Table Of Contentnews and views
These findings are significant because of from us as birds possess a mechanism for
the growing interest in the mechanisms that representing spatiotemporal events is enor-
underlie the formation of episodic memory. mously important, and could be a big step
Episodic memory is memory for events6, towards understanding how space, time and
each of which occurs in a unique setting of events are represented and remembered in
space and time. As such, it can be distin- the vertebrate brain.
100 YEARS AGO guished from semantic memory, which is Kathryn Jeffery and John O’Keefe are in the 8
Our present knowledge of the theory of memory for facts, or ‘knowledge’. Episodic Department of Anatomy and Developmental
errors receives an interesting addition at memory is disproportionately affected in Biology, University College London, Gower Street,
the hands of M. Charles Lagrange in the some types of amnesia, such as that seen in London WC1E 6BT, UK.
form of a contribution to the Bulletinde Alzheimer’s dementia, and it is thought to e-mails: [email protected]
l’AcadémieroyaledeBelgique(vol. xxxv. depend on the hippocampus7, an important [email protected]
part 6). Without going into details of a structure for spatial memory in both birds 1. Clayton, N. S. & Dickinson, A. Nature395,272–278 (1998).
purely mathematical nature, certain of M. and mammals. To store an episodic memory, 2. Shettleworth, S. J. & Krebs, J. R. J. Exp. Psychol. Anim. Behav.
Lagrange’s conclusions appear to be some method is needed for temporally Process. 8,354–375 (1982).
3. O’Keefe, J. & Nadel, L. TheHippocampusasaCognitiveMap
sufficiently important to be worth noticing. ordering the sequence of happenings that (Clarendon, Oxford, 1978).
In taking the arithmetic mean of a number make up an event. Perhaps episodes can be 4. Sherry, D. F. Anim. Behav. 32,451–464 (1984).
of observations as the most probable temporally ordered in the episodic memory 5. Clayton, N. S. Neuropharmacology37,441–452 (1998).
value of the observed quantity, common of non-humans, as well as humans. In this 6. Tulving, E. OrganizationofMemory(Academic, New York,
1972).
sense suggests that any observations light, the finding that animals as different 7. Vargha-Khadem, F. etal. Science277,376–380 (1997).
differing very widely from the rest should
be left out of count as being purely Mathematics
accidental, and thus likely to vitiate the Magic squares cornered
result. But as it is impossible to draw the
line from theoretical considerations
Martin Gardner
between values retained and values
omitted, any such omission would
necessarily be unjustifiable. This Dame Kathleen Ollerenshaw, one of no evidence that the pattern was known
discrepancy between theory and common England’s national treasures, has before the fourth or fifth century BC.) The
sense is, to a large extent, reconciled by solved a long-standing, extremely Chinese identify it with their familiar yin-
M. Lagrange’s “theory of recurring difficult problem involving the construction yang circle. The even digits, here shown shad-
means.” According to this theory, the and enumeration of a certain type of magic ed, are linked to the dark yin; the Greek cross
weightto be assigned to any observation square. The solution comes in a book* of odd digits is linked to the light yang. For
is inversely proportional to the square of written with David Brée. centuries the lo shu has been used as a charm
the error of the observed value relative to First, some background on magic on jewellery and other objects. Today, large
the most probable value. … The weighted squares, and their hierarchy of perfection. passenger ships often feature the lo shu on
mean is then taken as a second For many centuries, mathematicians — their main deck as a pattern for the game of
approximation to the most probable value. especially those concerned with combina- shuffleboard.
This mean determines a fresh series of torics — have been challenged by magic At order 4, the number of magic squares
weights to be assigned to the squares. These are arrangements of n2dis- jumps to 880. Among them is a special subset
observations by which a new weighted tinct integers in an n2narray such that each of 48 squares called pandiagonal, which have
mean … is found, and so on ... These row, column and main diagonal has the same three amazing properties. This is illustrated
successive means are called by M. sum. The sum is called the magic constant, by the example in Fig. 2, whose constant is 30.
Lagrange “recurring means,” and by their and n is called the square’s order. Traditional First, each broken diagonal also adds up
use the effects of sporadic errors are, to magic squares are made with consecutive to 30. The sequences 0, 3, 15, 12 and 7, 13, 8, 2
all practical purposes, eliminated, since integers starting with 0 or 1. If it starts with are examples. This can be expressed in
the weight assigned to the corresponding 0 it can be changed to a square starting with 1 another way: imagine an endless array of this
observations soon becomes relatively simply by adding 1 to each cell. square placed side-by-side in all directions to
small. No order-2 square is possible. The order- make a wallpaper pattern. Then every 424
From Nature15 September 1898. 3 square (Fig. 1) barely exists. Why? Because square drawn on the pattern will also be a
there are just eight different triplets of dis- pandiagonal magic square — in other words,
50 YEARS AGO tinct digits from 1 to 9 that add up to 15, the every straight line of four numbers will add
In the possession of the Science square’s constant. Each triplet appears as one up to 30. Second, every 222 square on the
Museum, London, there are six beautifully of the square’s eight straight lines of three wallpaper also adds up to 30. Third, along
engraved buttons, classified as diffraction numbers. The pattern is unique — except for every diagonal, any two cells separated by
gratings, which are still regarded as rotations and mirror reflections, which are one cell add up to 15.
masterpieces. They were the work of Sir only trivially different. In general, a magic square is called pan-
John Barton, deputy comptroller of the This little gem of combinatorial number diagonal if all its broken diagonals add up to
Royal Mint in the early part of the theory was called the lo shu in ancient China, the magic constant. Such squares can be con-
nineteenth century, about whom little is meaning ‘Lo River writing’. Legend has it that structed of any odd order above three and of
known personally, but who must have in the 23rd century BC, a mythical King Yu saw any order that is a multiple of four. If a pan-
been an ingenious inventor and capable the pattern on the back of a sacred turtle in the diagonal square also has similar properties to
engineer, for in 1806 he invented a River Lo. (Modern historians, however, find the order-4 pandiagonals, it is called ‘most
differential screw measuring instrument perfect’: for example, the most-perfect
capable of measuring 10–5inch. *Most-Perfect Pandiagonal Magic Squares: Their Construction and order-8 square in Fig. 3 has a magic constant
From Nature18 September 1948. Enumeration. Due for publication on 1 October by The Institute of of 252, and its 222 sub-squares add up to
Mathematics and its Applications, Catherine Richards House,
16 Nelson Street, Southend-on-Sea, Essex SS11EF, UK. 126, and any two numbers that are n/2 = 4
216 NATURE|VOL 395|17 SEPTEMBER 1998
Nature © Macmillan Publishers Ltd 1998
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0 62 2 60 11 53 9 55
2 7 6
15 49 13 51 4 58 6 56
8
9 5 1
16 46 18 44 27 37 25 39
4 3 8
31 33 29 35 20 42 22 40
Figure 1Lo shu, the only 323 magic square.
52 10 54 8 63 1 61 3
0 13 6 11
59 5 57 7 48 14 50 12
7 10 1 12
36 26 38 24 47 17 45 19
9 4 15 2
43 21 41 23 32 30 34 28
14 3 8 5
Figure 3A most-perfect square of order eight. Its rows, columns, diagonals and broken diagonals all
add up to 252, and all 222 sub-squares add up to 126. Kathleen Ollerenshaw and David Brée have
Figure 2A 424 magic square that is found a way to construct most-perfect pandiagonal magic squares of any order.
pandiagonal — the broken diagonals also add up
to its magic constant of 30. because she was 85 when she and Brée finally to experience. That this should have been
proved the conjectures she had earlier made. afforded to someone who had, with a few
cells apart add up to n211 = 63. In her own words, “The manner in which each exceptions, been out of active mathematics
Although all order-4 pandiagonals have successive application of the properties of the research for over 40 years will, I hope, encour-
been known to be most perfect for three binomial coefficients that characterize the age others. The delight of discovery is not a
centuries, little has been known about most- Pascal triangle led to the solution will always privilege reserved solely for the young.”
perfect squares of higher orders. There was remain one of the most magical mathematical Martin Gardner is at 3001 Chestnut Road,
no method of constructing them all, or even revelations that I have been fortunate enough Hendersonville, North Carolina 28792, USA.
of determining the number of squares of a
given order. Bacterial infection
These questions are finally settled by For whom the bell tolls
Kathleen Ollerenshaw and David Brée in
their new book. The authors have devised
Craig Gerard
a method for constructing all the most-
perfect squares of any order, and a way of
calculating their number. “Nowthisbelltollingsoftlyforanother, saysto Gram discovered that the outer membranes
Unlike the ordinary pandiagonals, there me, Thoumustdie.” of bacteria may be classified as Gram nega-
are no most-perfect squares with odd order, tive or Gram positive, based on the ability of
so the only possible orders are multiples of The 1623 Meditation 17 by the English acetone/alcohol to decolorize cells stained
four. At each leap in order, the number of metaphysical poet John Donne was with Gram’s iodine and crystal violet. Gram-
essentially different most-perfect squares probably inspired by the church bells negative organisms contain a complex
increases rapidly: from 48 squares of order that tolled to announce death by the plague. glycolipid in their outer membranes, known
four, to 368,640 of order eight, to 2.22953 Death in this and many other infectious as lipopolysaccharide (LPS) or endotoxin. In
21010 of order 12. When you reach order 36, diseases typically follows septic shock, often picogram quantities, this substance can
the number is 2.7675421044 — around a caused by so-called Gram-negative bacteria. induce mammalian white blood cells to
thousand times the number of pico-pico- The path that leads from these organisms to secrete cytokines which, if left unchecked,
seconds since the Big Bang. septic shock has been under investigation for can lead to fever, coagulation defects, lung
This solution of one of the most frustrat- over a century, and, on page 284 of this issue, dysfunction, kidney failure and circulatory
ing problems in magic-square theory is an Yang etal.1report that a cell-surface receptor, collapse.
achievement that would have been remark- Toll-like receptor 2 (TLR2), may be part of How are trace quantities of this glycolipid
able for a mathematician of any age. In Dame the long-awaited solution to the puzzle. recognized, and the signals necessary for the
Kathleen’s case it is even more remarkable, In 1884, the Danish physician Christian production of toxic cytokines transduced?
NATURE|VOL 395|17 SEPTEMBER 1998 217
Nature © Macmillan Publishers Ltd 1998
