Santa is known for many traits such as his love of red clothing and his long white beard. A well-kept secret is that Santa’s beard grows exactly 1 cm per week.
What is charming for some, however, causes extra work for others. Santa himself does not worry much about his beard – he prefers spending his days overseeing the workshop, eating cookies, and sipping hot chocolate, ginger shots, and the occasional mulled wine.
The elves, on the other hand, must ensure he remains presentable when going out.
The longer the beard grows, the more crumbs it catches and the larger the stains from hot drinks. Beyond a certain length Santa even begins to trip over it, and a stumbling Santa is not fit to steer a sleigh. In short: an overgrown beard means more work for the elves and could even endanger Christmas.
Since life would be much easier if Santa’s beard stayed close to the mandatory minimal length for Christmas-staff of 20 cm, the elves propose two possible trimming schemes to be applied every Sunday:
Barbara’s Proposal: Each Sunday, Barbara tosses a fair coin:
- Heads: the beard is cut by 2 cm,
- Tails: the beard is left untouched for another week.
Harry’s Proposal: Each Sunday, Harry rolls a fair six-sided die:
- If a 6 appears: the beard is cut down to exactly 20 cm,
- If 1–5 appear: nothing is done for that week.
Santa, overwhelmed by several recent “beard-related incidents” (including the notorious pasta-sauce disaster), must choose one of the two schemes right now. Since today is Sunday, the first coin toss / die roll / spinner spin will happen immediately after his choice.
Assuming all random devices are fair, and that Santa wants his beard to be as long as possible on average, which proposal should he choose?
a. 🎲✂️🪙 He should choose Barbara’s proposal.
b. 🎲🎯🧔♂️ He should choose Harry’s proposal.
c. 🎡✂️🎄 It does not matter which proposal he chooses.
d. 📏🤷♂️🔀 When today his beard is shorter than 25 cm, he should choose Harry’s proposal; otherwise Barbara’s.
e. 🧹⏳✨ All three proposals yield the same average length, but Harry’s requires the least grooming time on average, so he should be chosen.
f. 🧔🎅❌ A man’s beard is a man’s beard. Even if Christmas must be cancelled, he should not cut it ever.
Related control theory topic: stochastic systems