Prove that a square cannot be dissected into an odd number of triangles of equal area.

Got to read about the question and its history in "Algebra and Tiling Homomorphisms in the Service of Geometry by Sherman Stein and Sándor Szabó

quite an interesting question, neither my doubt nor my homework, just want to see the various ideas that people here would come up with it.

edit: the proof i have seen involves higher mathematics which I have not yet completed in my college courses. Posting here mainly to see if anyone can come up with an idea that would be understood by me or a similar pre college student

  • 1
    $\begingroup$ Do you know how to prove it yourself? If so, what's your question? $\endgroup$
    – TonyK
    Sep 7, 2011 at 11:18
  • $\begingroup$ @tony edited my q $\endgroup$
    – Bhargav
    Sep 7, 2011 at 11:22
  • $\begingroup$ Is the sentence fragment beginning "Got to read" an instruction to read before answering or an explanation of where you came across the problem? $\endgroup$ Sep 7, 2011 at 12:22
  • $\begingroup$ its just an explanation mate so that comments dont start pouring in saying what i have worked out regarding the question. And you can surely have a read once if u want , its got an interesting history $\endgroup$
    – Bhargav
    Sep 7, 2011 at 13:13

1 Answer 1


The only proof I know of is the $p$-adic one due to Monsky.

EDIT: I found some confirming evidence at Math Reviews. The review of Charles H. Jepsen and Paul Monsky, Constructing equidissections for certain classes of trapezoids, Discrete Math. 308 (2008), no. 23, 5672–5681, MR2459386 (2009h:05053), written by Sherman Stein, says, in part,

The simple question, "Can a square be cut into an odd number of triangles of equal areas,'' raised some forty years ago, has generated more questions, only some of which have been answered. The answers so far have used valuations and algebraic integers.


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