maths - stumped

[doorknob]

A rectangle may be subdivided ino a finite number of smaller rectangles by drawing a number of line segments in the rectangle parallel to the sides. It is not required that these line segments should pass from one side of the rectangle to the other, so some of these line segments may start or end on any of the internal line segments drawn perpendicular to the given line segment. Thus it may turn out that leaving some but not all the internal line segments of a subdivision of a rectangle may result in a different subdivision of the give rectangle into a smaller number of rectangles: if this can occur for some choice of line segments to be removed, call the subdivision complex: or it may be that this is impossible, in which case call the subdivision simple.

Prove that there are no simple subdivisions of a rectangle into 3 or 4 smaller rectangles.

Find simple subdivisions of a rectangle into 7 and 8 rectangles.

Are there two essentially different ways of finding a simple subdivision of a rectangle into the same number of rectangles.

........ help.....just don't understand!

[Gef]

I don't understand the question either.
Surely a rectangle can be divivded into four smaller rectangles - look at the St. George's cross.

[mibn2cweus]

While your waiting fault free rectangles

[newtron]

Yes Gef, but that subdivision is a complex subdivision because if you remove any line segment, the rectangle is still subdivided into two or three smaller rectangles.

This is an interesting problem. I'll work on it and get back to you.

[Gef]

Thanks newtron I now understand - no wine tonight!

[newtron]

Well, I really don't have the time to solve this problem, but it is intriguing. I was playing around and I found that drawing a simple subdivision is quite complex. LOL. I finally was able to construct one however. Begin by drawing a line perpendicular to the right side of your rectangle, begining at about 3/4 the length of the side from the bottom. Make this line about 3/4 the length of the top side of the rectangle. Now draw a line perpendicular from the top side that intersects you first line at a 90 degree angle. Again, make this second line about 3/4 the length of the left side of the rectangle. Draw a line perpendicular to the left side that intersects the second line at a 90 degree angle. Draw a fourth line perpendicular to the bottom of the rectangle that intersects the third and first lines at 90 degree angles. You'll end up with 5 smaller rectangles in the original rectangle. You will see a spiral- like pattern with one of the rectangles being in the center and not sharing a side with the original rectangle. I don't think you can end up with a smaller number of rectangular subdivisions by removing any one line segment. Check to make sure that is right. I hope this makes sense. Hopefully this might provide som clues on how to proceed in solving this problem. Come on ABers, this is a challenging problem!!

[newtron]

Actually, for the subdivision described above, you can remove any number of internal line segments and you will not be left with only rectangular subdivisions.

[heathfield]

I hope there's a practical reason somewhere for all this?