Authors of this question: Damiano Rotondo, Institutt for data- og elektroteknologi (IDE), Universitetet i Stavanger, Norway & Helem Sánchez, Hafrsfjord Skole, Norway
During their playful winter breaks, Santa’s elves love to ride their sleds down the icy hills surrounding Santa’s house and workshop. However, their safety is at risk! The area is full of dangerous trolls and goblins lurking in the North-West, North-East, South-West, and South-East regions, ready to ruin the elves’ fun and destroy their toys and sleds if they slide too far in the wrong direction.
A map spanning 5 candyards (the elves’ unit of length) in the four directions North, South, West, and East, is provided below. Santa’s house is located at the center of the area shown on the map. Directly in each of these directions from Santa’s house, there are four hills, each with a height of a little bit more than 1.8 candyards. The topography of the region shows that the hills rise steadily to these peaks, and beyond them, the terrain slopes back down. Ice covers the entire landscape, making it incredibly slippery, so once an elf starts sliding, their sled will always move towards decreasing values of height. However, the direction of descent from any given point on the ice depends on the subtle variations in the topography, making the elves’ path unpredictable.
The elves want to know: from which starting points on the icy hills can they safely slide without ending their ride in the arms of the trolls or goblins? Help the elves determine their safe zones. Can you help the elves determining the largest region of guaranteed safety, ensuring they never slide into danger?
a. 🛑🤷♂️ 🎲 Every point is unsafe, as one can never be sure that the sled will ultimately finish the ride at Santa’s house due to unpredictability of movement.
b. 🏔️🏠 👍 As long as the elves start from the top of one hill, they are safe, provided the initial velocity of the sled is directed towards Santa’s house.
c. 🔵✅ 📏 The elves should start from the blue ellipse surrounding Santa’s house (less than 0.2 candyards), as only that height is safe for sliding with the sled.
d. 🟢🗺️ 🛷 The elves can start safely from any point at a height of 1 candyard, which corresponds to the green subset extending continuously across the map, separating the hills and valleys.
e. 🏟️🛷 🏠 The elves can start safely from any point below the height of 1 candyard, corresponding to the largest stadium-shaped region surrounding Santa’s house.
