https://mindyourdecisions.com/blog/2016/09/04/the-hardest-easy-geometry-problem-sunday-puzzle/ Refer this link for problem diagram

We can formulate a system of 4 equations involving angles CEF, AFE, AEF, BEF, and specify ranges for each angle (greater than 0, less than 180), using this we can get new ranges for each angle between which we can choose any value for one of these angles and put it in these 4 equations and get the values of other 3, which means that x is not unique.

If you search for this question, others have used unnecessary derivations to prove x is 30 degrees (which in my case is one of the infinite solutions)


Edit: only angles unknown in this figure are: AEF, AFE, CFE, BEF (Which is x) Now below equations can be easily derived:

BEF + CFE = 110,

AFE + AEF = 160,

AFE + CFE = 130,

BEF + AEF = 140

Now for angles to remain in range 0-180, below conditions must be satisfied:

0 < BEF < 110,

0 < CFE < 110,

30 < AEF < 140,

20 < AFE < 130

Now choose value of any one angle in above range and use above equations to find the values of other three.

(BEF=30 is one of the infinite solutions)


  • 1
    $\begingroup$ What exactly are your system of 4 equations? $\endgroup$ – The Demonix _ Hermit Oct 17 '19 at 9:56
  • 2
    $\begingroup$ Just because your system of equations does not uniquely determine the solution does not automatically mean that there are multiple solutions. It can also mean that you have chosen the equations poorly. $\endgroup$ – Rahul Oct 17 '19 at 10:13
  • $\begingroup$ I have edited the post, please check again, all I am trying to say is that angle BEF can be anywhere between 0-110, and remaining 3 unkown angles can be found $\endgroup$ – Chaudhary Oct 18 '19 at 5:41

Angle $x$ is unique because the figure is determined up to similarity: Start with a line segment $BC$. Find the unique $F$ such that $\angle CBF=80^\circ $ and $\angle ECB=50^\circ$. Find the unique $E$ such that $\angle ECB=80^\circ$ and $\angle CBE=60^\circ$. Now $x=\angle FEB$ is uniquely determined.

  • $\begingroup$ I have edited the post, please take a look again. Thanks :) $\endgroup$ – Chaudhary Oct 18 '19 at 5:42

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