A magma is a group of molten rocks in a volcano. A group is a magma that is invertible, is associative, and has an identity element.

Property 1 Property 2 Property 3 Property 4 Property 5  
          Rectangle group w/ reflections (\(D_2\))
          {true} under logical and
          3D vectors under cross product
          Booleans under implies
          Octonions except 0 under multiplication
          Cube group (\(S_4\))
          {-1, 0, 1} under \(f\), where \(f(x, y) = x^3y^2\)
          Integers under addition

Properties

  • Associative
  • Commutative
  • Idempotent
  • Identity element
  • Invertible
  • No identity element
  • Zero element


  • If two properties are opposites, exactly one of them appears in the list.
  • The one property referenced by the clue directly below this one appears last in the list.
  • \(\forall x,y,z, x(yz) = (xy)z\) does not appear earlier than 3rd.
  • Commutative appears somewhere before the property alluded to by the output required from a cross product for it to be commutative.
  • Each property referenced by the clue directly above this one appears somewhere after each property referenced in the clue directly below that actually appears in the list.
  • Between the properties that contain the word “identity”, each one that appears in the list appears somewhere before each other property that starts with “I” and appears in the list.
  • Each property that implies \(\forall y, ∃x, xx = x\) does not appear in the list, unless contradicted by a clue somewhere above this one.