Combining Squares

You are given two squares with edge lengths a and b, such that a > b > 0. Describe exactly how you can cut the squares with two straight lines, allowing that a single line may cut both squares, so that the cut pieces can be reassembled into a single larger square with total area a2 + b2.

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Picking a Trusted Fiduciary

Apologies to readers, as the wording of this puzzle had a flaw, pointed out by John Berglund and Jeff Subeck. Two of the conditions should have been:

  • Excluding Delilah (rather than “Delilah says”), over half of the other candidates are truthful.
  • Excluding Leni (rather than “Leni says”), fewer than half of the other candidates are truthful.

Jeff Subeck submitted the following solution for the corrected wording:

  • Excluding Leni, there are at most four truthful.
  • Excluding Delilah, there are at least five truthful.

Since adding Leni and removing Delilah increases the number of truthful candidates, it must be that Leni is truthful and Delilah is not. Thus, Deckard picks Leni.

That is all that is needed to solve the puzzle, but it turns out that there are only two possibilities as to which candidates are truthful.

Due to the statements above, there are exactly four truthful, excluding Leni and Delilah.

  • Since Irma says that Squeaky is truthful, either both are truthful or both are liars.
  • Since Medea, Salome and Squeaky are either all truthful or all liars, the same can be said for Medea, Salome, Squeaky and Irma.

Thus, the list of truthful candidates is either Leni, Irma, Squeaky, Medea and Salome or Leni, Eve, Gomer, Ilse and Jezebel.

Solutions were also submitted by Shyam Bihari Agarwal, John Berglund, Scott Brown and David Skurnick.