You have 10 identical-looking bottles of identical-looking pills (let’s say there’s a hundred pills in each bottle). One of those 10 bottles contains poison. The only way to differentiate poison from real pills is the weight – while real pills weigh 1 g each, the poison is only 0.9 g. You have one scale that shows the exact weight (down to the milligram) of whatever is weighed.
How can you tell which bottle contains fake pills with just 1 weighing?
You have 100 doors in a row that are all initially closed. You make 100 passes by the doors starting with the first door every time. The first time through you visit every door and toggle the door (if the door is closed, you open it, if its open, you close it). the second time you only visit every 2nd door (door #2, #4, #6). The third time, every 3rd door (door #3, #6, #9), etc, until you only visit the 100th door.
What state are the doors in after the last pass? Which are open which are closed?