Classic Mind-benders

A drawer has 10 black socks and 10 white socks mixed together, in a dark room. Maansi thinks getting a matching pair (two socks of the same color) for certain requires pulling out at least 11 socks. Her brother says just 3 socks are enough. Who is right?

A bat and a ball cost Rs 110 together. The bat costs Rs 100 more than the ball. Ravi says the ball costs Rs 10. Meena says the ball costs Rs 5. Who is right?

A monk starts up a mountain path at dawn and reaches the top by evening. The next day he walks back down the same path, starting at dawn again. Anaya says there is no reason he would ever be at the exact same spot at the exact same time of day on both trips. Her friend says there must be such a spot. Who is right?

10 prisoners are lined up, each wearing a black or white hat. Each prisoner can see the hats of everyone in front of them, but not their own or those behind. Starting from the back, each one, in turn, must say only "black" or "white" to guess their own hat color — everyone else can hear, and a plan can be agreed beforehand. Ojas thinks that no matter what plan is made, everyone's chance of being right stays a flat 50%. His teacher says a clever plan can guarantee almost everyone survives. Who is right?

3 monks and 3 demons want to cross a river. The boat can carry at most 2 people (someone must row it). On either bank, if any monks are present, demons must never outnumber them (or the demons overpower them). Krish thinks this is completely impossible, since the boat only seats 2 while 6 people need to cross. His sister says it is possible with the right order. Who is right?

100 doors start closed, numbered 1 to 100. Over 100 rounds: in round k, every door whose number is evenly divisible by k gets toggled (e.g. round 3 toggles doors 3, 6, 9, 12...). Ananya thinks that after 100 rounds, a random-looking scattering of doors will be open, with no clean pattern. Her friend says the open doors will follow a clean mathematical pattern. Who is right?