A Mathematician's Holiday Oxford University

Given four bottles with indistinguishable liquid, one of which is a vital medicine, two containers and a test that can be done only once, how can you determine which of the the bottles contain the medicine?

In a restaurant where you can order tasting plates of 10 items, what is the smallest number of plates you can order to identify all 10 items on a menu?

By drawing on a piece of paper, can you connect three houses to three utilities (gas, electricity, water) without any of the lines crossing?

How can you get three people to the upper floor of the hotel if two of them can never be left alone?

What is the quickest route to get from where you are standing, collect some water from a river and get to the hotel?

Three bags contain 2 t-shirts or 2 hoodies or 1 hoodie and 1 t-shirt, and none are labelled correctly. Can you tell which back belongs to whom by only taking one (random) item from one bag?