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1 vote
0 answers
52 views

'Self-similar and perfect' partitions of planar regions

Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually non-congruent may be called a self-similar perfect partition. A classical example ...
Nandakumar R's user avatar
  • 5,979
4 votes
1 answer
438 views

Perfect squaring of rectangles

A perfect squaring of a rectangle may be defined as a partition of the rectangle into finitely many squares all of which are mutually non-congruent. https://en.wikipedia.org/wiki/Squaring_the_square ...
Nandakumar R's user avatar
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9 votes
0 answers
187 views

Cubing the cube - as 'perfectly' as possible

Ref: https://en.wikipedia.org/wiki/Squaring_the_square A perfect cubing of a cube is a partition of the cube into some finite number of smaller cubes that are pair-wise non-congruent. The above page ...
Nandakumar R's user avatar
  • 5,979
1 vote
0 answers
196 views

Squares as sum of squares

For which positive integers n is $n^2$ the sum of precisely n smaller positive squares? Of these n x n squares, which can be actually cut into n smaller squares?
Bernardo Recamán Santos's user avatar
34 votes
1 answer
3k views

Tiling a square with rectangles

Is it possible to completely tile a square with different rectangles of integer sides but all with the same area? The original problem, not requiring integer sides for rectangles, was proposed by Joe ...
Bernardo Recamán Santos's user avatar
6 votes
3 answers
1k views

Consecutive Integer Squared Square

Is it possible to construct a squared square out of consecutive integer squares? Be it 1,2,3,...n or k,k+1,k+2,...n.
Matt Watson's user avatar