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[null,null,"Le problème de Freudenthal"]
[]
[null,"L'énoncé se trouve sur la page \"GV\""]
[]
["La feuille \"GV\""]
[null,"La feuille du résumé du contenu du site de Gérard Villemin, qui sert de guide."]
[null,"Guide très utile malgré quelques imperfections. Voir les remarques en colonne F"]
[]
["La feuille \"Tables\""]
[null,"On trouve sur cette feuille :"]
[null,null,"La table des sommes en B4:AI21. Son intérêt est juste de voir la zone d'étendue des couples (m, n) qui répondent à m + n = s < 100"]
[null,null,"La table des produits \"mn\" en B23:CU72"]
[null,null,"La liste (linéaire) des sommes et produits des couples (m, n) en CY3:DC2357"]
[null,null,"La même liste est reproduite en DF3:DJ2357, classée par ordre des produits de manière à pouvoir les compter"]
[null,null,"Le nombre de chacun des produits identiques est reporté, colonne DK et DC"]
[null,null,"Lorsque ce nombre est égal à 1, la cellule correspondante du tableau des produits en B23:CU72, est colorée en jaune. Les cellules non colorées contiennent un produit ambigu présent dans plusieurs cellules"]
[null,null,"Les colonnes DM:DO listent les produit ambigus ou mm (multiplication multiple). Liste en colonne DO"]
[null,null,"Colonnes DQ:FI, les sommes et produits des couples sélectionnés"]
[null,null,null,"Dont colonnes FF à FI qui comptent les produits identiques"]
[]
["Sélections basées sur les ambiguités"]
[null,"Ambiguité est le nom donné aux produits p et aux sommes s issus de plusieurs possibilités pour les deux nombres m et n ;  s = m + n ; p = mn"]
[null,"Exemples :"]
[null,null,"p = 4 et s = 4 proviennent sans ambiguité de m = 2 et n = 2 car il n'y a qu'une seule possibilité"]
[null,null,"s = 5 avec m =2 et n = 3 sans ambiguité"]
[null,null,"s = 6 est ambigu, car on peut faire 2 + 4 ou 3 + 3"]
[null,null,"p = 10 provient de m = 2 et n = 5 sans ambiguité"]
[null,null,"p = 12 est obtenu avec 2 x 6 ou 4 x 3, produit ambigu"]
[null,"Le rôle des facteurs pris dans la liste des nombres premiers intervient fortement pour reconnaître un produit ambigu, mais cela ne suffit pas"]
[null,"Il a été préféré ici, dans les limites données par l'énoncé (s < 100),  d'établir la liste complète des produits, de les classer et de les compter. Un produit qui n'apparait qu'une fois est sans ambiguité le résultat d'un seul couple."]
[]
[null,"On ne doit raisonner ici que sur les couples ambigus, puisque, en première affirmation, ni Sandrine, ni Paul, ne peuvent trouver LA solution"]
[null,"Chacun dans son coin peut imaginer ce que l'autre pense et ainsi, autant l'un que l'autre, ils peuvent faire des listes de sommes et produits sur lesquels il faut se polariser"]
[]
[null,"La liste des sommes retenues est celle des couples dont le produit est ambigu."]
[null,null,"Pour chaque somme, l'ensemble des couples doit avoir un produit ambigu."]
[]
[null,"La liste des produits est celle des couples étudiés lors de l'établissement de la liste des sommes à conserver"]
[null,"Il ne reste à la fin qu'à compter les produits identiques"]
[null,"On constate ainsi que dans la liste des produits correspondants à la somme s = 17 il n'y a qu'un seul produit qui n'apparait qu'une fois dans tous les produits"]
[null,"Voir DU8:DW8 qui donne la solution :"]
[]
[null,null,null,"m = 4","n = 13","s= 17","p = 52"]
[]
[]
["Calculs"]
[null,"Dans cette feuille \"Tables\", il y a des calculs par formule et des calculs par macro-instructions"]
[null,"Les formules appliquées à une colonne ne sont conservées que sur la ligne 4"]

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[null,"Page de Gérard Villemin","http://villemin.gerard.free.fr/aJeux1/Nombre/SomProd.htm#anniv",null,null,"Remarques"]
[]
[null,"Problème de Freudenthal\nLes deux nombres à deviner (résumé)"]
[]
[null,"Problème","Conversation"]
[null,"Deux nombres (m et n) ;","Paul : Je ne connais pas ces nombres."]
[null," m>1 ; n>1 ; m+n <100.","Sandrine : je m'en doutais bien, mais c'est mon cas également."]
[null,"Paul connait p = mn,","Paul : dans ce cas, je sais quels sont les deux nombres."]
[null," Sandrine connait s = m + n.","Sandrine : bravo, mais alors moi aussi."]
[]
[null,"La réponse est :","m = 4 ; n = 13 ; s = 17 ; p = 52"]
[]
[null,"Approche de la solution","Idée n° 1 : ne peuvent pas répondre à cause de plusieurs solutions (ambiguité)"]
[null,null,"Idée n° 2 : chacun fait le chemin mental de l'autre"]
[]
[null,"Cheminement progressif","Version A : limiter la somme à une petite valeur (non développé)."]
[null,null,"Version B : tentons de faire le raisonnement en connaissant les valeurs s = 17 et m = 52"]
[null,null,"Version C : solution complète sans connaitre s et p, avec exploration des possibilités pour toutes les sommes."]
[null,null,"Version D : sans connaissance préalable avec la théorie des nombres."]
[]
[null,"Version B"]
[]
[null,"On est quand même obligé","Paul sait que p = 52 = 2 x 26 = 4 x 13 ; ce produit est ambigu ; Il ne peut pas conclure."]
[null,"d'explorer toutes les possibilités,","Sandrine sait que s = 17 = 2 + 15 = 3 + 14 = 4 + 13 = 5 + 12 = 6 + 11 = 7 + 10 = 8 + 9 ; Elle ne peut pas conclure"]
[null,"même en connaissant s et p.","Sandrine réfléchit au raisonnement de Paul, elle examine les produits possibles pour chacune de ses sommes :"]
[null,null,"   avec m = 2 et n = 15 ;     p = 30 = 2 x 15 = 5 x 6 ; produit ambigu ; solution à conserver."]
[null,null,"   avec m = 4 et n = 13 ;     p = 52 = 4 x 13 = 2 x 26 ; produit ambigu ; solution à conserver."]
[null,null,"Dans la liste des solutions conservées par Sandrine 2 x 26 n'existe pas, mais 4 x 13 existe, d'où la conclusion de Paul"]
[null,null,"Mais il faut que Paul examine toutes les possibilités pour chaque somme de 5 à 100.",1,1,"En réalité de 6 à 99."]
[]
[null,"Famiiarisation, Indices"]
[]
[null,"Pour Paul p = mn = m'n' ; ambigu","Exemple : p = 15 = 3 x 5 de façon unique ; donc m = 3 et n = 5 ; paire (3, 5)"]
[null,null,"Exemple : p = 12 = 2 x 6 = 3 x 4. On ne peut pas choisir l'une ou l'autre : (2, 6) ou (3, 4)."]
[null,null,"La multiplication est unique si les deux facteurs sont des nombres premiers.",2,2,"Attention : 2 x 4 est aussi unique,"]
[null,"Sandrine fait la même déduction :","Elle élimine toutes les additions de nombres premiers :",3,3,"De même que tous les p = x3."]
[null,null,"Dans s = 12 = 2 + 10 = 3 + 9 = 4 + 8 = 5 + 7 ; la paire (5, 7) est à éliminer."]
[null,null,"Sandrine sait que s = 17 : chacun des couples correspond à un produit p :"]
[null,null,"  2.15 = 30 ; 3.14 = 42 ; 4.13 = 52 ; 5.12 = 60 ; 6.11 = 66 ; 7.10 = 70 ; 8.9 = 72 ; Laquelle est la bonne ?"]
[]
[null,"Sandrine se dit : et si Paul continuait","Avec p = 30, il aurait trouvé : (2, 15) ; (3,10) ; et (5, 6)     Pour (2, 15) ; s = 17 = 2 + 15 = 3 + 14 = … = 8 + 9."]
[null,"le même raisonnement …"]
[]
[null,"Les sommes possibles"]
[null,null,null,null,4,"Attention il faut ajouter 16 et tous les x4."]
[null,"Nécessité de faire un balayage\nde toutes les possibilités","Examen de tous les nombres m et n",null,5,"Faux (2, 3) et (2, 4) ne sont pas ambigus.\nLa solution minimale est (2, 5)."]
[null,"Table de Paul qui a un produit ambigu","Consulter la table des produits ambigu notés mm : {12, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52 …}.",4,6,"L'explication n'est pas claire"]
[null,"Sandrine se limite","Solution minimale : (2, 3).",5,7,"Il est sûr que, dans l'exploration"]
[null,"Somme < 100 ; alors","Copie exacte : Si s >= 55, alors : s et p peuvent s'écrire : s = 53 + (s - 53) et p = 53(s - 53) . Or 53 étant un nombre premier.",6,null,"de tous les couples (m, n) pour "]
[null,"le plus grand n <= 55 (53 + 2)","L'autre facteur, s - 53 qui est positif, est supérieur à 53. Dans ce cas s est plus grand que 100. ",7,null,"une somme s donnée,"]
[null,null,"Impossible selon l'hypothèse. P aurait donc une factorisation unique.",null,null,"on va rencontrer au moins une fois"]
[null,"Sandrine établit sa propre liste\ndéduite de celle de Paul","Exemple : avec m + n = 17, une possibilité (2, 15) ; p = 30 qui est dans la liste de Paul",null,null,"le couple (53, s - 53)"]
[null,"Elle recherche les m + n < 56 avec\nmn dans la liste de Paul","Les sommes possibles sont : {11, 17, 23, 27, 29, 35, 37, 41, 47, 53}",null,null,"dont le produit est supérieur à 100,"]
[null,"Il faut analyser toutes\nles sommes de 5 à 55.",null,null,null,"mais pas la somme."]
[null,"Exemple avec s = 10","10 = 2 + 8 (p = 16 ambigu ; A = 1) = 3 + 7 (p 21  ; A = 0) = 4 + 6 (p = 24 ; A = 1) = 5 + 5 (p = 25 ; A = 0) : Ambiguité globale Ag = 0",null,null,"L'examen systématique montre "]
[null,"Exemple avec s = 11","11 = 2 + 9 (p = 18 ; A = 1) = 3 + 8 (p = 24 ; A = 1) = 4 + 7 (p = 28 ; A = 1) = 5 + 6 (p = 30 ; A = 1) ; Ag = 1 ; somme à conserver.",null,null,"qu'effectivement,"]
[null,"Un seul produit sans ambiguité\nlève l'ambiguité (somme à éliminer)",null,null,null,"il n'y a pas de somme > 53 à retenir."]
[null,null,null,null,null,"Il faudrait plutôt chercher à démontrer"]
[null,"Les produits possibles",null,null,null,"que le couple (53, s - 53) est"]
[null,null,null,null,null,"un produit unique."]
[null,"La liste des sommes possibles","L'ensemble des produits pour S11 est {18, 24, 28, 30} ; produits issus des couples (2, 9), (3, 8), (4, 7), 5, 6) ; dont la somme est 11",null,null,"A retenir : Certains produits contiennent"]
[null,"est accessible autant à Paul qu'à Sandrine","S17 = {30, 42, 52, 60, 66, 70, 72}",null,null,"un facteur hors du champ de l'étude,"]
[null,null,"S23 = {42, 60, 76, 90, 102, 112, 120, 126, 130, 132}",null,null,"ce qui amène le produit ambigu"]
[null,null,"S27 = {50, 72, 92, 110, 126, 140, 152, 162, 170, 176, 180, 182}",null,null,"à non ambigu."]
[null,null,"S29 = {54, 78, 100, 120, 138, 154, 168, 180, 190, 198, 204, 208, 210}"]
[null,null,"S35 = {66, 96, 124, 150, 174, 196, 216, 234, 250, 264, 276, 286, 294, 300, 304, 306}"]
[null,null,"S37 = {70, 102, 132, 160, 186, 210, 232, 252, 270, 286, 300, 312, 322, 330, 336, 340, 342}"]
[null,null,"S41 = {78, 114, 148, 180, 210, 238, 264, 288, 310, 330, 348, 364, 378, 390, 400, 408, 414, 418, 420}"]
[null,null,"S47 = {90, 132, 172, 210, 246, 280, 312, 342, 370, 396, 420, 442, 462, 480, 496, 510, 522, 532, 540, 546, 550, 552]"]
[null,null,"S53 = {102, 150, 196, 240, 282, 322, 360, 396, 430, 462, 492, 520, 546, 570, 592, 612, 630, 646, 660, 672, 682, 690, 696, 700, 702}"]
[]
[null,"Solution"]
[]
[null,"Paul peut déclarer connaître la solution","Car 52 n'apparait qu'une fois dans la liste",8,8,"Une fois dans la liste S17"]
[null,"Sandrine connait elle aussi la solution","Car il n'y a plus qu'un seul produit jaune correspondant aux sommes qu'elle a retenues"]
[null,null,"p = 52 ;  m = 4 ; n = 13 ; S = 52",9,9,"S = 17"]
[]
[]
[]
[null,"Gérard Villemin propose une table des produits ambigus"]
[]
[null,null,"Voir : http://villemin.gerard.free.fr/TABLES/SSPMM.htm ",null,null,"Les SP4 (semi premiers à la puissance 4)"]
[null,null,null,null,null,"doivent être classés dans les mm,"]
[null,null,null,null,null,"multiplications multiples."]
[]
[null,null,null,null,null,"Exemple 28 = 16 = 2 x 8 = 4 x 4"]

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[null,null,"Le problème de Freudenthal",null,null,null,null,null,null,null,null,null,"Sommes m + n",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Liste couples (m, n)",null,null,null,null,null,null,"Par ordre croissant de p",null,null,null,null,null,null,"mn ambigus",null,null,null,"Sommes et produits ambigus",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Tous"]
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