2013 SMI Special Sessions Part I - Getting Medieval with IBL

1. 2013 SMI Special SessionsPart I - Getting Medieval with IBL Randall E. Cone, Ph.D. – Virginia Military Institute – 2013 HCPS/MCPS SMI Abstract:“Are you a knight or a knave? Deciding whether or not coursework is suitable for use as IBL material within a particular mathematics class involves the consideration of several related domains. We explore this decision-making process through examples, discussions, and activities in a series of mathematical puzzles.”

2. Rules of the Island: Knights always tell the truth. Knaves always lie.

3. Work in groups to solve the following problems. Designate a runner for your group’s written answers.

4. You arrive at the island. While looking for your long-lost Friend, you happen upon a strange castle. At the castle gate are two guards, dressed in different colors. You ask them: “are you knights or knaves?” The Red Guard glares, the Blue Guard says: “We are both knaves.” What do you know about the guards?

5. Reasoning: The Blue Guard says: “We are both knaves.” Suppose the Blue Guard is a knight. Then he is not allowed to lie and call himself a knave.  Suppose the Blue Guard is a knave. Since he is a knave, his claim “we are both knaves” is a lie. Hence, the Blue Guard is a knave, and the Red Guard a knight. 

6. Neither guard knows your friend, so you move on. Past the castle you come upon a long figure staring at the grave of his departed brother. He is murmuring something repeatedly... “We were always of the same type... cut from the same cloth...” You notice on the gravestone are written the words: “Of different types were we, my brother and I – he was cut from a different cloth.” What do you know about the brothers?

7. Clearly, there is a contradiction here, as both statements cannot be true simultaneously. If the brothers are both knights, then the deceased brother would be lying.  If the brothers are both knaves, then the muttering brother would be truthful.  The brothers must be of different types. The deceased brother must be a knight and the muttering brother a knave. 

8. Neither brother seems to want to talk with you, so you move on. Before long, you happen upon two Valkyrie, guarding a fork in the road. At the left fork is the Blue Valkyrie, at the right the Golden Valkyrie. Legend has it that one of the Valkyrie is a knight and one is a knave. Together they say: “one way leads to your Friend, the other: Death.” “You may ask us a single question, answerable by ‘yes’ or ‘no’.”

9. There are many possible questions that will work. This particular puzzle is an example of the Nelson Goodman Principle. Here are two valid questions: Ask of Blue Valkyrie: “Will the Golden Valkyrie tell me your fork will lead me to my friend?” Ask of either Valkyrie: “Are you the type who could claim my friend lies along your fork?” What would answers for either of these questions tell you?

10. Reasoning: Asked of Blue Valkyrie: “Will the Golden Valkyrie tell me your fork will lead me to my friend?” Suppose friend is on BV fork. If BV is a knight, GV is a knave and would lie and say ‘no’. Therefore BV would be truthful and say ‘no’. If BV is a knave, GV is a knight and would truthfully say ‘yes’. Therefore BV would lie and say ‘no’. Suppose friend is on GV fork. If BV is a knight, GV is a knave and would lie and say ‘yes’. Therefore BV would be truthful and say ‘yes’. If BV is a knave, GV is a knight and would truthfully say ‘no’. Therefore BV would lie and say ‘yes’. Conclusion: ‘no’ implies friend is on BV fork, ‘yes’ implies GV fork.

11. Reasoning: Ask of either Valkyrie (we’ll choose to question the Blue Valkyrie): “Are you the type who could claim my friend lies along your fork?” Suppose friend is on BV fork. If BV is a knight, then BV is the type who could claim your friend is on her fork and would say ‘yes’. If BV is a knave, then BV is not the type who could claim your friend is on her fork, but would lie and say ‘yes’. Suppose friend is on GV fork. If BV is a knight, then BV is the not type who could lie and claim your your friend is on her fork and would therefore say ‘no’. If BV is a knave, then she is the type (a liar) who could claim your friend is on your fork, but must lie and say ‘no’. Conclusion: ‘yes’ implies friend is on BV fork, ‘no’ implies GV fork.

12. You walk down the correct fork in the path, which will lead you to your Friend. At the edge of a small town, you recognize a fair lady standing there. She is a friend of your Friend’s and you know her name is Gwyneth or Gwendolyn, but can’t remember which. You’d like to ask her if she can help you find your friend, but you’d first like to know if she is a knightess or a knave-ess. To start conversation, you greet her by saying “Good day, M’Lady. I seem to recall you from somewhere, may I ask your name?” She answers with “Good day, M’Lord. My name is Gwendolyn.” You don’t know for certain if she is a knight or a knave, but with a very high probability you know her type. Why?

13. If she were a knave-ess, there would be a very low probability that she would choose “Gwendolyn”, one of the two names you are thinking of, to be her (false) name out of all possibilities. Hence, it is very likely that she is a knightess and would answer your other questions with veracity. After speaking with her for a bit, she indicates that you might your Friend at the town’s square, as it is an oft-frequented place by all of the island citizens.

14. After making your way to the town’s square, you there find a curious pair of gentlemen standing near a doorway. They are pointing at you and whispering to one another... ...And they are both wearing masks of the type your Friend wears! You know one of them must be your friend. The Left Figure points at the other and says: “He is your friend and he is a knave!” The Right Figure points at the other and says: “He is not your friend, but he is a knight!” Have you found your friend?

15. Reasoning: The Left Figure points at the other and says: “He is your Friend and he is a knave!” The Right Figure points at the other and says: “He is not your Friend, but he is a knight!” Suppose the LF is a knight. Then RF is your Friend and a knave. ButRF must lie, and so his claim LF is a knight is not possible. Suppose the RF is a knight. Then RF is your Friend. ButRF must be truthful, and LF cannot be a knight. Hence, both figures are knaves. Is one of them your Friend? Yes! The figure on the left must be your Friend!

16. “Either mathematics is too big for the human mind or the human mind is more than a machine.” Mathematical relevance of these “puzzles”? Without the precise forms of mathematical logic, such as is found and used in these puzzles, we would not have mathematics. -- Kurt Gödel

17. “Is IBL appropriate for use in teachingmy class’ material?” This question is about error correction.* The following decision tree may be applied: Does the target audience have the mathematical maturity to self-correct effectively (i.e. recognize when a problem “breaks”)? If so, then most problems are suitable for reconfiguring as IBL problems for your class.

18. Does the target audience have the mathematical maturity to self-correct effectively (i.e. recognize when a problem “breaks”)? If not, then there need to be explicit mechanisms put into place to correct students, the correction mechanisms reacting reasonably quickly for most problems (be the problems procedural or otherwise). This correction can be done through monitored group work and/or presentations and guided worksheets, as well as within other activities.

19. “If I have seen further than others, it is by standing upon the shoulders of giants.” -- Sir Isaac Newton Bibliography: Logical Labyrinths by Raymond M. Smullyan “The MacTutor History of Mathematics Archive” at Univ. of St. Andrews (Almost) all images courtesy of: VMI Digital Archive wikipedia.org www.isaacnewton.org.uk

20. Thanks to: The Harvest House and the Superintendents of HCPS and MCPS VMI Mathematics Department You, the Audience!