Bit-shifting

import java.math.BigInteger

object Board {

    fun getGrainCountForSquare(number: Int) =
            if (number < 1 || number > 64)
                    throw IllegalArgumentException("square must be between 1 and 64")
            else BigInteger.ONE.shiftLeft(number - 1)

    fun getTotalGrainCount() = BigInteger.ONE.shiftLeft(64).subtract(BigInteger.ONE)
}

An object declaration is used to define Board as essentially a singleton object instantiation of the class. This is sufficient, since there is no object state that needs to change with each call of the getGrainCountForSquare or getTotalGrainCount methods.

The getGrainCountForSquare method is implemented by an if/else expression. Although the function has multiple lines, it consists only of the one if/else expression, so it is defined using single-expression function syntax, with the curly braces omitted from the function call and the return type inferred.

An IllegalArgumentException is thrown if the number is out of bounds.

Otherwise, to calculate the grains on a square you can set a bit in the correct position of a BigInteger value.

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

You use the BigInteger.shiftLeft() method to set 1 (i.e. BigInteger.ONE) 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 square is 1 for square One, you subtract square 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 square is 2 for square Two, you subtract square by 1 to get 1, which will move it 1 position to the left. The result is binary 10, which is decimal 2.

For getTotalGrainCount 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 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.