Tricky Paper Puzzle

Originally posted by Gastel
You do a tri-fold, that is you move both edges away from each other but towards the folds that the other edge is making??? Crease the page lightly, then unfold. Refold the paper in half (the creases should align perfectly, if not, redo the first fold). Cut along the lined-up creases. This will yield 3 equi-sized pieces.

The question is: If the page wa ...[text shortened]... square has 4 equal lengthed sides and four equal angle corners. A rectangle is NOT a square.

well lets see i start by cutting off the bottom 8.5x8.5 square so i have 2.5x8.5 then i cut three 2.5x2.5 sections leaving 1x2.5, two more 1x1 sections and i have 0.5x1 which i cut in half (0.5x0.5) for a total of
1+3+2+2=8 sections would probably be the minimum number of squares. then i can take any square and cut it into 4 equal squares so
8+3x of course i could also cut into 9 equal pieces or n^2 equal pieces so how about 8+sum[(n^2)-1] for any subset N of the natural numbers.

Originally posted by aginis
well lets see i start by cutting off the bottom 8.5x8.5 square so i have 2.5x8.5 then i cut three 2.5x2.5 sections leaving 1x2.5, two more 1x1 sections and i have 0.5x1 which i cut in half (0.5x0.5) for a total of
1+3+2+2=8 sections would probably be the minimum number of squares. then i can take any square and cut it into 4 equal squares so
8+3x of course ...[text shortened]... pieces or n^2 equal pieces so how about 8+sum[(n^2)-1] for any subset N of the natural numbers.

You cannot have any squares that are the same size, so you would have to convert your same sized squares into other squares.

Originally posted by Gastel
You cannot have any squares that are the same size, so you would have to convert your same sized squares into other squares.

oh i thought you meant they don't have to be the same size.
well then i'm not sure its possible if they CANT be the same size

Originally posted by aginis
oh i thought you meant they don't have to be the same size.
well then i'm not sure its possible if they CANT be the same size

There is a branch of mathematics dedicated to this type of problem.

Originally posted by Gastel
There is a branch of mathematics dedicated to this type of problem.

which branch is that? my most advanced course so far is number theory
this feel like geometrical analysis or something.

assume that the smallest number of squares a rectangle can be cut into is n>2. Therefore there exists a rectangle which can be cut into n unequal squares call it R.
Case 1: R is of size axb a < b cut off axa and get a smaller rectangle (b-a)xa which can also only be cut into at the minimum n unequal squares. but we already made one square so we need at least n+1 squares. contradiction.
cut off cxc < a then you have 2 rectangles (b-c)xa and (a-c)xb both overlap in a square so we will only look at (a-c)xb still requires at least n squares so total is at least n+1. contradiction
Case 2: R is really a square axa we cut a smaller square bxb out of it but that leaves two rectangles (a-b)xb, (a-b)xa at least one now falls into the category of case 1. contradiction.

Thus no rectangle can be cut into a finite number of unequal squares.

Brooks, Smith, Stone, Tutte, Duke Mathematical Journal vol. 7 (1940), p312-340
Croft, Falconer, Guy, Unsolved Problems in Geometry (1991), p81
Gale, The Mathematical Intelligencer vol. 15 no. 1 (1993) p48-50
Gardner, More Mathematical Puzzles and Diversions (1963)

The theory behind what is known as 'squaring' a rectangle or square has been more discovery than proof. In 1909 Moron (yes that is his name... there is an accent over the second o), discovered the first squared rectangle. The rectangle was 33 x 32 and the squares had sides 1, 4, 7, 8, 9, 10, 14, 15 and 18. Earlier Max Dehn proved that any tiling by squares must be commensurable (all dimensions are integer multiples of a single number). Therefore, the first step would be to convert 8.5 x 11 to dimensions of 17 x 22 (each unit is 0.5"😉 or some multiple of. There is an interesting solving method that uses electrical circuit analysis (more specifically Kurchoff's Current and Voltage Laws and Smith diagrams).

Originally posted by Gastel
Brooks, Smith, Stone, Tutte, Duke Mathematical Journal vol. 7 (1940), p312-340
Croft, Falconer, Guy, Unsolved Problems in Geometry (1991), p81
Gale, The Mathematical Intelligencer vol. 15 no. 1 (1993) p48-50
Gardner, More Mathematical Puzzles and Diversions (1963)

The theory behind what is known as 'squaring' a rectangle or s ...[text shortened]... circuit analysis (more specifically Kurchoff's Current and Voltage Laws and Smith diagrams).

gcd (8.5, 11)=0.5 so thats the smallest unit i had that figured out, but the rest is over my head...way over.
Is there an infinite number of proportionally different rectangles that can be squared?

Originally posted by aginis
gcd (8.5, 11)=0.5 so thats the smallest unit i had that figured out, but the rest is over my head...way over.
Is there an infinite number of proportionally different rectangles that can be squared?

I don't know. Not all rectangles can be, but again as the theory is more dicovered than proved, very few definative answers exist.

The theory becomes even more interesting when the rectangles are permitted to become rings, mobius strips or klein bottles. The solutions always look extremely difficult.

Originally posted by Gastel
I don't know. Not all rectangles can be, but again as the theory is more dicovered than proved, very few definative answers exist.

The theory becomes even more interesting when the rectangles are permitted to become rings, mobius strips or klein bottles. The solutions always look extremely difficult.

do you mean topology?