Santa wants to upgrade his legendary Christmas Hot Cocoa Boiler so it can keep up when he changes his mind — which he often does — about how sweet the cocoa should be.
To help him, the elves designed a clever new system where sweetness no longer changes in a poof of magic, but slowly drifts up or down thanks to:
- A set of chocolate-brown levers – pulling one will make sweetness creep upward by 1 unit per minute;
- A set of snow-white levers – pulling one will make sweetness drift downward by 1 unit per minute.
In the proposed system, pulling two identical levers will make sweetness change twice as fast (i.e., 2 units per minute). If instead at any moment a brown-lever elf and a white-lever elf are both pulling, their actions cancel, and both elves step aside to go drinking a ginger ale.
And when & how shall an elf decide to do something? The proposal is that every minute, a new elf arrives and decides what to do based on the average sweetness of the previous minute:
- If the cocoa was too bland, the elf pulls one chocolate-brown lever.
- If it was too sweet, the elf pulls one snow-white lever.
Santa is happy, but you are worried that this strategy may hide a serious problem, especially when Santa suddenly asks for a sweeter cocoa. Which one?
a. 🍫🚀🧊 Watch out!
If Santa asks for a sweeter cocoa, the system could run away and push sweetness all the way to its maximum — and get stuck there forever!
b. ☕🎢📈 Watch out!
If Santa asks for a sweeter cocoa, the sweetness may shoot too high at first and overshoot the new level before settling.
c. 🧝♂️🔥⚙️ Watch out!
If Santa asks for a sweeter cocoa right when a new elf arrives, the poor machine might panic, overheat, and jingle itself to pieces!
d. ❄️⏳☕ Watch out!
If Santa asks for a sweeter cocoa, sweetness might briefly drift downward first — because a white-lever elf might react before any brown-lever elf jumps in.
e. 🧝🎧❄️ Watch out!
Tiny random fluctuations in sweetness levels when the elf arrives might confuse the arriving elves, making them pull the wrong lever again and again.
f. 🎅🍪🤪 Watch out!
If Santa tastes too many test samples, he might lose his sense of sweetness entirely and give the elves completely contradictory commands!
Related control topic: integral control