You and a fellow traveler have been imprisoned by a mad king with an unaccountable penchant for logic puzzles (as mad kings do). Can the two of you solve his riddle and escape with your lives—more importantly, can you do it without communicating with one another?
This week’s puzzle was suggested by reader Olivier H. “I came across a puzzle that is fairly difficult,” writes Olivier, “but nevertheless directly aims at the understanding of logic.” Neither of us is sure where the puzzle originated, but we think you’ll enjoy it.
SUNDAY PUZZLE #36: Surveying The Mad King’s Kingdom
You and a fellow traveler are caught trespassing through the kingdom of the fear-instilling Mad King. As a punishment, the king imprisons you both in separate cells and presents you with a riddle, which you must solve correctly in order to save your lives.
“From your separate cells, you can each see half the land of my kingdom,” says the king, “across which are distributed either 10 or 13 villages. Each day at 5 p.m., I will give each of you an opportunity to tell me the number of villages in my kingdom. If your answer is correct, you will both be freed. But if your answer is wrong, you will both be killed.”
On the fifth day, the two of you are freed. How many villages are there in the king’s kingdom, and how many villages did you each see? How do you know?
- The villages may be spread unequally across the kingdom.
- You cannot see the villages that your fellow traveler sees and vice versa.
- There is, of course, no communication between you and your fellow traveler, though you each know each other to be wise individuals.
- A wise individual is one who understands logic.
We’ll be back next week with the solution—and a new puzzle! Got a great brainteaser, original or otherwise, that you’d like to see featured? E-mail me with your recommendations. (Be sure to include “Sunday Puzzle” in the subject line.)
SOLUTION To Sunday Puzzle #35: Timing Toast
Last week, I asked you to determine the shortest amount of time in which three slices of bread could be toasted and buttered.
It’s an optimization problem—and boy did you all optimize! There were charts and spreadsheets and lots of really great collaboration in last week’s comments. My favorite thread was this one, started by hawkingdo, in which several of you troubleshoot your way to an optimized solution of 117 (or is it 120?) seconds.
The solution arrived at in last week’s comments is, most likely, the one first provided by Martin Gardner when he originally posed this puzzle several decades ago. “I originally published a solution showing how the three slices of bread could be toasted in two minutes,” he writes in My Best Mathematical and Logic Puzzles. But several of his readers eventually surprised him by slashing the time to 111 (or 114, depending on when you stop the stop watch—when the last slice has finished toasting, or when the last slice of toasted bread has been removed from the toaster). “I had overlooked the possibility of partially toasting one side of a slice, removing it, then returning it later to complete the toasting,” Gardner writes.
It’s worth mentioning that the 111-/114-second solution involves buttering one of the slices before it has completed toasting on both sides. Does this violate the rules of the puzzle? In my opinion, no. Does it violate some unspoken rule of how bread should be toasted and buttered? Very possibly. Here is the full sequence, as scanned from Gardner’s book: