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Mathematics and Cultures Across theChessboard: The Wheat and ChessboardProblem

Alberto Bardi

In number did outmillion the accountReduplicate upon the chequer’d board(Dante Alighieri, Divine Comedy,Paradise 28, 93; transl. Henry Francis Cary)

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Mathematics and the Invention of Chess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Mathematics and the Origins of Chess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Geometric Progressions and Chess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Arabic Sources on the Computation 264 − 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Greek Sources on the Computation 264 − 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Western Sources on the Computation 264 − 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Abstract

This chapter is an introduction to the wheat and chessboard problem and theinterplay between chess and mathematics in the several authors and culturalcontexts that have inherited, faced, and modified this problem, ranging from

A. Bardi (�)Polonsky Academy for Advanced Study in the Humanities and Social Sciences, Van LeerJerusalem Institute, Jerusalem, Israele-mail: [email protected]

© Springer Nature Switzerland AG 2019B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences,https://doi.org/10.1007/978-3-319-70658-0_82-1

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Antiquity to the Renaissance, considering a selection of Arabic, Persian, Greek,Latin, Italian, Romance, and Germanic sources.

KeywordsChess · Chessboard · Geometric progression of 2 · Mathematics · Wheat andchessboard problem

Introduction

Walking through Union Square or Washington Square in New York City, one willfind several street chess players. They sit by round tables set up with chessboards,ready to play against passersby for just a few dollars. Such recreational and socialactivity represents only one aspect of the wide range of features and capabilitiesencompassed by the chessboard. The chessboard is indeed an attractive andfascinating object: not only is it the basic material field of an internationally famousgame of strategy and intelligence, known worldwide under the name chess, withofficial rules and an official federation (FIDE 2000); but it has also inspired tasks,puzzles, and mathematical challenges. For instance, the legend of the origins ofthe game of chess is related to a calculation on the chessboard, which is knownto mathematicians to this day as the wheat and chessboard problem. The legendhas been transmitted through several redactions, in different languages, and fromdifferent cultural traditions. At this introductory stage, it is helpful to remindourselves of the standard features of this legend. The ruler of India discovered thegame of chess at his court and was fascinated by this original idea and the multitudeof possible combinations for placing the pieces on the board. He learned that theinventor of this game was at his court, so he summoned him and congratulated himon his wonderful idea. As reward, the ruler promised the inventor to grant any wishhe might express. The inventor’s wish was for a quantity of wheat grains to becounted on a chessboard: one grain for the first field of the board, two grains forthe second, and so on, so that each field contained twice as many grains as the onepreceding it. The ruler ordered his men to grant the inventor his wish. The followingday, the court mathematicians told their lord that such wish could not be fulfilled,for it would require more wheat than the world could supply. The quantity of grainsrequested by the inventor of the chess is equal to

1 + 2 + 22 + · · · + 263 = 264 − 1 = 18446744073709551615.

At first glance, the game of chess might appear to us as a mere street game,but upon closer inspection, it proves to be more than simply a recreational activity:the unexpected conclusion of the tale about the invention of chess shows the greatmathematical possibilities hidden in the game. For instance, particular mathematicalproperties of the chessboard lead to its so-called magic squares. A magic square oforder n is a squared table of n × n dimensions, which contain the integers from 1 to

Mathematics and Cultures Across the Chessboard: The Wheat. . . 3

n2. The numbers contained in the table are positioned in such a manner that the sumof the numbers of each row, each column, and the two major diagonals is equal toone other. The fascination of this mathematical object went beyond pure mathemat-ics. For example, the German Renaissance mensch Albrecht Dürer was so impressedby the magic squares that he reproduced one in his famous engraving Melancholy.(We use “mensch” to avoid sexist connotations, as suggested by Sriraman 2009, 75.)By means of experimenting with a chessboard and chess pieces, first-rate modernmathematicians such as Luca Pacioli (c. 1445–1517), Leonhard Euler (1707–1783),and Carl Friedrich Gauss (1777–1855) formulated solutions to problems of recre-ational mathematics related to the chessboard. Pacioli composed a treatise on chessproblems at the beginning of the sixteenth century, entitled De Ludo Schacorum(“On the Game of Chess”). The Knight’s Tour Problem, which deals with the searchfor a knight’s tour in chessboards of different sizes, was of particular interest toLeonhard Euler: he provided remarkable solutions to this for an 8 × 8 chessboardin 1759 (Watkins 2004, 3–8). Carl Friedrich Gauss worked brilliantly on the so-called 8-queens problem (Watkins 2004, 164–169), which focuses on how to placeeight queens on an 8 × 8 chessboard so that none of them attacks any of the others,elaborating a new arithmetical solution. In the twentieth century, the game of chesswas a source of inspiration for new branches of mathematics, namely, set theory andgame theory (Zermelo 1913; Von Neumann and Morgenstern 1944). For instance,chess is the field of application for the theory of so-called zero-sum two-persongames (Von Neumann and Morgenstern 1944, 124–125), and it sparks discussionson finiteness and the applicability of the minimax theorem (Ewerhart 2002).

These are only a few examples to indicate how mathematics is intertwinedwith the chessboard and the game of chess. For a more in-depth consideration ofthese interconnections, one might consult Gik (1986), Petković (1997), and Watkins(2004). Moreover, the influence of chess is not confined to the mathematics itinspires, for its long history involves strategical, moral, civic, poetic, and literaryissues, several aspects of which are outlined masterfully in the groundbreaking workon the history of chess by Harold J. Murray (1913, but see also earlier works on thehistory of chess, to which Murray is indebted: Hyde 1694; Van der Linde 1874,1881).

The current chapter is an introduction to the entanglements between chess andmathematics in relation to the wheat and chessboard problem. Due to the recentdiscovery of unpublished sources on the wheat and chessboard problem, it is worthreconsidering its main cultural and mathematical aspects, their developments, andtheir diachronic interplay.

Mathematics and the Invention of Chess

The history of chess – to make a very long story short – is believed to have its originin India, between the third and the sixth century BCE, during the Gupta Empire.From India it was introduced to Persia, during the Sassanid Empire, and then entered

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the Islamic world after the Islamic conquest of Persia. In the seventh century CE, itpossibly spread to China too. It was transmitted to the West through exchanges withthe Arabs, who had settled in Northern Africa, Spain, Southern France, Sicily, andSouthern Italy before the tenth century.

Mathematics and chess are connected in the earliest sources on the history of thegame. As far as we can reconstruct from the extant written sources, the legend of theinvention of chess indeed provides a version of the wheat and chessboard problemwithin its narrative (Wiedemann 1908; Ruska 1916; Wieber 1972, 88–102; Tropfke1980, 630–632).

The ninth-century Arab historian Ibn Wad. ı̄h. provides one of the mostancient written accounts of the tale of the wheat and chessboard problem withinthe wider context of a story about the legend of the invention of chess. The storygoes as follows (Wiedemann 1908, 43–44; Ruska 1916, 280–281; Tropfke 1980,631). An Indian philosopher named Qaflān developed a game (chess) which wouldfunction as a war simulation and protected the soul from participating concretelyin cruelty and bloodshed. The queen summoned him and praised his wisdom andinvention and promised him any reward he wished. Qaflān asked for a quantity ofcorn grains to be counted on a chessboard: one grain for the first field of the board,two grains for the second, and so on, so that each field contained twice as manygrains as the one preceding it. The grain stores of that reign were not enough togrant Qaflān’s wish, but he said he did not need such a quantity, adding that for himjust a small piece of lawn was enough. The queen then asked him about the numberof the grain corns he wished. He therefore explained the computation: the first rowhas 255, the second one 32768, the third 8388608, the fourth 2147483648, the fifth549755813888, the sixth 140737488355323, the seventh 36028797018963968, andthe eighth 9223372036854975808. The total of the corn grains on the chessboard isequal to 18446744073709551615.

The Arab historian Al-Khāzinı̄ (about 1130) also provides this tale in his Bookof the Balance of Wisdom (Khanikoff 1858) but ascribes the problem to an Indianphilosopher called S. is.s.a ben Dāhir (Wiedemann 1908, 45–54; Wieber 1972, 96;Tropfke 1980, 631). Instead of grains of corn, his version deals with the doubling ofpieces of gold. A tale similar to Al-Khāzinı̄’s was transmitted by the thirteenth-century Arabic historian Ibn Khallikān (Ruska 1916, 276) in his biographicaldictionary of Arab scholars (Slane 1961). Another Arab historian, (about946), reports a similar tale and agrees on the Indian origins of chess (Ruska 1916,280; Tropfke 1980, 631).

The renowned Arab scholar Al-Bı̄rūnı̄ (973–1048) mentions the wheat andchessboard problem in his Chronology of Ancient Nations. His account concernsonly the solution, that is, the computation of the sum of the grains on the 64 squaresof the chessboard and the properties of such computation; he does not provide anytale on the invention of chess (Schau 1879, 134–136). This constitutes the mostancient source which provides an account of the sum of the grains on the chessboard,but it omits the tale about the invention of chess.


Mathematics and the Origins of Chess

Given the well-documented link between chess and the wheat and chessboardproblem, exploring the history of this problem could shed new light on the origins ofthe game of chess. The widespread distribution of the wheat and chessboard problemmakes it difficult to trace with certainty its origins and subsequent dissemination.Nevertheless, Arabic sources on this problem agree on one point: the inventor ofchess was an Indian philosopher, and the game to be played on a chessboard of 64squares was invented by him.

It is very likely that this game reached the Arabic-speaking world through Persianintermediaries (Murray 1913, 207–210). A Persian source, however, ascribes theinvention of chess to the Persian king Khosrow I Anushiruwān (501–579). This textis written in Pahlavi (a variant of Middle Persian), and the history goes as follows:the Indian king Devasārm sent an embassy to the king Khosrow Anushiruwān withthe game of chess, which the Persian king had invented, asking for an explanationof the game. The Indian ambassadors learnt that this was to be understood as awargame and in contrast to the chance-governed game nard, which was dependenton astrology (specifically on combinations of planetary positions and zodiacal signs)(Ruska 1916, 280). In this instance, chess and mathematics represent order andcivilization versus chance and chaos. However, the Persian source does not mentionany version of the wheat and chessboard problem.

Since all the Arabic sources that refer to the invention and the chessboardproblem agree on the Indian origins of chess, it is therefore likely that the wheatand chessboard problem – in the form whereby the solution must be the sum of aunity redoubled 64 times on a chessboard – shares the same origin. Moreover, allsources agree on the mathematics involved in the solution to the problem, which is,in more technical terms, the sum of a geometric progression of reason 2.

Geometric Progressions and Chess

Geometric progressions, that is, sequences of numbers where each term after thefirst is found by multiplying the previous one by a non-zero number (properly calledcommon ratio), were known about and used extensively in problem-solving beforethe invention of the chessboard and the game of chess. Applications of geometricprogressions were already an object of recreational mathematics in Babylonian andEgyptian mathematics in Antiquity. The most ancient versions of the wheat andchessboard problem can be traced back to before the ninth century and are thereforemore ancient than Al-Bı̄rūnı̄’s and versions, which constitute, as hasbeen suggested, the earliest testimony about the wheat and chessboard problem ona 64-squared chessboard (Ruska 1916, 282; Høyrup 1994, 96–99). The differencebetween this version and the most ancient versions is twofold: the latter give aredoubling that differs from the 64 times and are not included as part of legendson the invention of chess. The most ancient account of the wheat and chessboard

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problem is a text on Babylonian mathematics from the eighteenth century BCE.This version deals with corn grains which are to be doubled 30 times (Soubeyran1984).

As for geometric progressions unrelated to the wheat and chessboard problem,their use is attested by written sources that go back to ages and contexts moreancient than the invention of chess. For instance, the renowned Rhind papyrus,found in Egypt and originating from around 1700 BCE, is a document redacted inhieratic script which provides a collection of mathematical problems: a geometricprogression with common ratio 7 (7 + 72 + 73 + 74 + 75) is part of the contentof the papyrus (Newman 1952, 27; Boyer and Merzbach 1991, 11–16). A fragmentof a Greek papyrus of unknown provenance, originating from the third to the fourthcentury BCE, contains an application of a geometric progression of ratio 2 with thenumber 5 as the initial term, which should be doubled 30 times. The sum of theprogression constitutes the solution to a currency conversion problem of ancientcoinages, that is, for drachmae and talents (Boyaval 1971, 165–168). Treatiseson geometric progressions and their properties, such as Euclid’s Elements BookIX (esp. Prop. 35 and 36), redacted between the fourth and the third centuriesBCE, can be traced back to Antiquity. These topics would be reprised in Thābitibn Qurra’s renowned treatise On the Determination of Amicable Numbers (tenthcentury) (Rashed 2015, 337, 399–410).

As for our wheat and chessboard problem, this takes place on an 8 × 8chessboard and must be solved by computing the sum of the geometric progression1 + 2 + 22 + . . . + 263, which is equal to 264 − 1. The method of solving the wheatand chessboard problem is usually provided in the extant sources by explanationsin textual form or by means of displaying the geometric progression in a column. Ifthe tale of the invention of chess is provided, the computation always follows that.The column displays the number of a field of the squares and the correspondingresults of the reduplication. Some partial results of the progression are usuallyconverted into different units of measurement in order to make the computationeasier, so that whoever is computing has to deal with quantities smaller than theywould otherwise be. We will see some examples below. Some sources also providerules to demonstrate the validity of the computations alongside considerations of theproperties of the progression.

While Arabic and Western sources (in Latin and in Romance and Germaniclanguages) were extensively examined in the scholarship on the history of the wheatand chessboard problem (Wiedemann 1908; Ruska 1916; Wieber 1972; Tropfke1980, 630–633; Sesiano 2014, 138–141), only recently have examinations of Greekmanuscripts brought to light new evidence on the dissemination of the sum of thegeometric progression to solve the wheat and chessboard problem. (This scholarlygap was pointed out long ago: “There is one branch of the later Greek literature,fairly circumscribed in extent, which might possibly give us some reference tochess earlier in date than any I have cited. The mathematical problem known as‘the doubling of the squares of the chessboard’ may have been known to the laterGreek mathematicians, as we find it included in the oldest Western mediaeval MSS.on mathematics. The Greek MSS. have not so far been examined for this purpose”


(Murray 1913, 167).) In what follows, we provide sections on Arabic, Western, andGreek sources on the sum of the geometric progression to solve our problem, that is,on the computation of 264 − 1. The sections on Arabic and Western sources offer anintroductory account on this topic, displaying significant sources alongside a briefcommentary. The section on Greek sources, given the lack of scholarship on thistopic, constitutes the most up-to-date account concerning the wheat and chessboardproblem in Greek literature.

Arabic Sources on the Computation 264 − 1Several Arabic sources present the wheat and chessboard problem within books ofalgebra. The most ancient example of our problem in this genre of mathematicalliterature is in a lost book of the mathematician and polymath Al-Khwārizmı̄ (780–850 CE). We learn this from a book of algebra by the mathematician Abū Kāmil(ninth century), who mentions Al-Khwārizmı̄ as one of his sources (Rashed 2012,724–726; Sesiano 2014, 139). Later on, this problem is contained in many books ofalgebra, such as that by Al-Kashı̄ (mid-fifteenth century) (Luckey 1951, 26–27, 61).

The simplest computational method to solve the wheat and chessboard problemis the one mentioned in the legend of (see above). It consists in the sumof the quantities that the reader obtains at each final square (i.e., on the extreme rightcorner) of each row. Therefore:

The first row has 255The second 32768The third 8388608The fourth 2147483648The fifth 549755813888The sixth 140737488355323The seventh 36028797018963968The eighth 9223372036854975808

The total of the grains on the chessboard is equal to 18446744073709551615 (i.e.,264 − 1).

Among the several primary sources for this geometric progression (of which thereader can find a comprehensive overview in Wieber 1972, 103–119), the Arabicmanuscript Orientalis A 1343, preserved in the Gotha Research Library, deservesa special mention, for it is entirely devoted to the mathematics of the wheat andchessboard problem (Pertsch 1878–1892, 3: 14–15; Wiedemann 1908, 54–58).

In the following, we provide a sample of a computation of a geometric progres-sion related to the wheat and chessboard problem from the manuscript Orientalis A1343. The quantities in the third column are not provided in the manuscript, but wehave added them to allow the read to become acquainted with the raw results of thereduplications.

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Square Unit of measurement Quantity

1 1 h. abba 12 2 h. abba 23 4 h. abba 44 8 h. abba 85 16 h. abba 166 32 h. abba 327 64 h. abba 648 128 h. abba 1289 256 h. abba 25610 512 h. abba 51211 1024 h. abba 102412 2048 h. abba 204813 4096 h. abba 409614 8192 h. abba 819215 16384 h. abba 1638416 32768 h. abba = 1 qadah. 3276817 2 qadah. 6553618 4 qadah. 13107219 8 qadah. 26214420 16 qadah. = 1 waiba 52428821 2 waiba 104857622 4 waiba 209715223 8 waiba = 1 irdabb + 2 waiba 419430424 2 irdabb + 4 waiba 838860825 5 irdabb + 2 waiba 1677721626 10 irdabb + 4 waiba 3355443227 21 irdabb + 2 waiba 6710886428 42 irdabb + 4 waiba 13421772829 85 irdabb + 2 waiba 26843545630 170 irdabb + 4 waiba 53687091231 341 irdabb + 2 waiba 107374182432 682 irdabb + 4 waiba 214748364833 1365 irdabb + 2 waiba 429496729634 2730 irdabb + 4 waiba 858993459235 5461 irdabb + 2 waiba 1717986918436 10722 irdabb + 4 waiba 3435973836837 21845 irdabb + 2 waiba 6871947673638 43670 irdabb + 4 waiba 13743895347239 87381 irdabb + 2 waiba 27487790694440 164761 irdabb + 4 waiba = 1 šūna 54975581388841 2 šūna 109951162777642 4 šūna 219902325555243 8 šūna 439804651110444 16 šūna 8796093022208

(continued)


45 32 šūna 1759218604441646 64 šūna 3518437208883247 128 šūna 7036874417766448 256 šūna 14073748835532849 512 šūna 28147497671065650 1024 šūna = 1 madı̄na 57294995342131251 2 madı̄na 114589990684262452 4 madı̄na 229179981368524853 8 madı̄na 458359962737049654 16 madı̄na 916719925474099255 32 madı̄na 1833439850948198456 64 madı̄na 3666879701896396857 128 madı̄na 7333759403792793658 256 madı̄na 14667518807585587259 512 madı̄na 29335037615171174460 1024 madı̄na 58670075230342348861 2048 madı̄na 117340150460684697662 4096 madı̄na 234680300921369395263 8192 madı̄na 469360601482738790464 16384 madı̄na 9387212036854775808

The conversions in the second column are intended to facilitate the computations.It is Ibn Khallikān who claims to have learnt such methods from a mathematicianfrom Alexandria in Egypt (Wieber 1972, 106). A brief explanation of this is thatthe first sequence of reduplications occurs from the 1st to the 16th square, wherewe obtain 32768 h. abba (grain of corn), which is equal to 1 qadah. (cup); from the16th to the 20th, we obtain 16 qadah. , which is equal to 1 waiba (dry measure of20 units/mass of wood); from there to the 40th square, we have irdabb (buschel)and šūna (store), so that 164761 irdabb + 4 waiba, that is, 174762 + 2/3 irdabb = 1šūna. From the 40th square to the 50th, we have 1024 šūna, which corresponds to 1madı̄na (town). By redoubling until the 64th house, we obtain 16384 madı̄na, whichcorresponds to 263 = 9223372036854775808.

The manuscript Orientalis A 1343 contains further geometric progressionsrelated to the reduplications on the chessboard. In the Arabic versions of thecomputation 264 – 1, it is common to read conversion methods which are designed tomake the computations easier. There are many variants in the units of measurement(Wieber 1972, 103–119).

It is worth reporting Al-Bı̄rūnı̄’s chapter on the wheat and chessboard problem,for it constitutes the most extensive source on the mathematical properties ofthis problem in Arabic literature. Al-Bı̄rūnı̄ takes for granted that the reader isacquainted with the tale on reduplication of grains on the chessboard and providestwo rules for it. The first rule is:

The square of the number of a check x of the 64 checks of the chessboard is equal to thenumber of that check the distance of which from the check x is equal to the distance of thecheck x from the first check.

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For example: take the square of the number of the 5th check, i.e. the square of 16(162) = 256, which is the number belonging to the 9th check. Now, the distance of the 9thcheck from the 5th is equal to the distance of the 5th check from the first one. (TranslationSchau 1879, 134–136)

Al-Bı̄rūnı̄’s second rule on the doubling of the chessboard is:

The number of a check x minus 1 is equal to the sum total of the numbers of all the precedingchecks. Example: The number of the 6th check is 32. And 32 − 1 is 31, which is equal tothe sum of the numbers of all the preceding checks, i.e. of 1 + 2 + 4 + 8 + 16 (=31). If wetake the square of the square of the square of 16, multiplied by itself [ . . . ], this is identicalwith taking the square of the number of the 33rd check, by which operation the number ofthe 65th check is to be found. If you diminish that number by 1, you get the sum of thenumbers of all the checks of the chessboard. The number of the 33rd check is equal to thesquare of the number of the 17th check. The number of the 17th check is equal to the squareof the number of the 9th check. The number of the 9th check is equal to the square of thenumber of the 5th check. And this (i.e. the number of the 5th check) is the abovementionednumber 16. (Translation Schau 1879, 134–136)

After the second rule, we find further considerations of the problem and ofgeometric progressions, taken from another work by Al-Bı̄rūnı̄, entitled “Book ofCiphers.” The text is self-explanatory:

I shall explain the method of the calculation of the chess problem, that the reader mayget accustomed to apply it. But first we must premise that you should know, that in aprogression of powers of 2 the single numbers are distant from each other according toa similar ratio. (Lacuna?) If the number of the reduplications, i.e. the number of the singlemembers of a progression is an even one, it has two middle numbers. But if the number ofthe reduplications is an odd one, the progression has only one middle number.

The multiplication of the two ends by each other is equal to the multiplication of thetwo middle numbers. (In case there is only one middle number, its square is equal to themultiplication of the two end numbers.) This is one thing you must know beforehand. Theother is this: −

If we want to know the sum total of any progression of powers of 2, we take the doubleof the largest, i.e. the last number, and subtract therefrom the smallest, i.e. the first number.The remainder is the sum total of these reduplications (i.e. of this progression).

Now, after having established this, if we add to the checks of the chessboard one check, a65th one, then it is evident that the number which belongs to this 65th check, in consequenceof the reduplications of powers of 2, beginning with 1, is equal to the sum of the numbersof all the checks of the chessboard minus the 1st check, which is the number 1, the firstmember of the progression. If, therefore, 1 is subtracted from this sum, the remainder is thesum of the numbers of all the checks of the chessboard.

If, now, we consider the 65th check and the 1st as the two ends of a progression, theirmedium is the 33rd check, the first medium.

Between the checks 33 and 1, the check 17 is the medium, the second medium.Between the checks 17 and 1, the check 9 is the medium, the third medium.Between the checks 9 and 1, the check 5 is the medium, the fourth medium.Between the checks 5 and 1, the check 3 is the medium, the fifth medium.Between the checks 3 and 1, the check 2 is the medium, the sixth medium, to which

belongs the number 2.Taking the square of 2 (22), we get a sum which is a product of the multiplication of the

number of the 1st check by that of the 3rd check (1 × 4 = 22). The number of the 1st checkis 1. This product, then, is the fifth medium, the number of the 3rd check, i.e. 4.

The square of 4 is 16, which is the fourth medium of the 5th check.


The square of 16 is 256, which is the third medium in the 9th check.The square of 256 is 65536, which is the second medium in the 17th check.The square of 65536 is 4294967296, which is the first medium in the 33rd check.The square of 4294967296 is 18446744073709551616.If we subtract from this sum 1, i.e. the number of the first check, the remainder is the

sum of the numbers of all the checks of the chessboard. I mean that the number which at thebeginning of this digression we have used as an example (of the threefold mode of numeralrotation). (Translation Schau 1879, 135–136)

Given the immense quantity of the final result of the sum of the grains, Al-Bı̄rūnı̄provides a method to better grasp the number 18446744073709551615 by means ofdividing it by dry measurements based on the Arabic coinage, the dirham (A dirhamcoin is made of silver (Miles 2012)):

The immensity of this number cannot be fixed except by dividing it by 10000.Thereby it is changed into Bidar (sums of 10000 dirhams).The Bidar are divided by 8. Thereby they are changed into (loads).The are divided by 10000. Thereby the mules, that carry them, are formed into

(herds), each of them consisting of 10000.The are divided by 1000, that, as it were, they (the herds) might graze on the bordersof Wâdîs, 1000 kids on the border of each Wâdî.The Wâdîs are divided by 10000, that, as it were, 10000 mountains should rise out of eachWâdî.In this way, by dint of frequently dividing, you find the number of those mountains to be2305. But these are (numerical) notions that the earth does not contain.(Translation Schau 1879, 136)

Further Arab authors were aware that the final sum was too big to be easilycomprehended and therefore developed methods to allow the reader to visualizethis quantity (Wieber 1972, 116–119). For instance, Al-Khāzinı̄ expounds a methodusing conversions of units of measurement into dirhams, in order to distribute264 − 1 throughout the surface of the Earth by means of coins. This can be explainedbriefly as follows. Given the diameter of the Earth of 6490 + 10/11 miles and thecircumference of the Earth of 20400 miles, the surface of the Earth correspondsto the multiplication 6490 + 10/11 miles × 20400 miles, which gives a result of132416400 miles2. This number corresponds to 2118662400000000 cubits2. If wedivide 264 − 1 dirhams by 2118662400000000 cubits2, we obtain a result of 8708dirhams pro cubit2.

The weight of a cubit of silver corresponds to circa 420867 dirhams pro cubit3. Ifwe take a cube with edge length of 1 cubit (which consists of dirhams), we have 75dirhams + a fraction, that is, the length of a cubit in dirhams3. The square of this is5625. If we divide 8708 dirhams by 5625, we obtain 1/48 cubit, that is, the quantityof silver that represents the dirhams stretched out along the surface of the Earth. Ifwe distribute this throughout the inhabited part of the Earth, the quantity is equal to1/12 cubit (Wiedemann 1908, 53–54; Wieber 1972, 117).

Later on, the tenth-century Arab astrologer and mathematician Al-Qabisiemploys a very similar method to distribute 264 − 1 along the surface of theEarth in dirham coins. His result of the computation of the surface of the Earth incubits2 is the same as Al-Khāzinı̄’s, that is, 2118662400000000. The procedures

12 A. Bardi

after this passage are slightly different. According to his computations, a cubit2 isequal to 500 dirhams; therefore the surface of the Earth is equal to the multiplicationof 2118662400000000 × 500, that is, 1059331200000000000 dirhams. The total ofthe reduplication of the squares of the chessboard is about 17 + 2/5 times higherthis sum (Sesiano 2014, 148–149).

Greek Sources on the Computation 264 − 1Alongside Greek sources on geometric progression, such as Euclid’s Elements IXand later mathematical literature in Greek, such as collections of mathematicalproblems originating from the Byzantine era (Hunger and Vogel 1963; Vogel 1968;Chalkou 2006; Deschauer 2014), we have only to look to the Greek scientifictradition of the fifteenth century to find applications of geometric progressions thathelp to solve the wheat and chessboard problem. Up until now, only one Greeksource has been detected (Heiberg 1899, 168–169), but we have also found twoadditional sources (more below). These sources do not contain any tales aboutthe wheat and chessboard problem or the invention of chess, but simply providea geometric progression of reason 2, which deals, as usual, with a unity (withoutfurther specification of unit of measurement) to be doubled 64 times.

It is likely that the geometric progression on the squares of the chessboard wasbrought into Greek mathematics through Arabic or Persian intermediaries. A lexicalinvestigation supports this interpretation. In fact, the Greek word for chessboardis zatrikion, which stems from the Sanskrit catur-aṅga (“four members”) and wasmediated into Greek through shatranij, an occurrence detectable both in Persian andArabic literature. Greek historiographical works agree in assigning Eastern (Persianor Assyrian) origins to the game of chess (Murray 1913, 162–163).

The Greek source on the doubling of the chessboard that we already knowabout is provided in the fifteenth-century manuscript Vindobonensis phil. gr. 65(henceforth V), which contains a selection of mathematical problems and wasredacted at around 1436 (Hunger 1961, 182–184). The progression is arranged intotwo columns and is entitled “doubling of the chessboard.” The Greek numerals formα to θ, supplemented with a sign for the “zero,” and are used as we use the numeralsfrom 1 to 9, supplemented with 0; this results in a positional notation for numbers.This is evidence of the influence of Arabic or Indian notation. Such a system wasalready known in Byzantium before the fifteenth century, for Indian numerals werethe object of a mathematical treatise by Maximos Planudes, a Byzantine scholaractive in the half of the thirteenth century (Allard 1981).

We found another geometric progression of the wheat and chessboard problemvery similar to V in the fifteenth-century Greek manuscript Ambrosianus I 112 sup.(henceforth A), preserved at the Ambrosiana Library in Milan (Martini and Bassi1906, 562–564). In the following, we provide a translation of the progression intoour notation according to the version in the manuscript V.


Doubling of the chessboard

1 1 29 2684354562 2 30 5368709123 4 31 10737418244 8 32 21474836485 16 33 42949672966 32 34 85899345927 64 35 171798691848 128 36 343597383689 256 37 6871947673610 512 38 13743895347211 1024 39 27487790694412 2048 40 54975581388813 4096 41 109951162777614 8192 42 219902325555215 16384 43 439804651110416 32768 44 879609302220817 65536 45 1759218604441618 131072 46 3518437208883219 262144 47 7036874417766420 524288 48 14073748835532821 1048576 49 28147497671065622 2097152 50 57294995342131223 4194304 51 114589990684262424 8388608 52 229179981368524825 16777216 53 458359962737049626 33554432 54 916719925474099227 67108864 55 1833439850948198428 134217728 56 36668797018963968

57 7333759403792793658 14667518807585587259 29335037615171174460 58670075230342348861 117340150460684697662 234680300921369395263 469360601482738790464 9387212036854775808

At row 50, both A and V share an error, that is, 249 = 572949953421312,when it should be 562949953421312. This is the cause of all the subsequenterrors, so that 263 becomes 9387212036854775808 and not as it should be,i.e., 9223372036854775808. On this account, both scribes made a mistake inmultiplying 248 × 2. Alternatively, it may be that the error was already presentin the manuscript from which they copied. In contrast to the Arabic sources which

14 A. Bardi

provide progressions in columns, no conversion between units of measurement isundertaken.

Both manuscripts provide two texts about the properties of the geometricprogression.

Demonstration of the doubled numbers for each houseSeek the doubling of a house you wish; from this subtract 1; the remaining number is

the sum of the above numbers of the doubled houses; in this way, the first house doubledup to the eighth becomes 128; subtract 1, 127 remains; and so this is the whole sum of thedoubling of the first house until the seventh; 127; in fact, the sum of 1 and 2 and 4 and 8and 16 and 32 and 64 becomes 127; the same happens also for the others. (My translation:cfr. A, f. IVr; V, f. 147r)

It is equivalent to this: 1 + 2 + 22 + 23 + 24 + 25 + 26 = 127 = 27 − 1.This rule can be applied to all the numbers provided in the two columnsof the doubling. Therefore, in general terms, we could write in this way,1 + 2 + 22 + 23 + 24 + 25 + . . . + 2n = 2n + 1 − 1.

There is also another demonstration for the group of five houses, namely the fifth, thetenth, the fifteenth, the twentieth, and the houses which follow and which are sought for5, namely 16, which lays beside the fifth house; multiply this by 32 and the multiplicationof these is 512; this is the doubling of the tenth house, 512; multiply this by 32 and themultiplication of these is 16384; this is the doubling of the fifteenth house, 16384; multiplythis by 32 and the multiplication of these is 524288; this is the doubling of the twentiethhouse, 524288; the same also happens for the other [numbers] that are sought for groups offive. (My translation; cfr. A, f. IVr; V, f. 147r)

The second text deals with the following property about the fifth house (24) ofthe progression: by multiplying the fifth house by the very next house (sixth), onegets the number corresponding to the tenth house; by multiplying the tenth houseby the sixth house, one gets the fifteenth house; and so on. This is equivalent to24 × 25 = 29, 29 × 25 = 214, 214 × 25 = 219, etc. The multiplication property with25 remains valid in all the terms of the geometric progression. In general terms, wecan write: 2n × 25 = 2n + 5.

We have found an unpublished geometric progression of 2 on the chessboard inthe fifteenth-century Greek manuscript Ambrosianus E 80 sup. (Martini and Bassi1906, 329–331). In contrast to the Greek and the Arabic geometric progressionsmentioned above, this example contains an unusual trait: it arranges the geometricprogression into an 8 × 8 chessboard and, similarly to the Arabic progressions incolumns, employs conversions between units of measurement.

On the one hand, the current progression provides no sum of the terms of theprogression, in exactly the same way as the progressions in columns mentionedabove. On the other hand, in this instance, the Greek numerals are employed in theirstandard notation, and no further text is provided after the display.

In the following, we provide the progression on the chessboard as it appears inthe manuscript Ambrosianus E 80 sup. Letters and numbers in bold are added tofacilitate the commentary.


Zatrikion

A B C D E F G H

1 [?] 1 2 4 8 16 32 64 1282 256 512 1024 2048 4096 8092 [?] 16384 23 4 8 16 32 [?] 64 [?] 2 4 84 16 32 64 128 256 512 1024 20485 4096 8192 [?] 16384 [?] 2 4 8 16 326 64 128 256 512 1024 2048 4096 81927 16384 32768 65536 131072 262144 524188 1048576 20971528 18619392 37238784 74477568 148955136 297910272 595820544

There is an error at cell F2, for it should be 8192, not 8092. This is the cause ofall the subsequent mistakes in the other cells, especially in the rows 7 and 8.

This geometric progression of 2 employs a conversion system similar to thesamples seen in the Arabic sources, but the abbreviations of the Greek language,which are located in the places where we have inserted question marks, have notyet been solved by expert Greek philologists and paleographers. New light willbe shed on this by solving those abbreviations. The abbreviated nouns have thesame function as the units of measurement encountered in the Arabic geometricprogressions in columns, which is to make computations easier by means of dealingwith numbers smaller than the raw results.

As usual, the number 1 (A1) is the initial term of the progression with thecommon ratio 2. The progression continues till 215 (G2). From here, the partialresult is converted into a different unit of measurement, and the progressioncontinues with the same ratio up to cell E3, where we find 43. Then the partialresult is converted into a different unit of measurement. At cell C5, we find anotherconversion, and then the progression continues until square F8.

Western Sources on the Computation 264 − 1The wheat and chessboard problem is discussed not only in the Middle East andthe Eastern Mediterranean but also in the medieval and premodern West, both incollections of recreational mathematics and treatises on the properties of numbers.

The famous collection of mathematical problems entitled Propositiones adacuendos juvenes (Problems to Sharpen the Young) constitutes the earliest Westernsource and is ascribed to the English polymath Alcuin of York (ca. 800) (Folkerts1978). This work is also the earliest collection of mathematical problems inLatin literature. Problem 13 (Proposition 13) contains a version of the wheat andchessboard problem in a similar fashion to the Arabic version provided in a bookof algebra by Abū Kāmil (Høyrup 1986). Instead of wheat or grain or gold, it dealswith the doubling of soldiers of the army of a king (Folkerts 1978, 51–52). The

16 A. Bardi

reduplication occurs 30 times. The following is a modern translation of Alcuin’sProposition 13:

A king ordered his servant to collect an army from 30 villages as follows: He should bringback from each successive village as many men as he had taken there. The servant went tothe first village alone; he went with one other man to the second village; he went with threeother men to the third village. How many men were collected from the 30 villages?

Solution.After the first village there are 2 men in the army, there are 4 after the second village,

there are 8 after the third village, . . . . So after the nth village there will be 2n men collectedfor the army. After the 30th village there will be 230 = 1073741824 men in the army.(Translation O’Connor and Robertson 2012)

As for the wheat and chessboard problem in the version with 64 reduplications,it has also survived in the West. Sources attest to its progression after Alcuin; wefind it in the late Middle Ages and in the Renaissance. An example of the standardwheat and chessboard problem is provided by the Italian mathematician LeonardoFibonacci (c.1170–c.1250) in his renowned Liber Abaci (1202 CE) (Boncompagni1857, 309–318; Sigler 2002, 435–445). The Liber Abaci, redacted in Latin, containsseveral problems concerning geometric progressions, sometimes accompanied bytales, which constitute an example of the application of the method; for instance, astory about a financial dispute (on the sum of debts between two parties) providesthe application of a geometric progression on a chessboard to show the usefulness ofsuch a computational method in a problem-solving context. The coinage mentionedby Fibonacci is the denarius (denaro), a silver coin originating in the third-centuryRome (Metcalf 2012, 300):

A certain man gave one denaro [denarius] at interest so that in five years he must receivedouble the denari, and in another five he must have double two of the denari, and thusforever from 5 to 5 years the capital and interest are doubled; it is sought how many denarifrom this one denaro he must have in 100 years; you divide the 100 years by the 5; thequotient will be 20; therefore the denaro is doubled twenty times. Whence 20 places ofthe chessboard carry a similarity; therefore if we shall double the denaro twenty times weshall have the amount to which the denaro increases in 100 years; or in another way youdouble the denaro; there will be 2 which is the number to which the denaro grows in thefirst five years; and the 2 again you multiply by itself; there will be 4 which is the numberto which the denaro increases in the second five years; this 4 you again multiply by itself;there will be 16 which is the amount to which the denaro increases in four times five years;this 16 you double; there will be 32 for the amount after five times five years; this 32 youmultiply by itself; there will be 1024 for the amount after ten times five years; this 1024you multiply by itself yielding 1048576 denari for the amount after twenty times five years,namely the 100 years; this is 4369 pounds and 16 denari. The same method works for a manwho sold 20 pairs of hides; from the first he had 1 denaro; for the second 2, for the third4, and thus forever doubling up to the last pair; the sum is the aforesaid amount minus 1denaro. (Translation Sigler 2002, 437–438)

Similarly to Arabic mathematicians, Fibonacci provides methods to help thereader visualize the immense quantity of the result of 264 − 1 (Sesiano 2014, 149–150). The first method describes how to fill a cash box with coins. The coinageemployed by Fibonacci in this text is the bezant (bizantius aureus), a gold coin


that originated in Byzantium (Grierson 1991, 1, 287). Fibonacci’s text is self-explanatory:

From the sum of two rows of the chessboard 65536 is summed, namely from 16 places, andfrom these one coffer [aura] is filled, and then in order this coffer is doubled, and thus weshall have in the seventeenth place, namely in the first place of the third row two coffers;in the second place of the same row there are 4 coffers, in the third 8, in the fourth 16, inthe fifth 32, in the sixth 64, in the seventh 128, in the last place of the same row 256. In thefirst place of the fourth row 512. In the second place 1024, in the third 2048, in the fourth4096, in the fifth 8192, in the sixth 16384, in the seventh 32768, and in the last place youwill have 65536 coffers; from this if we shall fill one house [domus], then we shall have inthe first place of the fifth row 2 houses. In the second 4, in the third 8, and thus doubling inorder we shall have in the last place of the sixth row 65536 houses. From these if one shallmake one city [civitas], and the remaining places we continue doubling, then we shall havein the last place of the chessboard 65536 cities; therefore the sum of all the numbers onthe chessboard reaches 65536 cities; each city has 65536 houses, and in each house thereare 65536 coffers, and in each coffer there are 65536 bezants; because of the abovesaiddemonstration one must have in one coffer 1 bezant less (translation Sigler 2002, 435–436).

Later on, Fibonacci explains the sum by means of ships that are to be filled withgrains of corn, eventually claiming “that the number of ships is effectively infiniteand uncountable is here easily observed”:

And you will wish to double beginning with one grain of corn on the first place, and youwish to know how many ships are needed to carry the corn if each ship will carry 500Pisan modia, each of which is 24 sestari, each of which weighs 140 pounds, each of whichweighs 12 ounces, and each ounce weighs 25 pennyweights; each pennyweight weighs 6carobs; each carob weights 4 grains of corn; all of these are disposed under a fraction inorder thus: 1/4 0/6 0/25 0/12 0/140 0/500, that is 18446744073709551615 and is the sum ofthe grains of corn on the chessboard, you divide by the above written parts that are underthe fraction, and whatever will remain over the 4 that is at the head of the fraction will begrains, and whatever will remain over the 6 will be carobs, and whatever is over the 25will be pennyweights, and whatever is over the 12 will be ounces, and whatever is over the140 will be pounds, and whatever is over the 24 will be sestari, and whatever will remainover the 500 will be modia; truly the integer which remains after the division will be thenumber of ships to be loaded, as here is shown: 3/4 3/6 0/25 6/12 115/140 13/24 123/5001725 028 445; that the number of ships is effectively infinite and uncountable is here easilyobserved. And you note that the 500 modia of each ship are seagoing modia, namely 16000Roman modia, or 8000 Syrian modia, or 4000 Sicilian salme. Truly in the second of thedoubling of the chessboard squares, namely when any place in the sequence of places isproposed to be the sum of all the preceding doubles, one can find them in two ways; thefirst is indeed from place to place up to the last number. The second truly is the way inwhich you take the 1 which is proposed for the first place, and you add it to the 2 that is putin the second place; there will be 3 which you multiply by itself; there will be 9 which isthe number of the sum of the first and second places, namely three. For example, if in thefirst place 1 is put, in the second two, and in the third 6, namely double the sum of the twopreceding places, the sum of them will be 9, as we said before; this 9, if it is multiplied byitself makes 81 which number is the sum of the first place and double the two followingplaces, namely 5 places. For example, if at the first place 1 is put, at the second 2, at thethird 6, at the fourth 18, at the fifth 54, undoubtedly they add to 81; if you multiply this81 by itself, then it makes 6561; this number is the sum of the first place and double thenn following places, namely 9 places. For example, the number at the first place is 1, thesecond 2, the third 6, the fourth 18, the fifth 54, the sixth 162, the seventh 486, the eighth

18 A. Bardi

1458, the ninth 4374; all added together they make 6561, which 6561 you multiply by itselfmaking 43046721; this above written number disposed is the sum of the doubles of thefirst place and double eight following places, namely 17 places. Whence if you multiplythe 43046721 by itself, then 1853020188851841 will result for the sum of the doubles ofthe first place and double 16 places, namely 33 places; this number multiplied by itselfyields 3433683820292512484657849089281 for the doubles of the entire chessboard andone place more; this place is double the entire chessboard; therefore it must be that a thirdpart of the above written number is the sum of the doubles of all the chessboard squares;therefore this number divided by 3 yields for the sum of all the doubles on the chessboardsquares 1144 561 273 430 837 494 885 949 696 427. (Translation Sigler 2002, 436–437)

Fibonacci’s account is the most comprehensive source on the wheat and chess-board problem in Latin literature.

A few centuries later, the Italian mathematician Luca Pacioli wrote his mas-terpiece Summa de arithmetica in Italian, which contains another account of thewheat and chessboard problem (Pacioli 1494). (Full title: Summa de arithmetica,geometria, proportioni et proportionalita, that is, Summary of arithmetic, geometry,proportions and proportionality.) Pacioli provides two methods for doubling grainsof corn on a chessboard. Here is an excerpt from Pacioli’s chapter on the wheat andchessboard problem:

To double a grain of corn as many times as there are white houses and black ones on thechessboard, which are in sum 64, can be understood in two ways. The other way means thatonly the following house doubles the preceding house and nothing more. In this way it doesnot grow up as much as the previous way. Now, in order to double a small grain in this way,be aware that if you add the first term to the sum of 4 houses, and you multiply the sum foritself, and from that product you subtract the first term you added, what remains is equal tothe sum of the doubling of 8 houses, for the doubling of 4 houses is 1 2 4 8, whose sumis 15. I say that if you add the first term, which is 1, it makes 16. Multiply this by itself, itmakes 256; subtract the first term, it remains 255. So much will be the sum of the housesmultiplied in the way mentioned. (My translation; cf. Pacioli 1494, f. 43 nr. 28)

Pacioli’s methods would be reprised in another important mathematical treatiseredacted in Italian, namely, General trattato di numeri et misure (General treatiseon numbers and measures; printed in 1556) by the Italian mathematician NiccolòTartaglia (c.1499–1557). His works contains a chapter devoted to geometric pro-gressions. We learn from his text that Tartaglia considers Pacioli an authority on thewheat and chessboard problem, for he explicitly refers to him in methods to solveit: “according to the rules of the friar Luca” (cf. Tartaglia 1556, book 1, f. 16r).

Further discussions of the wheat and chessboard problem are provided in Latinand in Romance or Germanic languages, for instance, in the so-called AlgorismusRatisbonensis, a fifteenth-century collection of problems in Latin and German,redacted by a monk (Frederick) in the Benedictine monastery of Regensburg (Folk-erts 1971, 71; Sesiano 2014, 9 n.12), and in the so-called Algorismus Columbia, afourteenth-century collection of problems similar to Fibonacci’s Liber Abaci (Vogel1977).

The Western sources that include the wheat and chessboard problem often have afinancial or trading focus, which sits alongside the mathematical nature of the workthat provides the context for the problem. This reflects Italy’s economic growth


between the thirteenth and sixteenth centuries. The works of Fibonacci, Pacioli, andTartaglia were the most advanced accounts of their age in computational techniques.These were useful not only for Italian scholars but also for merchants and traders,whose activities were indeed flourishing at the time when Fibonacci, Pacioli, andTartaglia redacted the treatises.

In contrast to Arabic sources, Western sources never present tales about theinvention of chess and give no hints as to the origins of the problem; they simplydeal with applications of the geometric progression on the squares of the chessboard,accompanied by methods and considerations about the properties of the progressionitself.

Moreover, the Western sources provide evidence of the dissemination of ourproblem in non-mathematical literature. For instance, the renowned Dante Alighieri(1265–1321) alludes to the wheat and chessboard problem in a verse of his Paradise(see the verses quoted above in the epigraph to this chapter). This demonstrates thatour problem was being discussed among learned Italians as early as the thirteenthcentury.

Number Theory

The mathematical possibilities triggered by the wheat and chessboard problemare not confined to the Renaissance; indeed, they embrace a modern branch ofmathematics known to date as number theory. By computing 264 − 1, as wellas any 2n − 1 of the geometric progression of the chessboard, we obtain a so-called Mersenne number. This name derives from Marin Mersenne (1588–1648), aFrench polymath who devoted some of his scholarship to studying the propertiesof numbers. Mersenne numbers are equal to a power of 2 minus 1. Given aninteger n, we will obtain a number Mn that is equal to 2n − 1. Therefore,Mn = 2n − 1. Following Mersenne’s puzzles, mathematicians have been challengedto find criteria to determine which integer n we must use so that Mn = 2n − 1results in a prime number. To obtain such special numbers, the integer n mustbe one of the following: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209,44497, 86243, 132049, 216091, etc. Such numbers are called Mersenne primes.Scholars have been using computers to find large Mersenne primes. Such a pursuithas given rise to significant collectives of mathematicians and number amateurs,thus extending this pursuit beyond the boundaries of academic mathematics intopopular culture. For instance, the Great Internet Mersenne Prime Search (GIMPS –see the official website: www.mersenne.org) comprises an international team ofvolunteers developing software to search for Mersenne prime numbers. The pursuitof Mersenne primes is still ongoing. To date, the largest known Mersenne primeis 282589933 − 1, which is the 51st known Mersenne prime. This discovery wasmade by Patrick Laroche and officially claimed on December 21, 2018 (https://www.mersenne.org/primes/?press=M82589933). Moreover, the properties of suchnumbers gave rise to new mathematical problems, some of which remain unsolved.

www.\penalty \z@ {}mersenne.\penalty \z@ {}orghttps://www.mersenne.org/primes/?press=M82589933https://www.mersenne.org/primes/?press=M82589933

20 A. Bardi

For instance, mathematicians are currently discussing the possibility of the existenceof an infinite set of Mersenne numbers (see Mersenne number. Encyclopediaof Mathematics. Available through http://www.encyclopediaofmath.org/index.php?title=Mersenne_number&oldid=36008 – accessed 4/9/2019).

Summary

The history of the wheat and chessboard problem proves to be a fil rouge throughcenturies and civilizations, leaving traces in several cultural contexts, crossing lan-guages and literary genres. Its earliest source stems from Babylonian mathematics ina version where the reduplication occurs 30 times and is not related to a chessboard.From that, the version featuring a unity to be redoubled 64 times on a chessboardwas developed, and this proves to be the most widespread version (in Arabic, Greek,and Western sources). The first reference to the latter version traces back to the ninthcentury and is transmitted by Arabic sources. This is precisely the age in which wehave evidence of the relationship between our problem and the game of chess andits invention. On this account, evidence of intrinsic connections between chess andmathematics can be traced back to the ninth century.

According to Arabic sources, the sum of the geometric progression 264 − 1constitutes the answer of an Indian philosopher who was invited by a queen or aking to express a personal wish, in reward for having invented the game of chess. Itis likely that this tale, like most ancient legends, was transmitted orally in the firstinstance; as such, it is reasonable to surmise that the mathematical possibilities ofchess would have been known to Indian or Arabic mathematicians before the taleabout the invention of chess was written down.

The Arabic and Latin literature on the wheat and chessboard problem present awide range of things to be redoubled. The tale about the invention of chess featuresgrains of corn or wheat, or pieces of gold, various coinages, houses, cities, soldiers,and an extensive variety of dry measurements. These units of measurement are usedby mathematicians to make the computations easier.

The transmission of the wheat and chessboard problem cuts across literarygenres. The earliest sources are historiographical works which present it throughtales about the invention of chess, where chess is considered as a war simulator orstrategical game at the disposal of kings and courtiers. Later on, it is transmittedin books of algebra, a mathematical genre typical of Arabic literature. In Westernsources, it is extant in chapters of recreational mathematics or general treatises oncomputations and the properties of numbers. Notably, Luca Pacioli’s account onthe wheat and chessboard problem is inserted in a book of mathematical problemswhich are likely meant to serve as stimulation for Renaissance financial accountants(bookkeepers). It is not accidental that Pacioli was the first scholar to systematize thetechnique of double-entry bookkeeping (Fischer 2000, 303–304). On this account,the history of the wheat and chessboard problem mirrors the cultural contexts inwhich it is employed.

http://www.encyclopediaofmath.org/index.php?title=Mersenne_number&oldid=36008http://www.encyclopediaofmath.org/index.php?title=Mersenne_number&oldid=36008


All of this proves that the wheat and chessboard problem has a great degree ofversatility. From a perspective of tale transmission, the most widespread versionof our problem was born through a combination of the invention of chess with atale that allows for the application of geometric progression. Later on, the secondcomponent become popular among Arabic mathematicians, and the invention of thegame of chess was left to one side. The application of the geometric progressionproved to be a field of experimentation: Arabic and Western mathematicians haveprovided not only extensive considerations of the properties of the progression buthave also explored how to best visualize the geometric progression and its finalresult. Apart from explanations in textual form, the most common representationalformat is the column with the number of the chessboard square alongside thecorresponding result of the reduplication. In this instance, the Greek literatureproves to be original inasmuch as it provides a representation of the progressioninto an 8 × 8 chessboard. Moreover, the final result of 264 − 1 inspired Arabic andWestern mathematicians to develop methods to visualize this huge quantity. In thiscase, one of the methods consisted in distributing 264 − 1 throughout the surface ofthe Earth, thus showing that only a tiny part of the total sum was enough to cover thewhole globe. This method includes measurements of the surface, the circumference,and the diameter of the Earth, thus testifying to an interplay between the wheat andchessboard problem and one of the mathematical problems that puzzled thinkersfrom Antiquity: measuring the surface of the Earth.

Cross-References

�Mathematics and Big Data�Mathematics and Economics/Social Choice Theory� Puzzles and Mathematics

Acknowledgments A special thank to Sandro Caparrini for fruitful discussions. For usefulsuggestions I am indebted to several anonymous reviewers as well as Roshi Rashed, Jens Hoyrup,Stefan Deschauer, Richard Kremer, and Reyhan Durmaz. This chapter was completed duringa research-stay at Dumbarton Oaks Research Library and Collection (Harvard University), aninstitution I do wish to thank. I am grateful to the Polonsky Academy and the Van Leer Institute aswell.

References

Allard A (1981) Maxime Planudes. Le grand calcul selon les Indiens. Université catholique,Louvain-La-Neuve

Boncompagni B (1857) Scritti di Leonardo Pisano. Liber Abbaci, vol 1. Tipografia delle scienzematematiche e fisiche, Roma

Boyaval B (1971) Le P. IFAO 88: Problemes de conversion monetaire. Z Papyrol Epigr 7:165–168Boyer C, Merzbach UC (1991) A history of mathematics. Wiley, New YorkChalkou MD (2006) The mathematical content of the Codex Vindobonensis Phil Graecus 65 (ff.

11–126). Introduction, edition and comments. Byzantine Research Centre, Thessaloniki

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Mathematics and Cultures Across the Chessboard: The Wheat and Chessboard ProblemContentsIntroductionMathematics and the Invention of ChessMathematics and the Origins of ChessGeometric Progressions and ChessArabic Sources on the Computation 264 − 1Greek Sources on the Computation 264 − 1Western Sources on the Computation 264 − 1Number TheorySummaryCross-ReferencesReferences

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