At a conference of logicians, five attendees (A, B, C, D and E) are each either a truth teller (always telling the truth) or a liar (always telling lies). They make these statements:
- A says, “Exactly one of us is a liar.”
- B says, “Exactly three of us are truth tellers.”
- C says, “Exactly three of us are liars.”
- D says, “Exactly one of us is a truth teller.”
- E says, “All five of us are liars.”
How many liars are there among the five attendees, and who are the liars?
Also, note there is a follow up challenge in the solution below to a previous puzzle.
Surface Volume Mismatch
If it is possible, then define a simply connected set in 3-dimensional Euclidean space that has volume 1 and infinite surface area. Alternately, if it is impossible then prove that such a set is impossible.
One solution is the well-known example of Gabriel’s Horn (a.k.a. Torricelli’s Trumpet). Using x,y,z coordinates take the set defined by z ≥ 1 and x2+y2 ≤ 1/z2 .
The volume is ∫1∞(π/z2)dz = π and the surface area is π+2π∫1∞√(1+(d/dz (1/z))2 ) dz>2π∫1∞dz = ∞ .
Follow up challenge: Can you construct such a set that is also bounded, or prove that is impossible.
Solutions were also submitted by Roger Bovard, Bob Conger, Edward Lotkowski, Jonas Meyer, Ron Miller, Jim Muza and Natalie Ramirez.
