical version with
different numbers than Dudeney. I don't have any other early versions of
the cylindrical problem.
SILHOUETTE AND VIEWING PUZZLES. The best known, though not the
earliest, example is to find an object which will plug holes in the shape of
a circle, a square and a triangle. FMP, p. 340 = FF 1957, probs. 70 & 71
= FF 1979, probs. 74 & 75 give problems with circle, square, cross and
with triangle, square, tee. I haven't seen earlier versions of these
NETS OF POLYHEDRA. In 1926, Dudeney gives the problem of finding all
ways of unfolding a cube into a flat network and he correctly finds 11
ways. In FMP, p. 179: To develop a cube, Perelman asks the same question,
but his answer says there are 10 solutions, but two can be turned upside
down, increasing the total to 12. The fact that Perelman has the wrong
answer has two interpretations. First, he had created the problem and
failed to get the right answer. Second, he had vague information about
Dudeney's problem and answer and was misled by it. Both cases are possible.
WHAT COLOUR WAS THE BEAR? These are problems involving travel near a
pole - e.g. man goes 10 miles south, then 10 miles east, then 10 miles
north to return to his starting point. Simpler versions - e.g. man starts
at the North Pole, goes 40 miles south and 30 miles east, how far is he
from his starting point? - occur in 1907, 1925, 1930s, but those with
triangular circuits occur in the 1940s. FF 1957, prob. 6: A dirigible's
flight = FF 1979, prob. 7: A helicopter's flight concerns a square circuit
and notes that going 500 km N, E, S, W doesn't get you back to where you
started. This may be the earliest version of the problem with a circuit,
though he doesn't ask the more interesting question of where you could be if
the square circuit does return to the origin. My next example with a square
circuit does ask this in 1958/59 but fails to get the complete solution,
which I have recently found.
CUTTING UP IN FEWEST CUTS. FF 1979, prob. 122: Sectioning a cube and
prob. 123: More sectioning, ask for the minimum number of cuts to divide a
3 x 3 x 3 cube into unit cubes and a 8 x 8 chessboard into unit squares.
Surprisingly, I don't recall seeing any other versions of these problems
before relatively recent times. These are not in the 1957 ed. - they are
two of the problems added in the 2nd ed.
HUNDRED FOWLS. FF 1957, prob. 37 = FF 1979, prob. 40: Hundred rubles
for five. Using 50, 20 & 5 kopeck coins, it is impossible to make 5, 3
or 2 rubles in 20 coins. Perelman describes this as a magician's come-
on. I don't recall any other versions of the problem which depend on there
being no solutions.
WATER IN WINE VERSUS WINE IN WATER. This has been a popular problem
since the late 19C. FMP has it on p. 215.
SKELETON ARITHMETIC. Again, these are popular problems, first
appearing c1900. FMP, p. 256, has the skeleton division of 11268996 by
124 yielding 90879 with only the 7 of the quotient given. This has 11
PAN_DIGITAL PRODUCTS. FF 1957, prob. 45 = FF 1979, prob. 48: Tricky
multiplication finds all the pan-digital (without 0) products. There are
2 of the form A x BCDE = FGHI and 7 of the form AB x CDE = FGHI.
Loyd gives a 10-digital problem involving the smallest result, but my next
version of the problem is 1934 when all 9 solutions are given. Hence
Perelman may be the earliest source for this, though I suspect some 19C
versions may turn up or it may appear in Dudeney.
FLOATING BODY PROBLEMS. Surprisingly, I haven't noticed any of these
prior to a possible Dudeney in the Daily Mail in 1905. Hence FMP, pp. 114
& 199 are among the earliest popular versions I know of. P. 114 asks
whether a bucket full of water is heavier or lighter than a similar bucket
with a floating block of wood in it. This is conceptually the same as
asking whether a glass full of water with floating ice will overflow when
the ice melts, which is the 1905 problem. FMP, p. 199 has a balanced
balance with iron versus stone - what happens when it is submerged?
NIM GAMES. Nim is first described by C. L. Bouton in 1902. He claimed
that it was widely played in America and was called Fan-Tan by the Chinese.
He later admitted that the identification with Fan-Tan was wrong. He later
admitted that he coined the word Nim from the German word 'nimm', the
imperative of take. Interestingly, Luo Jianjin and Siu Man-Keung tell me
there is a Chinese character, nian, pronounced 'nim' in Cantonese, which
means to pick up or take. However, there seems to be no historic connection
between these words.
Wythoff's Nim, described by Wythoff in 1907, has two piles and one can
take any amount from one pile or the same amount from both piles. A. P.
Domoryad's Russian book on mathematical games says this 'is the Chinese
national game of TSYANSHIDZI ("picking stones")'. I have only seen this in
English translation, so the original Chinese word is hard to determine.
Prof. Siu could not work out what the Chinese was. Winning Ways says it is
called Chinese Nim or Tsyan-shizi. Is there any evidence for any games of
this kind in China or eastern Russia, etc.?
FOX AND GEESE, ETC. These are board games with asymmetric forces. Fox
and Geese is supposed to be medieval, even 1st millenium, but Murray's
History of Chess cites a North Asian version of Bouge-Skodra (Boar's
Chess). Are there other early forms in this area?
TANGRAMS. These are traditionally associated with China of several
thousand years ago, but the earliest books are from the early 19C and appear
in the west and in China at about the same time. Indeed the word 'tangram'
appears to be a 19C American invention. A slightly different form of the
game appears in Japan by 1742 and there is an Utamaro woodcut of 1780
showing some form of the game (not yet seen by me). Needham says there are
some early Chinese books, and van der Waals' historical chapter in Elffers'
book Tangram cites a number.
I would be interested in seeing antique versions of the game itself.
The only historical antecedent is the 'Loculus of Archimedes', a 14 piece
puzzle known from about -3C to 6C in the Greek world. Could it have
travelled to China over the Silk Road? I found a plastic version of the
Loculus on sale in Xian, made in Liaoning province.
DEAD DOGS AND TRICK PONIES. There is a pattern of overlapping bodies
and heads so that the same head can be viewed as part of several bodies.
Examples are known from medieval Persia (Rza Abbasi (1587-1628)) and Edo
period Japan (17C - 19C). There are said to be other examples from India
and/or China. I would like to know of early examples. Perhaps some exist
in southern USSR.
THE JOSEPHUS OR SURVIVOR PROBLEM. This is the problem of counting out
every k-th person from a circle. It was a common medieval European problem
in the form where half of a group is to be eliminated. The usual form
involved 15 Christians and 15 Turks on a ship in a storm. The captain
announces that half of the passengers must go overboard and one of them says
that everyone should get in a circle and be counted off by 9s. Cardan
suggested that this might be the way in which Josephus survived. It appears
in the Japanese literature as early as 1627, with 15 children and 15
stepchildren counted by 10s, but with one child (the 15th) skipped,
until only one is left. Ahrens cites some indications that it may go back
to the 11C in Japan and believes that the problem arose independently in
Japan. Is there any Chinese or central Asian material on this? I have
recently seen an article which claims an Irish origin of the problem, c800,
and which gives early medieval forms called the Ludus Sancti Petri.
Murray's History of Chess mentions 10 diagrams of this in an Arabic chess
MS of c1370, possibly referring to a c1350 work. Murray asserts the problem
is of Arabic origin.
last Web revision:December 22, 1998