05-22-2015, 11:29 PM
Sorry, it was my fault regarding the second image... indeed, it does not make 5 cubes, it makes 9. I became confused when writing the text of the post. : - )
Micheus, to be honest I did not understand your presentation and, of course, I can not be sure as to its solution’s being correct or not. Besides, my math knowledge is very poor… whenever I approach such problems I do approach them mostly by an arithmetical way than in a mathematical way (I mean not by formulas, equations etc)… and using as an aid visualizations, as much as it is possible.
As I said, maybe your solution is ok, I can not be sure (I did not be able to find any solution to it yet using my own ways)… I do have, however, an impression that you did not calculate the overlapping cubes (the ones in the in between areas). So, for being more clear I am putting a 2D diagram below. The total quadrangles that can be derived from the initial-big quadrangle (‘a’), including the overlapping ones (those in images ‘d,e,f,g,h.i,j’), plus the initial-big quadrangle (‘a’) are 30 (a=1+b=16+c=4+d=2+e=2+f=1+g=1+h=1+i=1+j=1=30).
As you see there are not only the ‘regularly’ arrayed quadrangles (as those in images 'b' and 'c') in the table, there are also the overlapping ones, that’s to say the ones located between the regularly arrayed ones. So, if you try to find those too in a table constituted from much more quadrangles than the 16 of the diagram below, finding and calculating the overlapping ones becomes very difficult. Imagine, for example, a big quadrangle constituted from 64 small quadrangles… how many quadrangles can be found in such a table?
When the question is put forth in a two dimensional way -and having a quadrangle constituted from a total of a few smaller quadrangles- is easy to find the solution but… when it comes to the three dimensional plane and instead of a quadrangle you have a cube, things become too much complicated. : - )
![[Image: For%20Wings%20forum_total_a_zpsvjmlkmqe.jpg]](http://i1272.photobucket.com/albums/y390/Cloudydaylover/3Dworks/For%20Wings%20forum_total_a_zpsvjmlkmqe.jpg)
Micheus, to be honest I did not understand your presentation and, of course, I can not be sure as to its solution’s being correct or not. Besides, my math knowledge is very poor… whenever I approach such problems I do approach them mostly by an arithmetical way than in a mathematical way (I mean not by formulas, equations etc)… and using as an aid visualizations, as much as it is possible.
As I said, maybe your solution is ok, I can not be sure (I did not be able to find any solution to it yet using my own ways)… I do have, however, an impression that you did not calculate the overlapping cubes (the ones in the in between areas). So, for being more clear I am putting a 2D diagram below. The total quadrangles that can be derived from the initial-big quadrangle (‘a’), including the overlapping ones (those in images ‘d,e,f,g,h.i,j’), plus the initial-big quadrangle (‘a’) are 30 (a=1+b=16+c=4+d=2+e=2+f=1+g=1+h=1+i=1+j=1=30).
As you see there are not only the ‘regularly’ arrayed quadrangles (as those in images 'b' and 'c') in the table, there are also the overlapping ones, that’s to say the ones located between the regularly arrayed ones. So, if you try to find those too in a table constituted from much more quadrangles than the 16 of the diagram below, finding and calculating the overlapping ones becomes very difficult. Imagine, for example, a big quadrangle constituted from 64 small quadrangles… how many quadrangles can be found in such a table?
When the question is put forth in a two dimensional way -and having a quadrangle constituted from a total of a few smaller quadrangles- is easy to find the solution but… when it comes to the three dimensional plane and instead of a quadrangle you have a cube, things become too much complicated. : - )
![[Image: For%20Wings%20forum_total_a_zpsvjmlkmqe.jpg]](http://s1272.photobucket.com/user/Cloudydaylover/media/3Dworks/For%20Wings%20forum_total_a_zpsvjmlkmqe.jpg.html)