Let a sphere of radius R, shown here in green, we consider two other brown spheres of radius [math]R_1[/math] et [math]R_2[/math] which are tangent to each other and both internally tangent to the larger green sphere. [list=1] [*] What is the maximum radius M and the minimum radius m of a 4th sphere [math] S_1[/math]which would be tangent to the three preceding? [*] Let [math]r_1\in[m,M] [/math] the radius of [math]S_1[/math], how many solutions to accommodate a fifth sphere [math]S_2[/math]tangent to the previous four? [*] We can continue and thus build a necklace around both brown spheres made of spheres tangents to the first three, and by requiring that each sphere [math]S_{n+1}[/math] more added in the necklace is tangent to the previous one [math]S_n[/math]. Whatever, it's quite surprising that the construction of such a necklace perfectly closes with a last-added sphere[math]S_n[/math], always tangent to [math]S_1[/math]. Why this necklace contain always 6 spheres? [/list] Detailed Solution at: [url]http://mathmj.fr/geogebra/hexlet.pdf[/url] ( Writen in French, i need help to translate it in English. )