how many ways can you divide a square into four congruent shapes? Remember that the answer cannot be infinity because rotations of one pattern still result in the same four congruent shapes.


closed as unclear what you're asking by Alconja, Wen1now, Peregrine Rook, user58, Rubio Jun 27 '17 at 12:10

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • 8
    $\begingroup$ Huh? It should definitely be infinite. Draw two perpendicular lines which intersect at the center, and it will always partition the square into four congruent parts, no??? $\endgroup$ – greenturtle3141 Jun 11 '17 at 15:45

As pointed out in greenturtle3141's comment, there are indeed


ways to split the square into 4 congruent shapes without rotations.

Draw two lines crossing through the square's center, splitting it into 4 congruent shapes. If you were to pivot the lines around the square's center by any angle between 0 and 90 degrees (not inclusive), you end up with a new unique set of 4 congruent shapes. The range of unique rotations is just under 90 degrees; pivoting at least 90 degrees will result in a repeated split.

enter image description here enter image description here

The answer is "infinity" because the rotation angle can be infinitesimally small. Pivoting by just a tiny fraction of a degree will produce new shapes that are not rotations. The range of rotation (just under 90 degrees) is continuous, so there are infinite possible angles.

It is possible that there are other ways to draw these two lines to divide the square into 4 congruent shapes. However, it doesn't matter, because infinity plus any positive number is still infinity.


Not the answer you're looking for? Browse other questions tagged or ask your own question.