Mathématiques récréatives: Éclairages historiques et épistémologiques 9782759823192

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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¿TXH6RQREMHFWLIHVWGHIRXUQLUGHVUHVVRXUFHVpFODLUDJHV 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\SHVGHVXSSRUWVVXSSRUWV 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 *UDLQVGHEOpGRXEOHPHQWVVXUO¶pFKLTXLHU (QWUH©IpFRQGHQDWXUHªHWDQJRLVVHVPDOWKXVLHQQHV 'LUKDPVTXDQGO¶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¶$OFXLQªpOpPHQWVFRQWH[WXHOV ¬ODGpFRXYHUWHGHTXHOTXHVXQVGHV©SUREOqPHVG¶$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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Fig. 2 – Échiquiers des dames à la polonaise (Lucas 1879-1880 : 155) [© BnF].

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Fig. 4 – Le jeu de baguenaudier (Lucas 1880-1881a : 36) [© BnF].



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Fig. 10 – La première page de la « Table générale des quarrés de quatre » (non paginée) [© BnF].

 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 arithmeticaepGLWpVHQSXLVTX¶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.

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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 arithmeticaepGLWpVHQSXLVTX¶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.

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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¶pFKHFVGHGRPLQRVGHVFDUUpVPDJLTXHV8QH6RFLpWpGHVVFLHQFHVUpFUpDWLYHVYRLWOHMRXUHQDYHF&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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Les « transformations générales des carrés » par Lucas /D FLQTXLqPH UpFUpDWLRQ GX 3 4 YROXPH GHV Récréations mathé1 2 matiquesGH/XFDV V¶LQWLWXOH 4 3 ©ௗ/HVFDUUpVPDJLTXHVGH)HUPDWௗª 2 1 PDLV HOOH FRQFHUQH DXVVL OHV WDEOHV GH )UpQLFOH (OOH FRPPHQFH SDU OD Fig. 15 – Transformation par échanges de quartiers FRQVHUYDWLRQ GX FDUDFWqUH PDJLTXH dans un carré pair. SRXU GHV WUDQVIRUPDWLRQV JpRPpWULTXHV GH URWDWLRQ HW GH V\PpWULH DJLVVDQW VXU OHV FDUUpV GH F{Wp SXLV OD FRQVWUXFWLRQGXFDUUpPDJLTXHGHF{WpSDU)HUPDWTXLHVWDXVVLFHOOHTXH QRXVDYRQVYXHSOXVKDXWFKH])UpQLFOH /XFDV DERUGH HQVXLWH FH TX¶LO QRPPH ©ௗWUDQVIRUPDWLRQV JpQpUDOHV GHV FDUUpVௗª,OFRPPHQFHSDUOHVpFKDQJHVSDUTXDUWLHUVSRXUGHVFDUUpVGHF{Wp SDLU¿J ©ௗWRXWFDUUpSDLUUHVWHPDJLTXHVLO¶RQpFKDQJHVLPXOWDQpPHQW VDQVOHVWRXUQHUOHVTXDUWLHUVRSSRVpVௗª/XFDV  ,OH[DPLQHDXVVLOHFDVGHVFDUUpVLPSDLUV&HVWUDQVIRUPDWLRQVQHGRQQHQW SDV GH QRXYHDX[ FDUUpV SRXU OH F{Wp PDLV SRXU OH F{Wp RQ REWLHQW GHV FDUUpVGLIIpUHQWVGHFHX[TXLVRQWREWHQXVSDUURWDWLRQHWV\PpWULH,OpQRQFH VDQVOHGpPRQWUHUTXH/XFDV  7RXWFDUUpUHVWHPDJLTXHVLO¶RQpFKDQJHGHX[KRUL]RQWDOHVSXLV GHX[ YHUWLFDOHV TXL VRQW WRXWHV OHV TXDWUH j OD PrPH GLVWDQFH GXFHQWUH

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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).

0 1 2 3 4 5 6 7 8 9

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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. 7 – Les Nimbers des positions menant à une position finale sont égaux à 1.

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Fig. 8 – L’arbre de jeu de Kayles à partir de la position initiale 1-2, avec leurs Nimbers.

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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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