Varsity Math, Week 21
This week, the Varsity Math team has discovered an old book of classic math puzzles, and it has inspired them to create a couple of new twists.
Spiral Fleeing
Four dogs (Pooch, Rover, Spot and Tige) start at the corners of a square of side length equal to the answer to Tree-angle from last week. All four dogs trot straight toward the center of the square at a rate of one meter per second, where they will all meet and stop simultaneously. Pooch has a flea, but he knocks the flea off the instant they are starting. The flea always flies at a constant rate of five meters per second. It flies so that at any instant, it is heading straight toward Rover’s current location, until it gets to Rover. At that point, Rover knocks the flea off instantaneously, and the flea heads toward Spot in the same manner, who knocks it off instantaneously, so it heads toward Tige, and then back toward Pooch, and so on, visiting each dog and being knocked off instantaneously in a repeating loop. The flea stops and hops down to the grass when all the dogs have reached the center.
What is the length of the flea’s spiral path from Pooch’s original corner to the center of the square?
[asciimathsf]
Rational Water
An explorer is stranded in the desert, but fortunately comes upon a cache of water and the trail to safety. Rescue lies at the 11th way station along the trail. Hiking from the cache to the first way station or from one way station to the next requires drinking one liter of water to survive, but the explorer can only carry seven liters at a time and still navigate the treacherous trail. The explorer can, however, carry water to any of the way stations and leave some there as a smaller cache, making sure to always carry enough water for the next hike. For example, the explorer could start out with seven liters, hike for two way stations (drinking two liters), leave three liters there, and drink the remaining two liters to hike back to the original cache. In this way, the explorer can gradually work closer to the 11th way station and escape from the desert. The explorer anxiously counts how many liters there are in the cache, hoping there will be enough to reach safety.
What is the minimum number of liters of water required for the explorer to reach safety?
Solutions to Week 20
Plead Insanity. The main thing that solving this problem requires is carefully writing down the assertion made in each statement of the testimony, and the associated belief of the person that you can deduce based on the known fact that both triplets are criminals and hence liars. So:
Left says: I am insane = Left is not sane.
Therefore, Left believes: Left is sane.
Right says: No, my sibling is not [insane] = Left is sane.
Right believes: Left is not sane.
Left says: I do not believe what [Right said] = Left does not believe Left is sane.
Left believes: Left does believe Left is sane.
But now looking above, we see that it is in fact the case that Left believes that Left is sane. Hence, from the last line, we see that Left believes something we know to be true. Therefore, Left really is sane, by the definition of “sane” in Logiton: Left believes a true statement. So Left cannot be Dana, from which we conclude that the Right twin is Dana, the insane thief of the Trophy of Truth. At last, the true culprit is unmasked! A serious blow to Non Sequitur.
Recent Weeks
Week 19: Marble Jam & Cubic Conundrum, solution to Double Veracity
Week 18: Word Block & Double Veracity, solution to Beaver Bash
Links to all of the puzzles and solutions are on the Complete Varsity Math page.
Come back next week for answers and more puzzles.
