TypeError: list indices must be integers, not float

It looks like you are using Python 3.x. One of the important differences in Python 3.x is the way division is handled. When you do x / y, an integer is returned in Python 2.x because the decimal is truncated (floor division). However in 3.x, the / operator performs ‘true’ division, resulting in a float instead of an integer (e.g. 1 / 2 = 0.5). What this means is that your are now trying to use a float to reference a position in a list (e.g. my_list[0.5] or even my_list[1.0]), which will not work as Python is expecting an integer. Therefore you may first want to try using middle = (first + last) // 2, adjusting so that the result returns what you expect. The // indicates floor division in Python 3.x.