Here we will see some known and easy way of computing different mathematical and sequential things.

NOTE: This post is in continuation to the PART 1, however its not necessary to refer part I and you could take it as an independent post also.

**Calculate the average of a series**

# For an arbitrary sequence def average(seq, total=0.0): num = 0 for item in seq: total += item num += 1 return total / num # For a sequence type such as a list or a tuple def average(seq): return float(sum(seq)) / len(seq)

**Calculate the factorial of a number**

In mathematics, the factorial of a number (that cannot be negative and must be an integer) n, denoted by n!, is the product of all positive integers less than or equal to n.

*For example: 5! is 5*4*3*2*1*.

import operator def factorial(num): # the "1" initial value allows it to work for 0 return reduce(operator.mul, range(1, num + 1), 1)

** Calculate the sum over a Sequences**

A function that sums the values in a sequence with the built in python keyword, its name is ‘sum’.

>>> assert sum([1,2,3]) == 6

**Calculate a derivative**

Compute and print a derivative of the symbolic expression a x^2 + b.

*Let’s say, i want the value of derivative at x=5.*

from sympy import * import numpy as np x = Symbol('x') y = x**2 + 1 yprime = y.diff(x) yprime >>> 2⋅x f = lambdify(x, yprime, 'numpy') f(np.ones(5)) >>> [ 2. 2. 2. 2. 2.]

** ****Calculate digits of pi**

from sys import stdout scale = 10000 maxarr = 2800 arrinit = 2000 carry = 0 arr = [arrinit] * (maxarr + 1) for i in xrange(maxarr, 1, -14): total = 0 for j in xrange(i, 0, -1): total = (total * j) + (scale * arr[j]) arr[j] = total % ((j * 2) - 1) total = total / ((j * 2) - 1) stdout.write("%04d" % (carry + (total / scale))) carry = total % scale

** Calculate an integer square root**

def isqrt(n): xn = 1 xn1 = (xn + n/xn)/2 while abs(xn1 - xn) > 1: xn = xn1 xn1 = (xn + n/xn)/2 while xn1*xn1 > n: xn1 -= 1 return xn1

**Calculate distance between two points on a globe**

from math import radians, cos, sin, asin, sqrt def haversine(lon1, lat1, lon2, lat2): """ Calculate the great circle distance between two points on the earth (specified in decimal degrees) """ # convert decimal degrees to radians lon1, lat1, lon2, lat2 = map(radians, [lon1, lat1, lon2, lat2]) # haversine formula dlon = lon2 - lon1 dlat = lat2 - lat1 a = sin(dlat/2)**2 + cos(lat1) * cos(lat2) * sin(dlon/2)**2 c = 2 * asin(sqrt(a)) # 6367 km is the radius of the Earth km = 6367 * c return km

** Find a solution to a sudoku puzzle**

The current shortest Python sudoku solver program is 178 bytes long and fits on four lines with no line longer than 80 chars. It prints the first solution only. To run, the intial layout must be given as command line argument containing 81 characters from 0-9 (0 = unknown), and the solution is returned in the same format to standard error. The input must have a valid solution.

def r(a):i=a.find('0');~i or exit(a);[m in[(i-j)%9*(i/9^j/9)*(i/27^j/27|i%9/3^j%9/3)or a[j]for j in range(81)]or r(a[:i]+m+a[i+1:])for m in'%d'%5**18] from sys import*;r(argv[1])

To run, on a blank board:

python sudoku.py 000000000000000000000000000000000000000000000000000000000000000000000000000000000