Bit-shifting

using System;
using System.Numerics; // Allows using BigInteger

public static class Grains
{
    public static ulong Square(int n)
    {
        switch (n)
        {
            case int num when num > 0 && num < 65: return (ulong)1 << (num - 1);
            default: throw new ArgumentOutOfRangeException("n must be 1 through 64");
        }
    }

    public static ulong Total()
    {
        return (ulong)((BigInteger.One << 64) - 1);
    }
}

Instead of using math to calculate the number of grains on a square, you can simply set a bit in the correct position of a ulong value.

To understand how this works, consider just two squares that are represented in binary bits as 00.

You use the left-shift operator to set 1 at the position needed to make the correct decimal value.

• To set the one grain on Square One you shift 1 for 0 positions to the left. So, if n is 1 for square One, you subtract n by 1 to get 0, which will not move it any positions to the left. The result is binary 01, which is decimal 1.
• To set the two grains on Square Two you shift 1 for 1 position to the left. So, if n is 2 for square Two, you subtract n by 1 to get 1, which will move it 1 position to the left. The result is binary 10, which is decimal 2.

For Total we want all of the 64 bits set to 1 to get the sum of grains on all sixty-four squares. The easy way to do this is to set the 65th bit to 1 and then subtract 1. However, we can't do this with a ulong which has only 64 bits, so we need to use a BigInteger. To go back to our two-square example, if we can grow to three squares, then we can shift BigInteger.One two positions to the left for binary 100, which is decimal 4. By subtracting 1 we get 3, which is the total amount of grains on the two squares.

Shortening

When the body of a function is a single expression, the function can be implemented as an expression-bodied member, like so

public static ulong Total() =>
    (ulong)((BigInteger.One << 64) - 1);

or

public static ulong Total() => (ulong)((BigInteger.One << 64) - 1);