Newton's Bucket is one of the eight most important physics problems involving both buckets and Sir Isaac Newton. It is considerably more important than the Newton's Plastic Bucket For Making Sandcastle Components, and slightly more important than Newton's Bucket For Making Slightly Larger Sandcastle Components, but is not as important as Newton's Bucket of Unstable Chemicals Left Underneath Your Bed. You should probably move that one to somewhere else, perhaps a neighbors' yard, at a neighbors' winter home, in another state with more whimsical views of environmental protection.
The problem of Newton's Bucket is no less great for being one you can study without benefit of your own bucket, or even your own Newton, should you lack one. First, imagine you have a bucket. Those who have buckets are to be credited for vivid imaginations. Next, imagine it is partially filled with water, which can be obtained by burning amines and filtering out carbon dioxide and nitrogen or nitrous oxides, or use the tap if you don't want to be any fun.
As you start spinning the bucket, you swiftly spill water on your feet. Changing to clean socks and filling the bucket not quite so much, spinning makes the water spin and curls towards the bucket's side until your arm falls off. Maybe it's better to hang the bucket from a rope to spin, so when your arm falls off you don't spill the water again.
Now the big question: when you're spinning the bucket, how do you know the bucket and water aren't actually staying still while the entire universe spins around them, the way it works when you get on a carousel ride and the universe spins about? And then how does the water know it should spin and curl, instead of insisting it should stay where it is and you should spin and curl instead? For us this is easy to answer: you quickly get bored spinning the bucket and set it down on the table to do something more interesting. If the rest of the universe were spinning, then when you got bored you would have to set down the rest of the universe, and you don't have enough room on the table for that.
So how does the water know you don't have enough room to fit the universe on your table? Is it any of the water's business what you do with the non-water parts of the universe?
Now let's make it completely incomprehensible: suppose there were a universe which had this bucket and the water, but absolutely nothing else. There's no distant galaxies, no planets, no pillows decorated with flowers, no bumpy yet wiggly things. If the bucket is spinning, how do you know that it's the bucket instead of the entire rest of space? We can't rely on the ``getting bored'' standard, because if there isn't anything in the rest of the universe then all that nothing would fit on your table, if the table existed.
The answer is, of course: you don't know, because you don't exist, since there's nothing in that universe except the bucket and the water, and you're surely not the bucket, not with your sense of dignity. Can we be sure you're not the water? This is a harder question, because to say whether you were water requires pressing yourself into a sponge, pulling back out, and seeing if you can't. But as implied, there are no sponges in that universe, and no way to tell.
Yet if you are the water, then you know what you're should do and should feel no less responsible to your audience solely because it does not exist. So you behave appropriately, curling against the sides of the bucket or not, depending on whether you're spinning. Perhaps you can tell by how dizzy you are. And if you are dizzy then obviously you were spinning, or else stood up too fast, and the equivalence of all this dizziness is a part of how the universe works.
Thus the problem of Newton's Bucket teaches us how lucky we are to live in a universe which has sponges, as otherwise we would have never know which things are spinning.
Trivia: Sir Isaac Newton was interred in Westminster Abbey on 28 March 1726. Source: The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time, Jason Socrates Bardi.
Currently Reading: Imperial Earth, Arthur C Clarke.