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Combinatorial libraries & synthesis glossary & taxonomy
Evolving Terminologies for Emerging Technologies
Comments? Questions? Revisions?
Mary Chitty 
Last revised November 20, 2018

Technologies term index  Related glossaries include Assays & screening   Chemistry   Drug discovery & development   Drug discovery & development informatics  Drug targets    Molecular Imaging   

array synthesis: Form of parallel synthesis in which the reaction vessels are  maintained in a specified spatial distribution, e.g. the wells of a 96-well plate or pins held in a rack IUPAC COMBINATORIAL CHEMISTRY Related term spatially addressable.

asymmetric catalysis: See catalytic asymmetric synthesis

bead: (Normally spherical) particle of solid support. IUPAC Combinatorial Chemistry Related term: microspheres; bead arrays Microarrays

biased libraries: Libraries of compounds (putative drug leads) that exclude compounds unlikely to interact with particular protein family structures; these libraries are, therefore, “informed” by structure data.  Broader term: combinatorial libraries.

binning: Approach to classifying the diversity of a set of compounds by grouping related members in "bins" on the basis of common physical or structural features. Commonly applied to the analysis of a set for the completeness of coverage of the desired property space. IUPAC COMBINATORIAL CHEMISTRY

biomimetic synthesis: [Clayton Healthcock's] research has been concerned with developing strategies for streamlining the synthesis of complex, naturally occurring compounds. "A particularly powerful strategy," he said, "is 'biomimetic synthesis,' in which we guess how nature might assemble a particular molecule and then try to mimic this hypothetical route in the laboratory."  Heathcock's work in the total synthesis of natural products and the development of new synthetic methodology has been of major interest to the pharmaceutical industry as well as to the international academic community.  Univ. of California- Berkeley news release "Organic chemist Clayton H. Heathcock named Dean of College of Chemistry at the University of California, Berkeley" 8/12/99

building block: One of a number of  interchangeable reagents which can be used in combinatorial library synthesis, part of the structure of which becomes incorporated into the final product, i.e. its residue. See also diversity reagent, monomer. IUPAC Combinatorial Chemistry  Related terms: combinatorial chemistry, combinatorial synthesis, diversity, directed library, fully combinatorial, in situ scaffold formation, monomer, pool/ split, residue, sub- library, unbiased library

catalytic asymmetric synthesis: The 2001 Nobel Prize in Chemistry will be shared by three scientists who devised techniques for catalytic asymmetric synthesis -- the use of chiral catalysts to accelerate the production of single- enantiomer compounds for pharmaceutical use and a wide range of other applications. Stu Borman, Asymmetric Catalysis Wins, Chemical & Engineering News  79 (42): Oct. 15, 2001

chemical ligand studies: Drug targets

cleavage: Process of releasing compound from solid support, thereby permitting assay or analysis of the compound by solution- phase methods. Dissolution of the compound following , rather than the cleavage step itself, may be rate- limiting. IUPAC COMBINATORIAL CHEMISTRY Related term: solid support

click chemistry: Organic chemistry methodology that mimics the modular nature of various biosynthetic processes. It uses highly reliable and selective reactions designed to "click" i.e., rapidly join small modular units together in high yield, without offensive byproducts. In combination with COMBINATORIAL CHEMISTRY TECHNIQUES, it is used for the synthesis of new compounds and combinatorial libraries.  MeSH 2011

cluster: Group of compounds which are related by structural or behavioral properties. Organizing a set of compounds into clusters is often used in assessing the diversity of those compounds, or in developing SAR models. See also binning; and principal components analysis, recursive partitioning (Algorithms). IUPAC Combinatorial Chemistry

combichem: See combinatorial chemistry

combinatorial biosynthesis: Of focused libraries of natural products holds great promise for capitalising on hardwon natural product leads. Miniaturisation of screens is required to reduce the cost of screening combinatorial libraries. Developments in the processes preceding and following synthesis are required to enable the flow of increased numbers of compounds without new bottlenecks developing. The impact of combinatorial chemistry will be greatly enhanced by synergy with ongoing parallel developments in genetic technologies, screening technologies and bioinformatics. D. Brown "Future pathways for combinatorial chemistry" Molecular Diversity 2 (4): 217- 222 April 1997

combinatorial biology: Bioengineering & Biomaterials

combinatorial chemistry: Using a combinatorial process to prepare sets of  compounds from sets of  building blocks. IUPAC Combinatorial Chemistry

In the early 1990's it was believed that combinatorial chemistry would revolutionize the drug discovery industry. Ten years later the route from design and synthesis of compound libraries to identification of lead structures is still long and costly. Synthesis of an almost unlimited number of organic compounds covering as much of chemistry space as possible is no longer the most cost effective and time saving approach to hit identification. Creating libraries, using biological target structure to inform chemical design, facilitated by quantum advances in structural genomics and computational capabilities, is a smarter, more efficient way to produce good initial leads. Considering solubility, permeability and other drug- like properties early in library design and introducing both target and lead structural constraints in lead development are further ways to ensure more compounds make it to trial.  Note that there is not enough matter in the universe to prepare all possible combinatorial variations.  Related terms: combinatorial libraries, diversity, dynamic combinatorial, chemistry, microtiter plates, molecular diversity, fully combinatorial, pool/ split  Equivalent term?: diversity oriented synthesis

combinatorial chemistry techniques: A chemistry- based technology in which sets of reactions, for solution or solid- phase synthesis, are used to create molecular libraries for analysis of compounds on a large scale. MeSH, 2000

combinatorial library: A set of compounds prepared by combinatorial chemistry. May consist of a collection of pools, or sub- libraries. Its composition may be described by the chemset notation. IUPAC Combinatorial Chemistry  See combinatorial library definition in  IUPAC Biomolecular Screening  Narrower terms: combinatorial library SAR, combinatorial peptide libraries, combinatorial protein libraries,  compound library, hit optimization library, lead discovery library,  biased libraries, combinatorial antibody libraries, directed libraries, focused libraries, pool, pool/ split libraries, sub- library, random libraries, unbiased libraries; Broader term: library;   Related terms: combinatorial synthesis, fully combinatorial, scaffold.

combinatorial organic synthesis: A key feature of combinatorial techniques is that compound synthesis can be designed such that a range of structures can be produced simultaneously as mixtures in the same reaction vessel or individually in parallel using semi- automated synthesis. The repetitive nature of the synthetic processes involved in most combinatorial applications lends itself to automation or semi- automation. This key feature means that the bench chemist can single- handedly prepare tens, hundreds, or thousands of compounds of known structures in the time that it would take to prepare only a few pure entities by orthodox methodology.  NCI, CANCER DRUG DISCOVERY: DIVERSITY GENERATION AND SMART ASSAYS, RFA: CA-97-006, May 9, 1997

combinatorial synthesis: A process to prepare large sets of organic compounds by combining sets of building blocks. IUPAC Medicinal Chemistry

compound collection:  compound library, compound deck. Set of chemicals that has been assembled and annotated for easy storage and retrieval and that is available for screening.  Note: Generally consists of compounds synthesized by combinatorial or standard synthetic methods or purchased from commercial sources, or of natural products as pure samples. Compounds may be stored in dry powder form, or dissolved at a fixed concentration in a solvent, such as dimethylsulfoxide. IUPAC Biomolecular Screening recommendations 2011   Chemical library, compound library, compound collection: (1) Collection of samples (e.g., chemical compounds, natural products, over-expression library of a microbe) available for biological screening. (2) Set of compounds produced through combinatorial chemistry or other means, which expands around a single core structure or scaffold.  IUPAC Mec Chem 2013

DNA Library Gene categorie  See related gene library, genome library These are different types of libraries.

deconvolute: To render less complex; Process of optimizing an activity of interest by fractionating (normally by resynthesis, or by elaborating a partial library) a pool with some level of the desired activity to give a set of smaller pools. Repeating this strategy leads to single members with (ideally) a high level of activity and is termed iterative deconvolution.  IUPAC COMBINATORIAL CHEMISTRY

deconvolution: Wikipedia Narrower term: iterative deconvolution    Related term: unbiased library.

discrete sub- structural analysis DSA: See under privileged substructures

diversity: The "unrelatedness" of a set of, for example, building blocks or members of a combinatorial library, as measured by their properties such as atom connectivity, physical properties, computational measurements, or bioactivity. IUPAC Combinatorial Chemistry

The general understanding of this term is that diversity describes the degree of dissimilarity within a set of chemical structures. This Opinion article proposes that this understanding is superficial at best and irrelevant at worst. It is argued that relevant diversity can only be measured by the application of external criteria (such as a biological assay), which can discriminate the different structures by their different behaviour within this external context. According to this understanding, the diversity of a collection is highly dependent on the applied criteria. Therefore, a relevant diversity of chemical structures, per se, does not exist. HJ Roth, "There is no such thing as 'diversity'!" Curr Opin Chem Biol. 9 (3): 293- 295, June 2005

diversity- oriented synthesis: Commonly referred to as combinatorial chemical synthesis. is a process by which multiple compounds (chemical libraries) are generated simultaneously, in a predictable fashion, by using techniques that involve parallel chemical transformations. Diversity- oriented synthesis may use solid- or solution- phase reaction techniques. A library may be small (e.g., a few compounds) or large (e.g., thousands or even millions of compounds), and it may focus on a narrow or wide range of “diversity space.” When subjected to high- throughput biological screening. ... Leaders in the pharmaceutical industry (where diversity oriented synthesis is used extensively) view the limitations of current methodology as a problem of considerable urgency and see this as a significant impediment to the identification of drug candidates in new classes and with new mechanisms of action. NIGMS, Centers of Excellence in Chemical Methodologies and Library Development, RFA-GM-01-006, June 18, 2001 Related term: combinatorial chemistry

diversity space: The underlying philosophies that need to be contemplated before embarking on combinatorial studies have strong analogies in computational chemistry and molecular simulation. This parallel has been explored by Peter Coveney, until recently at Schlumberger Cambridge Research and now at Queen Mary and Westfield College, London. The whole idea of searching 'diversity space' is mirrored by the idea of searching for energy minima within conformational space in molecular systems. It is therefore clear that there are strong parallels between stochastic algorithms and parallel combinatorial search methods. Equivalents to the random search methods, such as Metropolis Monte Carlo, simulated annealing and genetic algorithms will all have their part to play in combinatorial problems, especially those that are particularly complex in nature or 'multivariate' as Coveney likes to call them. Andy Whiting, Discovery and Diversity, Chembytes E-zine, Royal Society of Chemistry, UK, 1999 See also under chemical diversity libraries Related term: Assays diversity screening

diversomers:  Solid- phase chemistry, organic synthesis, and an apparatus for multiple, simultaneous synthesis have been combined to generate libraries of organic compounds. Arrays of compounds were synthesized over two to three steps incorporating chemically diverse building blocks on a polystyrene- based solid support in a multiple, simultaneous manner. SH Dewitt et. al, Diversomers: an approach to nonpeptide, nonoligomeric chemical diversity. PNAS Proc Natl Acad Sci U S A; 90(15): 6909- 6913, Aug. 1, 1993

dynamic combinatorial chemistry:  Is based on the reversible combination of initial building blocks to form dynamic combinatorial libraries. It has recently emerged as an efficient strategy to detect and to evaluate affinity between the library products and a target molecule. Ivan Huc and Régis Nguyen "Dynamic Combinatorial Chemistry"  Combinatorial Chemistry & High Throughput Screening 4 (1): 53-74, 2001 

fragment libraries: The synthesis and screening of fragment libraries is at the leading edge of this trend toward more design-conscious compound library synthesis. Fragment-based approaches emerged from the understanding that a molecular weight >500 Daltons typically correlates with poorer oral bioavailability. Furthermore, drug compounds tend to be even larger than the leads from which they were derived as a result of lead optimization efforts. Vicki Glaser, When smaller is better 2008  
Related term: Assays fragment based screening

Free Wilson model: See under combinatorial library SAR

fully combinatorial: Containing, or designed to contain, all possible combinations of building blocks. Pool/ Split libraries are generally fully combinatorial while parallel synthesis libraries may not be. See also reagent efficiency. IUPAC COMBINATORIAL CHEMISTRY

gel phase: Description applied to certain 'solid' supports which display properties intermediate between solid and liquid phases, e.g. in the apparent mobility of the support as determined by NMR spectroscopy. IUPAC COMBINATORIAL CHEMISTRY

gene library, genomic library: Gene categorie  These are different types of libraries.

high throughput chemistry: Some years ago, combinatorial chemistry began emerging as a highly efficient means of generating as many compounds as desired, feeding the demand for compounds as a result of increasingly high throughput screening.  Now there is growing recognition of the need for greater quality and diversity over sheer numbers, and greater experience is making it possible to more intelligently design better libraries. 

high throughput organic synthesis: It appears that the real difficulties in drug discovery chemistry continue to be from high-throughput organic synthesis through purification. New tools, improved strategies, enhanced techniques, and new methods are beginning to revolutionize these segments of the drug discovery industry. 

I.R. Thermography Infrared thermography: Screening technique where the heat of reaction of a multitude of samples is simultaneously measured. Has been applied in particular to the screening of libraries of potential catalysts. IUPAC Combinatorial Chemistry

in situ scaffold formation: Process whereby a scaffold is formed during library production which contains residues of at least two building blocks; compare preformed scaffold. IUPAC COMBINATORIAL CHEMISTRY

iterative deconvolution: Multi- step application of deconvolution where successively smaller sub- libraries are prepared and tested to identify individual active members of a combinatorial library. IUPAC COMBINATORIAL CHEMISTRY Broader term deconvolution.

lead, lead likeness: Assays & screening

library: An unordered collection of clones (i.e., cloned DNA from a particular organism), whose relationship to each other can be established by physical mapping. [DOE] Narrower terms: cDNA libraries, chemical libraries, combinatorial library, compound libraries, hit optimization library, lead discovery library,  biased libraries, combinatorial antibody libraries, directed libraries, focused libraries, pool, pool/ split libraries, sub- library, random libraries, unbiased libraries; Related terms: Cell biology; gene library, genomic library; Sequencing: DNA library

linker: Bifunctional chemical moiety attaching compound to solid support or soluble support which can be cleaved to release compounds from the support. Careful choice of linker allows cleavage to be performed under appropriate conditions compatible with the stability of the compound and assay method. IUPAC COMBINATORIAL CHEMISTRY

MPS multiple parallel synthesis: See parallel synthesis

member: a) specific compound which is included in a library; b) the uncharacterized physical product of a library synthesis. IUPAC COMBINATORIAL CHEMISTRY

microwave chemistry: Emerging as a popular tool for manipulating chemical synthesis.

molecular diversity: Covers combinatorial chemistry, combinatorial libraries, solid- phase synthesis, robotics, molecular graphics: The visualization and manipulation of three- dimensional representations of molecules on a graphical display device. IUPAC Medicinal Chemistry Related terms combinatorial chemistry, drug design, high throughput organic synthesis; Drug discovery informatics  Broader term diversity

monomer: Member of a building block set which can be repeatedly incorporated into a library to give a set of compounds of repeating structure; e.g. amino acids in a peptide library. IUPAC COMBINATORIAL CHEMISTRY

orthogonality: (a) Property of protecting groups or linkers allowing removal, modification, or cleavage of one such without affecting others; (b) pooling strategy whereby library members are incorporated in more than one pool, and are mixed with a different set of other members in each pool. Thus a hit results in two or more active pools with only one member in common.  IUPAC COMBINATORIAL CHEMISTRY

One meaning of orthogonal is independent of disjoint, with no overlap.

parallel synthesis: Strategy whereby sets of discrete compounds are prepared simultaneously in arrays of physically separate reaction vessels or microcompartments without interchange of intermediates during the assembly process. Contrast pool/ split. IUPAC COMBINATORIAL CHEMISTRY

peptide library: A collection of cloned peptides, or chemically synthesized peptides, frequently consisting of all possible combinations of amino acids making up an n-amino acid peptide. MeSH, 1997

Used by Mario Geysen (1985 +) to map peptide epitopes or antigenic sites on proteins.  Numerous strategies have developed over the past 20 years to synthesize mixtures of thousands to millions of peptides and allow selection of those with the desired activities.  Related term: combinatorial peptide libraries

pool: (a) A sub-library; (b) process of  combining and mixing library components or sub- libraries. See pool/ split. IUPAC COMBINATORIAL CHEMISTRY

pool/split: Also split/ pool; split & mix; divide, couple, recombine; portion/ mix. Strategy for assembly of a combinatorial library. The solid support is divided into portions, each of which is subjected to reaction with a single building block. Pooling of these portions results in a single batch of solid support bearing a mixture of components. Repetition of the divide, couple, recombine processes results in a library where each discrete particle of solid support carries a single library member, and the number of  members is equal to the product of the number of building blocks incorporated at each step (i.e. fully combinatorial) IUPAC COMBINATORIAL CHEMISTRY

preformed scaffold: A scaffold which is incorporated into the library as a unit. Compare in situ scaffold. IUPAC COMBINATORIAL CHEMISTRY Broader term: scaffold.

privileged structure: Substructural feature which confers desirable (often drug- like) properties on compounds containing that feature. Often consists of a semi- rigid scaffold which is able to present multiple hydrophobic residues without undergoing hydrophobic collapse. IUPAC COMBINATORIAL CHEMISTRY

property space: Multidimensional representation of a set of compounds in which the axes represent quantifiable properties such as molecular weight, CLogP, molar refractivity, etc., and individual compounds are represented by a vector or set of coordinates. IUPAC COMBINATORIAL CHEMISTRY

random library: See unbiased library. IUPAC Combinatorial Chemistry

residue: (a) Portion of a chemical structure which can be identified as being derived from a particular building block, such as the alanine residue in the peptide (b) portion of a building block which is incorporated into the final product but is not part of the scaffold. IUPAC Combinatorial Chemistry Compare residue Proteins

resin: Insoluble polymeric material which allows ready separation from liquid-phase materials by filtration; can be used to carry library members (i.e. solid support) or reagents, or to trap excess reagents or reaction by-products (see scavenger resin) IUPAC Combinatorial

retrosynthetic analysis: A synthesis is planned in reverse, beginning with the final product. The investigator identifies a reaction that could afford a particular product and then deduces the structure of the required starting material. This process is repeated until a set of easily- procured starting materials is identified that can be converted, by using an appropriate sequence of reactions, to a complex target structure.  NIGMS, Centers of Excellence in Chemical Methodologies and Library Development, RFA-GM-01-006, June 18, 2001 Compare: chemical diversity libraries

ring-closing metathesis RCM; ring- opening metathesis polymerizaton ROMP: See under metathesis strategies

SAR  Cheminformatics structure activity relationship SAR

scaffold: Core portion of a molecule common to all members of a combinatorial library. IUPAC Combinatorial Chemistry  This is different from the scaffold defined in the Sequencing  Narrower terms:  in situ scaffold, preformed scaffold. Chemistry molecular scaffold

small molecule libraries: Large collections of small molecules (molecular weight about 600 or less), of similar or diverse nature which are used for high-throughput screening analysis of the gene function, protein interaction, cellular processing, biochemical pathways, or other chemical interactions. MeSH 2008

SPE: See solid phase extraction.

solid-phase extraction: Method for sample purification whereby either the desired or undesired components of a mixture have preferential affinity for a solid material. Adding the mixture to the solid material then allows facile separation of the desired material by filtration. Related terms sequestration- enabling reagent and scavenger  resin. IUPAC Combinatorial Chemistry

solid phase synthesis: In solid- phase approaches, pin or bead techniques permit the synthesis of different molecules on each pin (i.e., "one molecule- one bead"). The products of solid- phase synthesis can be cleaved from the backbone matrix for solution screening (which is essential when the screening target is a cell), or the most active molecules displayed on the polymer surface may be detected using molecular targets (receptor, enzyme, antibody) pre- tagged with a means of detection (visible color, fluorescence, radioactivity, chromophore, etc.) and then isolated and identified. NCI, CANCER DRUG DISCOVERY: DIVERSITY GENERATION AND SMART ASSAYS, RFA: CA-97-006, May 9, 1997  Related term: high throughput purification  Narrower term: solid phase- traceless

solid support:  Insoluble, functionalized, polymeric material to which library members or reagents may be attached (often via a linker) allowing them to be readily separated (by filtration, centrifugation, etc.) from excess reagents, soluble reaction by- products, or solvents. IUPAC Combinatorial chemistry Related terms: bead, high throughput organic synthesis, cleavage, gel phase, linker, pool/ split, resin, sort and combine

soluble support: An attachment, common to all library members, which renders the library components soluble under conditions for library synthesis, but which can be readily separated from most other soluble components when desired by some simple physical process. This process has been termed liquid- phase chemistry. Examples of soluble supports include linear polymers such as poly(ethylene glycol), dendrimers, or fluorinated compounds which selectively partition into fluorine- rich solvents (see fluorous synthesis). IUPAC COMBINATORIAL CHEMISTRY

solution phase synthesis:  Solution techniques utilize methods essentially similar to standard methods for the synthesis of single compounds, except that instead of utilizing one well- defined reaction partner of each type per reaction, mixtures of several known reaction partners are utilized as building blocks, thus resulting in mixtures of analogs. The type of the chosen chemical reaction and the number of reactants depend on the nature of the desired structures. These mixtures may not be purified elaborately but are directly subjected to screening against the desired target (e.g., receptor, enzyme, antibody, cell). The chemistry is then repeated a few times in batches using different but well defined mixtures of reaction partners, and the products isolated and screened. In order to identify the leads, the chemistry is repeated several times using fewer reaction partners in each run, and the products are screened. This deconvolution process is repeated until the most potent lead is identified.  NCI, CANCER DRUG DISCOVERY: DIVERSITY GENERATION AND SMART ASSAYS, RFA: CA-97-006, May 9, 1997  Related term: high throughput purification

sort and combine: Use of directed sorting to facilitate library assembly. Related to pool/ split protocol but more commonly applied to macroscopic solid supports (such as pins and related carriers) where each library member is found on only one, or a small number of carriers. IUPAC COMBINATORIAL CHEMISTRY

space: Narrower terms: chemical space, diversity space, property space. Related term: spatially addressable

spatially addressable: Having the ability to identify at least part of the structure of a library component or pool by noting its physical location in an array. IUPAC COMBINATORIAL CHEMISTRY

split/pool: See pool/split

split samples: See under aliquot.

sub-library: See also pool. A subset of a combinatorial library, physically separate from the rest of the library, generally with one or more fixed building block. IUPAC COMBINATORIAL CHEMISTRY

synthesis: Narrower terms: biomimetic synthesis, combinatorial biosynthesis, combinatorial synthesis,  high throughput organic synthesis, multi- component reactions, organic synthesis, parallel synthesis, solid phase synthesis, solution phase synthesis, thematic libraries- synthesis; array synthesis Related term: retrosynthetic analysis

target-oriented synthesis TOS: Modern drug discovery often involves screening small molecules for their ability to bind to a preselected protein target. Target- oriented syntheses of these small molecules, individually or as collections (focused libraries), can be planned effectively with retrosynthetic analysis. Drug discovery can also involve screening small molecules for their ability to modulate a biological pathway in cells or organisms, without regard for any particular protein target. This process is likely to benefit in the future from an evolving forward analysis of synthetic pathways, used in diversity- oriented synthesis, that leads to structurally complex and diverse small molecules. One goal of diversity- oriented syntheses is to synthesize efficiently a collection of small molecules capable of perturbing any disease-related biological pathway, leading eventually to the identification of therapeutic protein targets capable of being modulated by small molecules. Stuart l. Schreiber, Target- oriented and diversity- oriented organic synthesis in drug discovery, Science 287(5460): 1964- 1969, Mar 17, 2000

tether: See linker.

unbiased library: Library prepared from building blocks and scaffold chosen without bias towards a particular target. IUPAC COMBINATORIAL CHEMISTRY

Libraries & Synthesis resources
ChemDiv Glossary , 2018
Combichem Glossary, Combinatorial Chemistry Review, Professor Oleg Larin, 2004-2016 
IUPAC Glossary of terms in Biomolecular Screening 2011
IUPAC Combinatorial International Union of Pure and Applied Chemistry, Glossary of Terms Used in Combinatorial Chemistry, D. Maclean, J. J. Baldwin, V.T. Ivanov, Y. Kato, A. Shaw, P. Schneider, and E. M.. Gordon, Pure Appl. Chem., Vol. 71, No. 12, pp. 2349- 2365, 1999. 100+ definitions.
IUPAC Glossary of terms in combinatorial chemistry, revision of the 1999 Technical Report,

How to look for other unfamiliar  terms

IUPAC definitions are reprinted with the permission of the International Union of Pure and Applied Chemistry.

It is easy to lose sight of the numbers involved in combinatorial chemistry and useful to recall the old story of the inventor of chess and his patron, the emperor of China. The emperor had so fallen in love with his new game that he offered the inventor a reward of anything he wanted in the kingdom. "Just one grain of rice on the first square of the chessboard, and two on the next."  "That's all?" "Yes, and four grains of rice on the third square, and so on." This seemingly modest request was quickly granted. But the grains of rice required for each square ultimately required more rice fields than would cover twice the surface of the Earth, oceans included.

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