Bit-shifting

Grains
Grains in Python 
def square(number):
    if number < 1 or number > 64:
        raise ValueError('square must be between 1 and 64')

    return 1 << number - 1


def total():
    return (1 << 64) - 1

Instead of using math for calculation, you can set a bit in the correct position for the number of grains on a square.

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 number is 1 for square One, you subtract number 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 number is 2 for square Two, you subtract number by 1 to get 1, which will move it 1 position to the left. The result is binary 10, which is decimal 2.
Square Shift Left By Binary Value Decimal Value
1 0 0001 1
2 1 0010 2
3 2 0100 4
4 3 1000 8

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 number of grains on the two squares.

Square Binary Value Decimal Value
3 0100 4
Square Sum Binary Value Sum Decimal Value
1 0001 1
2 0011 3
4th Dec 2024 · Found it useful?

Other Approaches to Grains in Python

Other ways our community solved this exercise
def square(number):
    if number < 1 or number > 64:
        raise ValueError('square must be between 1 and 64')
    return 2 ** (number - 1)


def total():
    return 2 ** 64 - 1
exponentiation

Use exponentiation to raise 2 by a specified power. 

def square(number):
    if number < 1 or number > 64:
        raise ValueError('square must be between 1 and 64')
    return pow(2, number - 1)


def total():
    return pow(2, 64) - 1
pow

Use pow to raise 2 by a specified power.