When presented with this paradox initially, I started looking at the specific example of a heap. The example of the heap is just an example; the paradox has other applications. But I'm only going to consider the specific example of a heap of sand.

The conclusion I came to was that a heap isn't made up of anything. The word "heap" describes a specific geometric shape, much like the words "circle" and "square" describe geometric shapes. A heap is a three-dimensional shape similar to a bell curve. It's the result of what would happen when solid objects are dropped onto a plane in a similar location.

Thus, a heap of sand is actually an invisible geometric shape that is made visible by the sand occupying it. It doesn't matter if sand is removed from one side more than the other, or if multiple mounds are created by removing sand in a star pattern from the heap - heaps are three-dimensional shapes that can have longer tails on either end, and they can even overlap each other to create more complex shapes.

In programming, an array can be of length zero. It doesn't stop being an array just because it's empty. Instead, the important aspect of an array is what the array data structure is. A heap is like an array in the sense that is not what it contains but more the abstract concept it describes. An array is a data structure, and a heap is a shape.

Removing a single grain of sand from a heap of sand over and over simply reduces the invisible heap's size. To be technical, the minimum number of grains of sand necessary for a heap is probably 3. I say that because a heap is a 3 dimensional shape, and if 2 grains of sand support a third, then it is a small heap. But, if only two grains of sand remain, then the shape described by 2 data points on a plane would be a line. If they were somehow stacked, it would also be a line, just vertical.

The paradox deals with more of the absurdity of numbers and language than the specifics of a heap. When we say say a pile of clothes, how many clothes make a pile? When we say a pack of dogs, how many make a pack? In English, every "group" is usually three, as two is a pair and one is singular. If a heap of sand merely means a group of sand, then 3 is the minimum to be called a heap; the paradox is resolved with the number 3.

The reason this is a paradox is the reason for many paradoxes: the attempt to define common uses of language and words in terms of mathematics. The debate usually concerns what exactly a heap is, and it remains a paradox because no one can define a heap in terms of number. But no one would debate it if a heap were simply a number.

I gave my answer: 3 grains of sand is the minimum for a heap. And I also provided 2 definitions for a heap: a 3D geometric shape similar to a bell curve, and a group, with "group" being defined as 3 or more of a kind.