Bit-shifting

// Package grains is a small library for determing the squares of a number.
package grains

import (
	"errors"
)

// Square returns the number of grains on the square for the number passed in.
func Square(num int) (uint64, error) {
	if num < 1 || num > 64 {
		return 0, errors.New("square number must be 1 through 64")
	}

	return 1 << (num - 1), nil
}

// Total return the sum of the squares from 1 to 64, which is the number of squares on a chess board.
func Total() uint64 {
	return 1<<64 - 1
}

Instead of using math to calculate the number of grains on a square, you can set a bit in the correct position of a uint64.

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. To go back to our two-square example, if we can grow to three squares, then we can shift 1 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.