Suppose a cube of ice completely melts in exactly one hour. Throughout the melting process, the temperature inside the entire ice cube remains uniform at just a tiny amount below the freezing point. The external environment maintains a temperature just above the freezing point. The heat transfer rate is uniform across the surface of the cube. Under the same conditions, how long would the same amount of ice shaped like a regular tetrahedron take to melt? How long would melting take if the ice were shaped like a torus with a major radius twice its minor radius? Is there any specific shape that would take the longest time to melt? Is there any specific shape that would take the shortest time to melt?
War of Attrition
In this puzzle, the Blue Army and the Red Army fight until one side is completely annihilated. At each instant, the rate of casualties experienced by each army is equal to the surviving size of the other army times the other army’s firepower constant. The Blue Army wins with 85% of its initial force surviving. The questions are the following:
- What would have happened if the Red Army’s firepower constant had been tripled?
- What would have happened if the Red Army’s initial size had been tripled?
One way to solve this is with straightforward and simple (but not necessarily easy) calculus and algebra.
Suppose the fighting starts at t = 0. For the Blue Army, let B(t) be its size and b > 0 be its firepower constant. Similarly, let R(t) and r > 0 be the Red Army’s size and firepower constant, respectively. Since B‘(t) = – r ∙ R(t) and R‘(t) = – b ∙ B(t), it follows that b ∙ B‘(t) ∙ B(t) – r ∙ R‘(t) ∙ R(t) = 0. Integrating with respect to t leads to the equation b ∙ B(t)2 – r ∙ R(t)2 = C, where C is constant over time. Only the Blue Army can win if C > 0 and only the Red Army can win if C < 0. Since we know that Blue wins with 85% of its initial force surviving, it follows that C = b (0.852) (B(0)2) = 0.7225 b ∙ B(0)2. Since C = b ∙ B(0)2 – r ∙ R(0)2, it follows that r R(0)2 = 0.2775 b ∙ B(0)2.
If r is tripled, then r R(0)2 = 0.8325 b ∙ B(0)2 and, therefore, C = 0.1675 b ∙ B(0)2, which is still greater than 0. So Blue still wins, but at a much greater cost with only √(0.1675) ≈ 41% of its initial force surviving.
If R(0) is tripled, then r ∙ R(0)2 = 2.4975 b ∙ B(0)2 and therefore C = –1.4975 b ∙ B(0)2. Since C is now less than 0, the Red Army wins. At the winning time t when B(t) = 0, the equation indicates –r ∙ R(t)2 = C = –1.4975 b ∙ B(0)2. Since R(t)/R(0) = √(1.4975 / 2.4975) ≈ 77%, about 77% of the Red Army’s initial force survives.
Solutions were submitted by Bob Conger, Mark Goldfarb, Hana Jin, Dave Oakden and Brad Rosin.
Know the answer? Send your solution to ar@casact.org.
