Mathématiques récréatives: Éclairages historiques et épistémologiques 9782759823192
Apprendre les mathématiques par les jeux. Cette idée vous paraît farfelue ? Détrompez-vous : les jeux ont de tout temps
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French Pages 254 [290] Year 2019
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- Nathalie Chevalarias
- Michèle Gandit
- Marcel Morales
- Dominique Tournès
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Mathématiques récréatives Éclairages historiques et épistémologiques
Enseigner les sciences Collection dirigée par Michèle Gandit /DFROOHFWLRQ©ௗ(QVHLJQHUOHVVFLHQFHVௗªV¶DGUHVVHDX[HQVHLJQDQWVGHVSUHPLHU HWVHFRQGGHJUpVjFHX[GHO¶XQLYHUVLWpDX[IRUPDWHXUVDLQVLTX¶jWRXWHSHUVRQQHLQWpUHVVpHSDUOHVPDWKpPDWLTXHVVFLHQFHVHWWHFKQLTXHVO¶pGXFDWLRQRX ODIRUPDWLRQVFLHQWL¿TXH6RQREMHFWLIHVWGHIRXUQLUGHVUHVVRXUFHV pFODLUDJHV KLVWRULTXHVpSLVWpPRORJLTXHV¿FKHVGHWUDYDX[SUDWLTXHV¿FKHVSRXUO¶HQVHLJQDQW SRXUPLHX[H[SOLTXHUHQVHLJQHUFHVGLVFLSOLQHV,OV¶DJLWQRWDPPHQW GHYDORULVHUHWGLIIXVHUOHVWUDYDX[GHUHFKHUFKHDFWLRQGHV,5(0GH0DWKV jPRGHOHUDLQVLTXHFHX[TXLVRQWPHQpVGDQVWRXVOHVSD\VIUDQFRSKRQHV JUkFH DX UpVHDX GHV ,5(0 HW j VHV OLHQV DYHF O¶$IULTXH O¶$PpULTXH ODWLQH O¶$VLHOH4XpEHFHWO¶(XURSHIUDQFRSKRQH'LYHUVW\SHVGHWUDYDX[HQODQJXH IUDQoDLVHSHXYHQWrWUHVRXPLVDLQVLTXHG¶DXWUHVW\SHVGHVXSSRUWV VXSSRUWV QXPpULTXHV REMHWV SK\VLTXHV GH PpGLDWLRQ GHV VFLHQFHV V¶DFFRPSDJQDQW G¶XQHUpÀH[LRQVXUOHVVDYRLUVHQVHLJQpV
Mathématiques récréatives Éclairages historiques et épistémologiques
Sous la direction de Nathalie Chevalarias Michèle Gandit Marcel Morales Dominique Tournès
EDP SCIENCES UGA ÉDITIONS 2019
Photo de couverture : IStock/CreativaImages. ,OOXVWUDWLRQVG¶DSUqVFHUWDLQHV¿JXUHVGXSUpVHQWRXYUDJH Graphisme : Jean-Christophe Monnier. Maquette : Jean-Christophe Monnier et Gwenn Cognard. ÉGLWLRQ *ZHQQ&RJQDUGHW6WpSKDQLH7ULQH ISBN 978-2-7598-2318-5 ISSN en cours ,PSULPpHQ)UDQFHSDU3UpVHQFH*UDSKLTXH0RQWV
© EDP Sciences 17, avenue du Hoggar 3DUFG¶$FWLYLWpGH&RXUWDE°XI±%3 /HV8OLV&HGH[$±)UDQFH
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TABLE DES MATIÈRES
Introduction 9
Partie 1 – Jeux de société ou miroirs d’une société ? Le jeu des quinze croyants et des quinze infidèles : variations sur la violence 19 ,QWURGXFWLRQ 6RXUFHVODWLQHVPpGLpYDOHV 6RXUFHVHQODQJXHVJHUPDQLTXHV 6RXUFHVKpEUDwTXHVDUDEHVSHUVDQHVWXUTXHVDIULFDLQHV 6RXUFHVHQODQJXHVURPDQHV &RQFOXVLRQ 5pIpUHQFHVELEOLRJUDSKLTXHV 3UDWLTXHVSRXUO¶HQVHLJQDQWRXOHIRUPDWHXU
L’exponentielle, entre jeu mathématique et vision du monde 47 ,QWURGXFWLRQ *UDLQVGHEOp GRXEOHPHQWVVXUO¶pFKLTXLHU (QWUH©IpFRQGHQDWXUHªHWDQJRLVVHVPDOWKXVLHQQHV 'LUKDPV TXDQGO¶DUJHQWFUpHO¶DUJHQW 5pIpUHQFHVELEOLRJUDSKLTXHV 3UDWLTXHVSRXUO¶HQVHLJQDQWRXOHIRUPDWHXU
Partie 2 – Portraits de récréateurs en leur temps Didier Henrion, compilateur de récréations mathématiques des années 1620 65 +HQULRQXQLQFRQQXUpSXWp" /DFRQIXVLRQGHVLGHQWLWpV +HQULRQ&\ULDTXH+pULJRQH« /HV©4XHVWLRQVLQJHQLHXVHVªGDQVODCollection mathematique /HVFRPPHQWDLUHVVXUODRecreation mathematique &RQFOXVLRQ 5pIpUHQFHVELEOLRJUDSKLTXHV 3UDWLTXHVSRXUO¶HQVHLJQDQWRXOHIRUPDWHXU
Revenir aux mathématiques par les récréations : l’exemple de Henri Auguste Delannoy (1833-1915) 85 'HODQQR\XQPLOLWDLUHGHFDUULqUHGHj 'HVUpFUpDWLRQVGDQVXQHSUHVVHPLOLWDQWH /HVWDWXWGHVUpFUpDWLRQVPDWKpPDWLTXHV &RQWULEXWLRQVGH'HODQQR\ &RQFOXVLRQ 5pIpUHQFHVELEOLRJUDSKLTXHV
Les récréations mathématiques chez Charles-Ange Laisant : de la géométrie de situation à l’Initiation mathématique 113 &KDQJHPHQWG¶LWLQpUDLUHSRXU/DLVDQW eGRXDUG/XFDVDPLHWFROODERUDWHXU 9HUVO¶Initiation mathématique &RQFOXVLRQ 5pIpUHQFHVELEOLRJUDSKLTXHV 3UDWLTXHVSRXUO¶HQVHLJQDQWRXOHIRUPDWHXU
Partie 3 – Variations combinatoires et algorithmiques La rithmomachie, un « jeu pédagogique » du XIe au XVIe siècle 139 ,QWURGXFWLRQ 8QHEUqYHKLVWRLUHGXMHX /HVUDSSRUWVGHQRPEUHVVHORQ%RqFH 'HVFULSWLRQGXMHXYHUVLRQXVIeVLqFOH /HMHXDXXIeVLqFOH
/HMHXDXXXIeVLqFOH 5pIpUHQFHVELEOLRJUDSKLTXHV
Géométrie, combinatoire et algorithmes des carrés magiques 159 /HPpPRLUHDes quarrés ou tables magiquesGH)UpQLFOH /DFRPELQDWRLUHGHVFDUUpVPDJLTXHVFKH])URORY &DUUpVPDJLTXHVHWUpFUpDWLRQVPDWKpPDWLTXHVFKH]/XFDV &RQFOXVLRQ 5pIpUHQFHVELEOLRJUDSKLTXHV
Les jeux combinatoires ou comment tisser un lien entre mathématiques, algorithmique et programmation 181 ,QWURGXFWLRQ 1DLVVDQFHGHODWKpRULHGHVMHX[FRPELQDWRLUHV /LHQVDYHFO¶DOJRULWKPLTXHHWODSURJUDPPDWLRQ &RQFOXVLRQ $QQH[H$ OLVWHGHVLQVWUXFWLRQVVXLYLHVSDUOHSURJUDPPH GH'U1LPHQIUDQoDLV $QQH[H% WDEOHDXSRXUH[pFXWHUODOLVWHGHVLQVWUXFWLRQV 5pIpUHQFHVELEOLRJUDSKLTXHV 3UDWLTXHVSRXUO¶HQVHLJQDQWRXOHIRUPDWHXU
Partie 4 – Quand la récréation entre en classe Entre histoire et mathématiques : variations pédagogiques autour des problèmes d’Alcuin 205 /HVSURSRVLWLRQVGLWHV©G¶$OFXLQª pOpPHQWVFRQWH[WXHOV ¬ODGpFRXYHUWHGHTXHOTXHVXQVGHV©SUREOqPHVG¶$OFXLQª 9DULDWLRQVSpGDJRJLTXHVDXWRXUGHVSUREOqPHVG¶$OFXLQ FRPPHQWOHVUHSUHQGUHHWOHVRUJDQLVHU" &RQFOXVLRQ 5pIpUHQFHVELEOLRJUDSKLTXHV
Récréations mathématiques et algorithmique dans le Liber abaci de Fibonacci (XIIIe siècle) 225 ,QWURGXFWLRQ )LERQDFFL±TXHOTXHVpOpPHQWVFRQWH[WXHOV 'HVSUREOqPHVUpFUpDWLIVGDQVOHLiber abaci
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À propos des auteurs 253
Dominique Tournès
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REVENIR AUX MATHÉMATIQUES PAR LES RÉCRÉATIONS : L’EXEMPLE DE HENRI AUGUSTE DELANNOY (1833-1915)
Fig. 1 – Henri Auguste Delannoy [Collection du Musée de Guéret, photographie M.-A. Baldensperger].
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Revenir aux mathématiques par les récréations : l’exemple de Henri Auguste Delannoy (1833-1915)
Tableau 2 – Suite anglaise : d(p, q) = d(p − 1, q) + d(p, q − 1) + d(p – 1, q − 1) ; d(0, q) = d(p, 0) = 1. q
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Géométrie, combinatoire et algorithmes des carrés magiques
2. La somme des 4 nombres d’un « carré d’angle » de côté 2 égale celle du « carré d’angle » opposé. 3. De même pour les 4 nombres des angles des carrés de côté 3. 4. Si la somme des 4 nombres d’un des « carrés d’angle » de côté 2 vaut S, alors ce sera aussi vrai pour les autres « carrés d’angle » de côté 2 et pour les angles des carrés de côté 3. Ce sont ces propriétés qui vont orienter les recherches des auteurs du XIXe siècle désirant comprendre et poursuivre le résultat obtenu par Frénicle.
La combinatoire des carrés magiques chez Frolov L’héritage des carrés magiques dans les années 1870-1880 Leonhard Euler a consacré deux mémoires aux carrés magiques ©ௗDe quadris magicisௗªUpGLJpHQHW©ௗ5HFKHUFKHVVXUXQHQRXYHOOHHVSqFHGHFDUUpV PDJLTXHVௗªSXEOLpHQIUDQoDLVHQGDQVOHVMémoires de la Société des sciences de Flessingue. Signalons que les deux mémoires d’Euler paraissent dans le tome II des Commentationes arithmeticaepGLWpVHQ SXLVTX¶LOV VRQWUpLPSULPpVHQSRXUOHVOpera Omnia. (QGDQVODNouvelle correspondance mathématique, Édouard Lucas DERUGHO¶XQGHVSUREOqPHVG¶(XOHU¿JXUDQWGDQVOHVHFRQGPpPRLUH¬SDUWLU de cette date, il y a un regain d’intérêt pour les carrés magiques en France, en SDUWLFXOLHUORUVGHVFRQJUqVGHO¶$VVRFLDWLRQIUDQoDLVHSRXUO¶DYDQFHPHQWGHV VFLHQFHV $)$6 %DUELQ 1RWRQVTX¶HQXQSURIHVVHXUGHPDWKpmatiques, nommé A. Labosne, réédite l’ouvrage de Bachet, qui était très rare, HQOHVLPSOL¿DQWHWHQO¶DXJPHQWDQW %DFKHW Mikhail Frolow publie à Saint-Pétersbourg HQ HQ IUDQoDLV Le problème d’Euler et les carrés magiques SXLV j 3DULV HQ Les carrés magiques, nouvelle étude. Il est ingénieur et il publiera plus tard en France VRXVOHQRPGH0LFKHO)URORY %DUELQ 'DQVOHVHFRQGRXYUDJHDXTXHO collaborent Édouard Lucas et Henri Delannoy, il corrige son ouvrage précédent et il s’intéresse à la table de Frénicle.
Les carrés magiques de côté 4 chez Frolov 'DQVVRQRXYUDJHGH)URORYH[SOLTXHOHVPpWKRGHVTXLOXLRQWSHUPLVGH ©ௗWURXYHUOHVFDUUpVGHLQGpSHQGDPPHQWGHV7DEOHVGH)UpQLFOHௗªTX¶LO Q¶DYDLWSDVjVDGLVSRVLWLRQ©ௗPDLVHQFRUHGHOHVFODVVHUGDQVXQRUGUHPpWKRGLTXH HW FRPPRGH SRXU OHXU YpUL¿FDWLRQௗª )URORZ ,O FRPPHQFH
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2. La somme des 4 nombres d’un « carré d’angle » de côté 2 égale celle du « carré d’angle » opposé. 3. De même pour les 4 nombres des angles des carrés de côté 3. 4. Si la somme des 4 nombres d’un des « carrés d’angle » de côté 2 vaut S, alors ce sera aussi vrai pour les autres « carrés d’angle » de côté 2 et pour les angles des carrés de côté 3. Ce sont ces propriétés qui vont orienter les recherches des auteurs du XIXe siècle désirant comprendre et poursuivre le résultat obtenu par Frénicle.
La combinatoire des carrés magiques chez Frolov L’héritage des carrés magiques dans les années 1870-1880 Leonhard Euler a consacré deux mémoires aux carrés magiques ©ௗDe quadris magicisௗªUpGLJpHQHW©ௗ5HFKHUFKHVVXUXQHQRXYHOOHHVSqFHGHFDUUpV PDJLTXHVௗªSXEOLpHQIUDQoDLVHQGDQVOHVMémoires de la Société des sciences de Flessingue. Signalons que les deux mémoires d’Euler paraissent dans le tome II des Commentationes arithmeticaepGLWpVHQ SXLVTX¶LOV VRQWUpLPSULPpVHQSRXUOHVOpera Omnia. (QGDQVODNouvelle correspondance mathématique, Édouard Lucas DERUGHO¶XQGHVSUREOqPHVG¶(XOHU¿JXUDQWGDQVOHVHFRQGPpPRLUH¬SDUWLU de cette date, il y a un regain d’intérêt pour les carrés magiques en France, en SDUWLFXOLHUORUVGHVFRQJUqVGHO¶$VVRFLDWLRQIUDQoDLVHSRXUO¶DYDQFHPHQWGHV VFLHQFHV $)$6 %DUELQ 1RWRQVTX¶HQXQSURIHVVHXUGHPDWKpmatiques, nommé A. Labosne, réédite l’ouvrage de Bachet, qui était très rare, HQOHVLPSOL¿DQWHWHQO¶DXJPHQWDQW %DFKHW Mikhail Frolow publie à Saint-Pétersbourg HQ HQ IUDQoDLV Le problème d’Euler et les carrés magiques SXLV j 3DULV HQ Les carrés magiques, nouvelle étude. Il est ingénieur et il publiera plus tard en France VRXVOHQRPGH0LFKHO)URORY %DUELQ 'DQVOHVHFRQGRXYUDJHDXTXHO collaborent Édouard Lucas et Henri Delannoy, il corrige son ouvrage précédent et il s’intéresse à la table de Frénicle.
Les carrés magiques de côté 4 chez Frolov 'DQVVRQRXYUDJHGH)URORYH[SOLTXHOHVPpWKRGHVTXLOXLRQWSHUPLVGH ©ௗWURXYHUOHVFDUUpVGHLQGpSHQGDPPHQWGHV7DEOHVGH)UpQLFOHௗªTX¶LO Q¶DYDLWSDVjVDGLVSRVLWLRQ©ௗPDLVHQFRUHGHOHVFODVVHUGDQVXQRUGUHPpWKRGLTXH HW FRPPRGH SRXU OHXU YpUL¿FDWLRQௗª )URORZ ,O FRPPHQFH
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× Fig. 12 – Les diagrammes de Frolov et les « carrés médians ».
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Géométrie, combinatoire et algorithmes des carrés magiques
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Fig. 13 – La détermination combinatoire d’un carré de côté 4.
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Fig. 14 – Une transformation de carrés magiques chez Frolov.
Carrés magiques et récréations mathématiques chez Lucas /HVUpFUpDWLRQVPDWKpPDWLTXHVVRQWHQYRJXHjOD¿QGXXIXVLqFOHHQ)UDQFH 8Q ©ௗMRXUQDO GHV MHX[ G¶HVSULW HW GH FRPELQDLVRQVௗª WLWUp Les tablettes du chercheur HVW FUpp HQ SDU %'HFRORPEH 2Q \ WURXYH GHV SUREOqPHV G¶pFKHFVGHGRPLQRVGHVFDUUpVPDJLTXHV8QH6RFLpWpGHVVFLHQFHVUpFUpDWLYHVYRLWOHMRXUHQ DYHF&KDUOHV$QJH/DLVDQWFRPPHYLFH3UpVLGHQW YRLU%DUELQ /HV TXDWUH YROXPHV GHV Récréations mathématiques G¶eGRXDUG /XFDV SDUDLVVHQWGHj /XFDVHVWPDWKpPDWLFLHQHQWKpRULHGHVQRPEUHV HWHQVHLJQDQWGHPDWKpPDWLTXHVj3DULVDXO\FpH&KDUOHPDJQHSXLVDXO\FpH 6DLQW/RXLV YRLU 'HFDLOORW ,O pFULW DXVVL HQ La Fasioulette du 3URIHVVHXU 1&ODXV GH 6LDP TXL HVW OD SUHPLqUH VpULH GH ©ௗ-HX[ VFLHQWL¿TXHVSRXUVHUYLUjO¶+LVWRLUHjO¶(QVHLJQHPHQWHWjOD3UDWLTXHௗª/XLHWVRQ DPL/DLVDQWDWWULEXHQWXQU{OHpGXFDWLIDX[UpFUpDWLRQVPDWKpPDWLTXHV %DUELQ ௗ %DUELQ
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Géométrie, combinatoire et algorithmes des carrés magiques
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La composition géométrique des carrés de côté 4 /XFDVQRPPH©ௗpJDOLWpVjTXDWUHERXOHVௗªGHVpJDOLWpVGHVRPPHVGHTXDWUH QRPEUHV j OD ©ௗFRQVWDQWHௗª F¶HVWjGLUH OD VRPPH PDJLTXH GDQV XQ FDUUp PDJLTXH GH F{Wp /HXUV SODFHV VRQW GpVLJQpHV SDU GHV V\PEROHV TXL SHUPHWWHQWGHOHVUHSpUHUYLVXHOOHPHQW/XFDVGpPRQWUHSDUpFKDQJHGHVTXDUWLHUVHWSDUURWDWLRQTXDWUHpJDOLWpV ¿J /XFDV
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Fig. 17 – Le théorème II de Lucas.
Fig. 16 – Le théorème I de Lucas.
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Fig. 18 – Le théorème III de Lucas.
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Fig. 3 – Les cases hachurées sont les positions perdantes sur lesquelles il faut amener la reine pour pouvoir gagner la partie. Les cases noires et grises (clair et foncé) sont les cases qu’il faut éviter, au risque de laisser notre adversaire atteindre une case hachurée. Pour une analyse détaillée de la recherche des positions perdantes au jeu de la reine de Wythoff, et de leurs propriétés, voir (Rougetet 2016b).
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Fig. 4 – Illustration du jeu proposé par Samuel Loyd (Loyd 1914 : 232). C’est à Rip Van Winkle de jouer après qu’un nain a abattu la 2e quille, que doit-il jouer pour remporter la victoire ?
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Fig. 9 – Support en plastique rouge du Dr. Nim. Les billes sont placées initialement dans la gouttière en haut du support. Le joueur actionne la gâchette en bas à droite du plateau pour faire descendre la bille qui passe par le chemin déterminé selon la position des différents flip-flops.
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Fig. 10 – Position initiale des leviers pour jouer en version misère à Dr. Nim avec 15 billes placées initialement dans la gouttière. Le levier de droite ainsi placé permet au joueur qui défie Dr. Nim, qu’il joue en premier ou non, de prendre la main sur la partie (à condition de bien jouer, auquel cas Dr. Nim prend la main et gagne la partie à coup sûr) [© L'inventeur du jeu : John Thomas Godfrey ; Crédit photo : Andrew Beck].
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Fig. 11 – Liste des instructions suivies par le programme au cours d’une partie de Dr. Nim (1968 : 14) [© L'inventeur du jeu : John Thomas Godfrey ; Crédit photo : Andrew Beck].
Fig. 12 – Tableau pour exécuter les instructions suivies par le programme au cours d’une partie de Dr. Nim (1968 : 13) [© L'inventeur du jeu : John Thomas Godfrey ; Crédit photo : Andrew Beck].
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Fig. 12 – Tableau pour exécuter les instructions suivies par le programme au cours d’une partie de Dr. Nim (1968 : 13) [© L'inventeur du jeu : John Thomas Godfrey ; Crédit photo : Andrew Beck].
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