in `return (1 << 64) - 1` why must we subtract 1 to get the total of grains for all squares of a chessboard? ? in python

In the expression return (1 << 64) - 1, the << operator is a left shift operator used for bit manipulation. It shifts the bits of the number 1 to the left by 64 positions.

When we shift the bits of 1 to the left by 64 positions, we get a number that has 1 at the 65th bit and 0 in all other positions. This is because 1 in binary is represented as 000...001, and shifting it to the left by 64 positions adds 64 zeros to the right.

Therefore, (1 << 64) results in the number 2^64 or 18446744073709551616 in decimal.

By subtracting 1 from this value, we effectively get the total number of grains for all squares of a chessboard. This is because in a chessboard, the number of grains on each square doubles compared to the previous square. So, if we subtract 1 from 2^64, we are effectively subtracting 1 grain from the total, which accounts for the initial square.

To understand it better, imagine the following:

• On the 1st square, we have 1 grain.
• On the 2nd square, we have 2 grains.
• On the 3rd square, we have 4 grains.
• And so on, until the 64th square.

In total, if we sum the number of grains on each square, we get 2^64 - 1.

Therefore, (1 << 64) - 1 is used in this context to represent the total number of grains for all squares of a chessboard.