Varsity Math, Week 2
This week, the team is visiting the Ag School, so Coach Newton’s come up with some pastoral conundrums for you to plow through. Also, you’ll see your first relay event today—it needs answers handed off to it from earlier puzzles. Relays will pop up from time to time, with a final Varsity Relay problem every five weeks. And brace yourself for your first Varsity challenge below.
Farm Roundup | Varsity
Sullivan farms a circular plot of land with a 400-meter-long center-pivot irrigator, represented by the shaded region in the diagram above. Sullivan wants to add a buffer zone of drought-resistant turf grass within the semicircle stretching from the westernmost to the northernmost point of her field, but outside the irrigated region. (So the shape of the buffer zone, represented by the unshaded region in the diagram, will be like a semicircle with a “bite” taken out of it by the main field.) One more piece of information: a bag of drought-resistant grass seed will cover 1,000 square meters of ground.
What is the minimum number of full bags of seed that Sullivan needs in order to be able to completely plant the new buffer zone?
Carambola Orchard | Freshman Relay
A mathematician plants 16 carambola trees in an orchard so that they lie in 15 straight rows with four trees in each row. It turns out that in this configuration, there are two categories of trees: the “3” trees, which lie in exactly three of the 15 rows, and the “5” trees, which lie in exactly five of the 15 rows. As it happens, the number of trees in one of these two categories is equal to the answer to last week’s problem Urn Respect.
Which category is it? (Your answer will be either 3 or 5.)
Solutions to Week 1
Instant Interrogation. If Acey were the completely innocent suspect, then it would be impossible for Acey to confess like that – it would be a false statement, and in Logiton, innocent people can’t lie. On the other hand, if Acey were the real thief, then it would again be impossible for Acey to confess – it would be a true statement, and criminals can’t tell the truth. So therefore, Acey must be the third suspect. But that means Acey did not steal the Medal of Reason, so Acey’s statement is a lie. Therefore Acey has committed some crime, just not this one. (I suppose in Logiton, that’s enough reason to arrest Acey, too, for further questioning.) Next look at Basie’s statement. Since Acey did not steal the Medal of Reason, Basie lied. That means that Basie is a criminal, and the only criminal left to uncover is the thief, so Basie stole the Medal of Reason. By process of elimination, Casey is the innocent bystander.
Urn Respect. You’ll have to wait until the relay is over for the solution to this one—it’s used in this week’s Carambola Orchard problem, so we don’t want to give it away.
Find more puzzles at varsity.momath.org.