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Q73828-prove-that-no-cube-exists-whose-corners-are-located-on-all-faces-of-an-other-cube-




Question Number 74087 by mr W last updated on 19/Nov/19
(Q73828)  prove that no cube exists whose corners  are located on all faces of an other cube.
$$\left({Q}\mathrm{73828}\right) \\ $$$${prove}\:{that}\:{no}\:{cube}\:{exists}\:{whose}\:{corners} \\ $$$${are}\:{located}\:{on}\:{all}\:{faces}\:{of}\:{an}\:{other}\:{cube}. \\ $$
Commented by mr W last updated on 18/Nov/19
here is my proof:  the inner cube has 8 corners, but the  outer cubes has only 6 faces.  that means two of  its corners must lie on the same face  of the outer cube. say these two corners  of the inner cube are 1 and 2. they lie  on the face ABCD of the outer cube.
$${here}\:{is}\:{my}\:{proof}: \\ $$$${the}\:{inner}\:{cube}\:{has}\:\mathrm{8}\:{corners},\:{but}\:{the} \\ $$$${outer}\:{cubes}\:{has}\:{only}\:\mathrm{6}\:{faces}. \\ $$$${that}\:{means}\:{two}\:{of} \\ $$$${its}\:{corners}\:{must}\:{lie}\:{on}\:{the}\:{same}\:{face} \\ $$$${of}\:{the}\:{outer}\:{cube}.\:{say}\:{these}\:{two}\:{corners} \\ $$$${of}\:{the}\:{inner}\:{cube}\:{are}\:\mathrm{1}\:{and}\:\mathrm{2}.\:{they}\:{lie} \\ $$$${on}\:{the}\:{face}\:{ABCD}\:{of}\:{the}\:{outer}\:{cube}. \\ $$
Commented by ajfour last updated on 18/Nov/19
think over sir- it shouldn′t   mean that. (question  said on each   face of outer cube there is an inner  cube′s corner)     later i added at least, still two  corners may or maynot touch  the outer cube walls..
$${think}\:{over}\:{sir}-\:{it}\:{shouldn}'{t}\: \\ $$$${mean}\:{that}.\:\left({question}\:\:{said}\:{on}\:{each}\:\right. \\ $$$${face}\:{of}\:{outer}\:{cube}\:{there}\:{is}\:{an}\:{inner} \\ $$$$\left.{cube}'{s}\:{corner}\right)\:\:\: \\ $$$${later}\:{i}\:{added}\:{at}\:{least},\:{still}\:{two} \\ $$$${corners}\:{may}\:{or}\:{maynot}\:{touch} \\ $$$${the}\:{outer}\:{cube}\:{walls}.. \\ $$
Commented by mr W last updated on 18/Nov/19
Commented by mr W last updated on 18/Nov/19
so we have an edge 12 from the inner  cube in the plane ABCD.  other corners of the inner cube are  numbered as shown.  we know these edges are parallel:  34//56//78//12.  their projections in the plane EFGH  are also parallel.  it′s easy to see that the edge 56 must  lie in the opposite face EFGH from  the outer cube. (but this is not relevant  for the proof.)  face 2468 and face 1357 from the  inner cube are perpendicular to the  edge 12. (clear)  let′s image that we cut the outer cube  along these two faces from the inner  cube. both cut sections are perpendicular  to the face ABCD and EFGH.  the shape of both cut sections is  rectangle.
$${so}\:{we}\:{have}\:{an}\:{edge}\:\mathrm{12}\:{from}\:{the}\:{inner} \\ $$$${cube}\:{in}\:{the}\:{plane}\:{ABCD}. \\ $$$${other}\:{corners}\:{of}\:{the}\:{inner}\:{cube}\:{are} \\ $$$${numbered}\:{as}\:{shown}. \\ $$$${we}\:{know}\:{these}\:{edges}\:{are}\:{parallel}: \\ $$$$\mathrm{34}//\mathrm{56}//\mathrm{78}//\mathrm{12}. \\ $$$${their}\:{projections}\:{in}\:{the}\:{plane}\:{EFGH} \\ $$$${are}\:{also}\:{parallel}. \\ $$$${it}'{s}\:{easy}\:{to}\:{see}\:{that}\:{the}\:{edge}\:\mathrm{56}\:{must} \\ $$$${lie}\:{in}\:{the}\:{opposite}\:{face}\:{EFGH}\:{from} \\ $$$${the}\:{outer}\:{cube}.\:\left({but}\:{this}\:{is}\:{not}\:{relevant}\right. \\ $$$$\left.{for}\:{the}\:{proof}.\right) \\ $$$${face}\:\mathrm{2468}\:{and}\:{face}\:\mathrm{1357}\:{from}\:{the} \\ $$$${inner}\:{cube}\:{are}\:{perpendicular}\:{to}\:{the} \\ $$$${edge}\:\mathrm{12}.\:\left({clear}\right) \\ $$$${let}'{s}\:{image}\:{that}\:{we}\:{cut}\:{the}\:{outer}\:{cube} \\ $$$${along}\:{these}\:{two}\:{faces}\:{from}\:{the}\:{inner} \\ $$$${cube}.\:{both}\:{cut}\:{sections}\:{are}\:{perpendicular} \\ $$$${to}\:{the}\:{face}\:{ABCD}\:{and}\:{EFGH}. \\ $$$${the}\:{shape}\:{of}\:{both}\:{cut}\:{sections}\:{is} \\ $$$${rectangle}. \\ $$
Commented by ajfour last updated on 18/Nov/19
but Sir i believe that two left  out corners can just be inside  and not touch the outer cube,  such an orientation, might even  result...(here 2 can be bit below  the ceiling and 5 slightly above  the floor ..
$${but}\:{Sir}\:{i}\:{believe}\:{that}\:{two}\:{left} \\ $$$${out}\:{corners}\:{can}\:{just}\:{be}\:{inside} \\ $$$${and}\:{not}\:{touch}\:{the}\:{outer}\:{cube}, \\ $$$${such}\:{an}\:{orientation},\:{might}\:{even} \\ $$$${result}…\left({here}\:\mathrm{2}\:{can}\:{be}\:{bit}\:{below}\right. \\ $$$${the}\:{ceiling}\:{and}\:\mathrm{5}\:{slightly}\:{above} \\ $$$${the}\:{floor}\:.. \\ $$
Commented by mr W last updated on 18/Nov/19
Commented by mr W last updated on 18/Nov/19
since the face 2468 is an inscribed  square of the rectangle, the rectangle  must be also a square with side length a.  that means the both cut sections must  be of same size.
$${since}\:{the}\:{face}\:\mathrm{2468}\:{is}\:{an}\:{inscribed} \\ $$$${square}\:{of}\:{the}\:{rectangle},\:{the}\:{rectangle} \\ $$$${must}\:{be}\:{also}\:{a}\:{square}\:{with}\:{side}\:{length}\:{a}. \\ $$$${that}\:{means}\:{the}\:{both}\:{cut}\:{sections}\:{must} \\ $$$${be}\:{of}\:{same}\:{size}. \\ $$
Commented by mr W last updated on 18/Nov/19
Commented by mr W last updated on 18/Nov/19
we get the length of the cut section of  (√2)a−s and heigth of a.  (√2)a−s=a  ⇒s=((√2)−1)a   ...(i)  since square of length s is an inscribed  square of the square with side length a,  s≥(a/( (√2)))    ...(ii)  we can not find a value of s which  fulfills both (i) and (ii).  that means an inner cube doesn′t  exist which touches all 6 faces of an  other cube.
$${we}\:{get}\:{the}\:{length}\:{of}\:{the}\:{cut}\:{section}\:{of} \\ $$$$\sqrt{\mathrm{2}}{a}−{s}\:{and}\:{heigth}\:{of}\:{a}. \\ $$$$\sqrt{\mathrm{2}}{a}−{s}={a} \\ $$$$\Rightarrow{s}=\left(\sqrt{\mathrm{2}}−\mathrm{1}\right){a}\:\:\:…\left({i}\right) \\ $$$${since}\:{square}\:{of}\:{length}\:{s}\:{is}\:{an}\:{inscribed} \\ $$$${square}\:{of}\:{the}\:{square}\:{with}\:{side}\:{length}\:{a}, \\ $$$${s}\geqslant\frac{{a}}{\:\sqrt{\mathrm{2}}}\:\:\:\:…\left({ii}\right) \\ $$$${we}\:{can}\:{not}\:{find}\:{a}\:{value}\:{of}\:{s}\:{which} \\ $$$${fulfills}\:{both}\:\left({i}\right)\:{and}\:\left({ii}\right). \\ $$$${that}\:{means}\:{an}\:{inner}\:{cube}\:{doesn}'{t} \\ $$$${exist}\:{which}\:{touches}\:{all}\:\mathrm{6}\:{faces}\:{of}\:{an} \\ $$$${other}\:{cube}. \\ $$
Commented by ajfour last updated on 18/Nov/19
you might not be correct here Sir,  when you believe-  ′′if such an inner cube exists, two of  its corners must lie on the same face  of the outer cube.′′
$${you}\:{might}\:{not}\:{be}\:{correct}\:{here}\:{Sir}, \\ $$$${when}\:{you}\:{believe}- \\ $$$$''{if}\:{such}\:{an}\:{inner}\:{cube}\:{exists},\:{two}\:{of} \\ $$$${its}\:{corners}\:{must}\:{lie}\:{on}\:{the}\:{same}\:{face} \\ $$$${of}\:{the}\:{outer}\:{cube}.'' \\ $$$$ \\ $$$$ \\ $$
Commented by mr W last updated on 18/Nov/19
ajfour sir:  the original question requested that  on each of the 6 faces of larger cube  there is at least one corner from the  inner cube. that means the 8 corners  of the inner cube must touch all 6  faces of the outer cube.
$${ajfour}\:{sir}: \\ $$$${the}\:{original}\:{question}\:{requested}\:{that} \\ $$$${on}\:{each}\:{of}\:{the}\:\mathrm{6}\:{faces}\:{of}\:{larger}\:{cube} \\ $$$${there}\:{is}\:{at}\:{least}\:{one}\:{corner}\:{from}\:{the} \\ $$$${inner}\:{cube}.\:{that}\:{means}\:{the}\:\mathrm{8}\:{corners} \\ $$$${of}\:{the}\:{inner}\:{cube}\:{must}\:{touch}\:{all}\:\mathrm{6} \\ $$$${faces}\:{of}\:{the}\:{outer}\:{cube}. \\ $$
Commented by mr W last updated on 18/Nov/19
if not all corners of the inner cube  should be on the faces of the outer  cube, the situation is quite different.
$${if}\:{not}\:{all}\:{corners}\:{of}\:{the}\:{inner}\:{cube} \\ $$$${should}\:{be}\:{on}\:{the}\:{faces}\:{of}\:{the}\:{outer} \\ $$$${cube},\:{the}\:{situation}\:{is}\:{quite}\:{different}. \\ $$
Commented by ajfour last updated on 18/Nov/19
let us not be deceived by the  lack of clarity in diagram,  and instead focus on how the  question is worded, let us both  review it and Sir please see the  diagram believing that 2 is below  ceiling and not touching the  back face/wall and at the same  time 5 is above the floor, and  not touching the front face;  (this orientation even might  not exist but until it is proved  with analysis, i imagined this  was possible with a little range  of values of s. Hence asked for  s_(min) .
$${let}\:{us}\:{not}\:{be}\:{deceived}\:{by}\:{the} \\ $$$${lack}\:{of}\:{clarity}\:{in}\:{diagram}, \\ $$$${and}\:{instead}\:{focus}\:{on}\:{how}\:{the} \\ $$$${question}\:{is}\:{worded},\:{let}\:{us}\:{both} \\ $$$${review}\:{it}\:{and}\:{Sir}\:{please}\:{see}\:{the} \\ $$$${diagram}\:{believing}\:{that}\:\mathrm{2}\:{is}\:{below} \\ $$$${ceiling}\:{and}\:{not}\:{touching}\:{the} \\ $$$${back}\:{face}/{wall}\:{and}\:{at}\:{the}\:{same} \\ $$$${time}\:\mathrm{5}\:{is}\:{above}\:{the}\:{floor},\:{and} \\ $$$${not}\:{touching}\:{the}\:{front}\:{face}; \\ $$$$\left({this}\:{orientation}\:{even}\:{might}\right. \\ $$$${not}\:{exist}\:{but}\:{until}\:{it}\:{is}\:{proved} \\ $$$${with}\:{analysis},\:{i}\:{imagined}\:{this} \\ $$$${was}\:{possible}\:{with}\:{a}\:{little}\:{range} \\ $$$${of}\:{values}\:{of}\:\boldsymbol{{s}}.\:{Hence}\:{asked}\:{for} \\ $$$$\boldsymbol{{s}}_{{min}} . \\ $$
Commented by ajfour last updated on 18/Nov/19
the situation is quite different/  much interesting then, thank  you Sir, my solution attempt  is split in two posts. First  in the original question, and  then in a later post as a system  of simultaneous equations,  (Q.74024)  please give a glance Sir..
$${the}\:{situation}\:{is}\:{quite}\:{different}/ \\ $$$${much}\:{interesting}\:{then},\:{thank} \\ $$$${you}\:{Sir},\:{my}\:{solution}\:{attempt} \\ $$$${is}\:{split}\:{in}\:{two}\:{posts}.\:{First} \\ $$$${in}\:{the}\:{original}\:{question},\:{and} \\ $$$${then}\:{in}\:{a}\:{later}\:{post}\:{as}\:{a}\:{system} \\ $$$${of}\:{simultaneous}\:{equations}, \\ $$$$\left({Q}.\mathrm{74024}\right) \\ $$$${please}\:{give}\:{a}\:{glance}\:{Sir}.. \\ $$
Commented by mr W last updated on 19/Nov/19
now i understand your solutions. yet  it′s not easy to find s.
$${now}\:{i}\:{understand}\:{your}\:{solutions}.\:{yet} \\ $$$${it}'{s}\:{not}\:{easy}\:{to}\:{find}\:{s}. \\ $$

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