Cut the shape below in four congruent pieces. The gray area is a hole.

enter image description here

In non-mathematical terms, cut the white area in 4 pieces having the same shape, same size, possibly mirrored or rotated.

Note: the hole is not randomly placed. It is in the exact size, shape and position where it needs to be for the solution to work.

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    $\begingroup$ Do we need to know anything more about the hole? It looks like it's horizontally symmetric, with a linear top edge and circular left and right edges; is that accurate? Do we need to assume that the vertices form an equilateral triangle, or that the center of a circular arc is the opposite vertex? $\endgroup$ Commented Nov 12, 2014 at 23:15
  • $\begingroup$ For those with little knowledge of maths, what is 'congruent' exactly? Same form, same size... $\endgroup$
    – 11684
    Commented Nov 12, 2014 at 23:30
  • $\begingroup$ A quick google says: Geometry (of figures) identical in form; coinciding exactly when superimposed. $\endgroup$
    – rjdown
    Commented Nov 12, 2014 at 23:32
  • $\begingroup$ @MartynA I think you need to look at Jonathan's answer more carefully. $\endgroup$
    – jamesdlin
    Commented Nov 13, 2014 at 12:12
  • $\begingroup$ Ah! I beg your pardon. $\endgroup$
    – MartynA
    Commented Nov 13, 2014 at 12:14

1 Answer 1


Here's one solution:

enter image description here

I assumed the radius of the hole's curvature matches the curvature radius of the circle, the hole's straight side is equal to the circle's radius, and its curved edges meet the straight edge at right angles.

Used Flash Pro to solve this, of all things: it has the useful feature that as you paint while editing an object, it updates all instances. So I made 4 instances, then made incremental changes to the source outline and instance positions, and converged pretty quickly on a solution.

I could imagine a genetic or hill-climbing algorithm using the same process.

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    $\begingroup$ Congratulations! As far as I know, it is the only answer. $\endgroup$
    – Florian F
    Commented Nov 13, 2014 at 13:25

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