# Can you solve the wizard standoff riddle? – Dan Finkel

October 28, 2019

You’ve been chosen as a champion
two rival magic schools. Your opponents are fearsome. From the Newt-niz school, a powerful sorcerer wields a wand
that can turn people into fish, but his spell only works 70% of the time. And from the Leib-ton school, an even more powerful enchantress wields
a wand that turns people to statues, and it works 90% of the time. Lots are drawn, and you’re chosen
to cast the first spell in the duel. The Newt-niz magician will go second, and the Leib-ton enchantress third, after which you’ll repeat casting in
that order until only one of you is left. The rules of magic duels are strict, and anyone who casts out of
order immediately forfeits the duel. Also, to prevent draws, the rules stipulate that
if everyone’s still standing at the end of the first round, you’ll all be turned into cats. Now, you must choose a wand. Your wizarding house presents you
with three options: the Bannekar, which binds
one target with vines and casts effectively 60% of the time, the Gaussian,
which turns one target into a tree and works 80% of the time, and the incredibly rare Noether 9000, which banishes one target
to a distant mountaintop and casts perfectly 100% of the time. Your opponents are masters of strategy,
as well as sorcery, and you know they’ll make the choices that
maximize their own chances of success. Which wand should you choose and what strategy should you employ to have the greatest chance
of winning the duel? Pause the video now if you want
to figure it out for yourself! Answer in: 3 Answer in: 2 Answer in: 1 You reach for the Noether 9000 first. After all, it makes sense to enter
the duel with the most powerful wand. But before you pick it up, you consider
what would happen. As the most dangerous wizard, you’d also be the target
of the other two magicians, and you’d need to take
care of the most dangerous of them first. But afterward, there’s a 70% chance you’d
be struck down by the remaining wizard. That’s trouble. Maybe it’s better to take the Gaussian. It works 80% of the time, which means you wouldn’t be a target
until the enchantress was incapacitated. But if you succeeded in transforming her, you’d probably be turned
into a fish immediately after. If you transformed the sorcerer, the enchantress would almost
certainly turn you to stone. It would really be better if you missed. And that’s when you have an idea: what if you took the Gaussian,
then missed on purpose? Then, you would wait for the sorcerer
to attack the enchantress, and you’d have an 80% chance
of winning against the sorcerer. It’s a good idea, but there’s a problem; the sorcerer could also pass his turn and the enchantress, knowing that
she couldn’t pass without becoming a cat, would cast her spell on one of you. And since you’re the most dangerous
between you and the sorcerer, you’d be the target. And that’s when you see
what you really need to do: take the weakest wand, the Bannekar,
and miss on purpose. Now the sorcerer knows that
he’ll be targeted by the enchantress and he’ll have to try to turn her into
a fish to avoid being turned into stone. Seventy percent of the time he’d succeed and you’d have a 60% chance
of winning the duel at the beginning of the next round. If he fails, chances are he’ll be
turned to stone and you’d still have a 60% chance of
winning the duel against the enchantress. There’s a slim 3% chance
you’ll all be turned into cats, but when everything’s accounted for, you have better than even odds
of winning with this strategy. And that’s the best you can do. Here’s what the probability of winning
for the different strategies looks like. Who would’ve thought
that the best way to take your shot would be to throw away your shot?