.. escape-backslashes
.. default-role:: math

Chapter 3: ``if`` statements
=============================

:Authors:
    - Vincent Delecroix
:License: CC BY-SA 3.0

Comparisons and ``if``
----------------------

The comparison signs in Python and many other programming languages are
as follows

+--------+--------------------------+
| ``==`` | equality                 |
+--------+--------------------------+
| ``!=`` | difference               |
+--------+--------------------------+
| ``<``  | less than                |
+--------+--------------------------+
| ``>``  | greater than             | 
+--------+--------------------------+
| ``<=`` | lesser than or equal to  |
+--------+--------------------------+
| ``>=`` | greater than or equal to |
+--------+--------------------------+

Exercise 3.1
~~~~~~~~~~~~

Which number is the largest `1000^{1001}` or `1001^{1000}`?

::

   sage: 1000^1001 > 1001^1000
   True


Exercise 3.2
~~~~~~~~~~~~

Let us consider the following code:

::

    sage: a = # enter a value for a
    ....: if a != 2: 
    ....:     print('lost')
    ....: elif a == 3:
    ....:     print('an instant, please')
    ....: else: 
    ....:     print('you win')

What is the above program doing 

* when the variable ``a`` is 1?

::

    sage: a = 1# enter a value for a
    ....: if a != 2: 
    ....:     print('lost')
    ....: elif a == 3:
    ....:     print('an instant, please')
    ....: else: 
    ....:     print('you win')
    lost

* when the variable ``a`` is 2?

::

    sage: a = 2# enter a value for a
    ....: if a != 2: 
    ....:     print('lost')
    ....: elif a == 3:
    ....:     print('an instant, please')
    ....: else: 
    ....:     print('you win')
    you win

* when the variable ``a`` is 3?

::

    sage: a = 3# enter a value for a
    ....: if a != 2: 
    ....:     print('lost')
    ....: elif a == 3:
    ....:     print('an instant, please')
    ....: else: 
    ....:     print('you win')
    lost

* when the variable ``a`` is 15?

::

    sage: a = 15# enter a value for a
    ....: if a != 2: 
    ....:     print('lost')
    ....: elif a == 3:
    ....:     print('an instant, please')
    ....: else: 
    ....:     print('you win')
    lost

Exercise 3.3
~~~~~~~~~~~~

Two prime numbers `p` and `q` are said *twin* if `q = p + 2`. Find all
twin prime numbers below 10000.

::

    sage: TwinPrimeNumbers = [];
    ....: for p in prime_range(2,10000):
    ....:     if (p+2).is_prime():
    ....:         TwinPrimeNumbers.append((p,p+2))
    ....: TwinPrimeNumbers

::

    sage: len(TwinPrimeNumbers)
    205

Exercise 3.4
~~~~~~~~~~~~

Find the smallest and largest integers in the set

.. MATH::

    \{ a^b - b^a : a \in \{1, 2, \ldots, 5\}, b \in \{1, 2, \ldots, 5\} \}

::

    sage: min([a^b-b^a for a in range(6) for b in range(6)])
    -399

::

    sage: max([a^b-b^a for a in range(6) for b in range(6)])
    399

Exercise 3.5
~~~~~~~~~~~~

Recall that the method ``digits`` of an integer returns the list of
its digits:

::

    sage: 1527.digits()
    [7, 2, 5, 1]

Solve `Euler problem 56 <https://projecteuler.net/problem=56 (digits of a^b)>`_
by finding the maximal sum of digits of numbers of the form `a^b` with both
`a` and `b` lesser than `100`

::

    sage: MaximalSumDigits = 0;
    ....: for a in srange(100):
    ....:     for b in srange(100):
    ....:         Sum = sum((a^b).digits())
    ....:         if MaximalSumDigits < Sum:
    ....:             MaximalSumDigits = Sum
    ....: MaximalSumDigits
    972

Exercise 3.6
~~~~~~~~~~~~

Solve `Euler problem 4 <https://projecteuler.net/problem=4>`_ about
palindromes. A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is `9009 = 91 × 99`.

Find the largest palindrome made from the product of two 3-digit numbers.

::

    sage: LargestPalindrome = 0;
    ....: for a in srange(100,1000):
    ....:     for b in srange(a,1000):
    ....:         if Word((a*b).digits()).is_palindrome():
    ....:             LargestPalindrome = max(LargestPalindrome,a*b)
    ....: LargestPalindrome
    906609

Exercise 3.7
~~~~~~~~~~~~

Let us consider the following list of integers:

::

    sage: l = [123, 414, 264, 18, 689, 21, 5571, 28, 589, 12, 111, 231,
    ....: 158, 551, 250, 68, 5728, 2222, 4198, 571, 28, 518, 999, 444,
    ....: 112, 689, 672, 334, 680, 273]

Construct two lists ``leven`` and ``lodd`` that contain respectively
the even and odd elements of ``l``.

::

    sage: leven = []; lodd=[]
    ....: for x in l:
    ....:     if x%2 == 0:
    ....:         leven.append(x)
    ....:     else:
    ....:         lodd.append(x)


Using ``in`` and ``not in``
---------------------------

The condition of an ``if`` or ``elif`` statement is not necessarily a comparison.
Basically, any Python object would fit!

::

    sage: a = 5
    sage: if a:
    ....:     print('I am not zero')
    I am not zero

What happens under the hood is that the object ``a`` (here an integer) is converted
to a boolean value (``True`` or ``False``). You can see the boolean value of an
object by using ``bool``

::

    sage: bool(5)
    True

::

    sage: bool(0)
    False

::

    sage: bool([])
    False

::

    sage: bool([0])
    True

A useful construction is obtained with the keyword ``in``: the result of ``a in b``
is whether ``a`` belongs to the object ``b``. For example:

::

    sage: 2 in ZZ
    True

::

    sage: 2/3 in ZZ
    False

::

    sage: 2/3 in QQ
    True

::

    sage: 1 in [3, 5, 2, 1, 2, 8]
    True

::

    sage: 'a' in 'Saint-Flour'
    True

::

    sage: 'z' in 'Saint-Flour'
    False

To check that an element is not in a given object use ``a not in b``:

::

    sage: 10 not in Primes()
    True

::

    sage: 5/2 not in ZZ
    True

Exercise 3.8
~~~~~~~~~~~~

Using an ``if`` statement involving ``in`` inside a ``for`` loop, count the
number of vowels in the string:

::

    sage: s = 'How many vowels are present in this sentence?'

::

    sage: VowelsCount = 0
    ....: for i in range(0,len(s)):
    ....:     if s[i] in 'aeiouy':
    ....:         VowelsCount = VowelsCount + 1
    ....: VowelsCount
    14

Count the number of consonant in the string:

::

    sage: s = 'How many consonants are present in this sentence?'


::

    sage: ConsonantsCount = 0
    ....: for i in range(0,len(s)):
    ....:     if s[i] not in 'aeiou ?':
    ....:         ConsonantsCount = ConsonantsCount + 1
    ....: ConsonantsCount
    27

Exercise 3.9 (Pythagorean triples)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

A Pythagorean triple is a triple ``(a, b, c)`` of positive integers so that
`a^2 + b^2 = c^2`. An example is `3^2 + 4^2 = 5^2`. How many Pythagorean
triples are there with `a`, `b` and `c` smaller than 100?

::

    sage: PythagoreanTripleCount = 0
    ....: for a in range(100):
    ....:     for b in range(a,100):
    ....:         for c in range(b,100):
    ....:             if a^2+b^2 == c^2:
    ....:                 PythagoreanTripleCount += 1
    ....: PythagoreanTripleCount
    150

Solve `Euler problem <https://projecteuler.net/problem=9>`_ by finding the
unique Pythagorean triple so that `a + b + c = 1000`

::

    sage: for a in range(1000):
    ....:     for b in range(a,1000):
    ....:         for c in range(b,1000):
    ....:             if a^2+b^2 == c^2 and a+b+c==1000:
    ....:                 PythagoreanTriple = (a,b,c)
    ....: PythagoreanTriple
    (200, 375, 425)

Combining conditions ``or``, ``and`` and ``not``
------------------------------------------------

To make even more complicated tests you can combine them. The main operators
for this are ``or``, ``and``.

::

    sage: n = 17
    sage: if n.is_prime() and (n+2).is_prime():
    ....:     print('a twin number!')
    a twin number!

Exercise 3.10
~~~~~~~~~~~~~

Let us call a positive integer ``n`` a triple twin if all of ``n``, ``n+2``
and ``n+6`` are primes. How many triple twins are there smaller than `10000`?

::

    sage: TripleTwins = 0;
    ....: for p in prime_range(2,10000):
    ....:     if (p+2).is_prime() and (p+6).is_prime():
    ....:         TripleTwins += 1
    ....: TripleTwins
    55

The operator ``not`` is used for negation of a condition.

::

    sage: not True
    False

::

    sage: not False
    True


More exercises
--------------

For more exercises in the same veine you can challenge yourself with

* `Euler problem 30 <https://projecteuler.net/problem=30>`_ (sum of certain numbers)
* `Euler problem 33 <https://projecteuler.net/problem=33>`_ (digit cancelling fractions)
* `Euler problem 34 <https://projecteuler.net/problem=34>`_ (numbers which are sum of factorials of their digits)
* `Euler problem 35 <https://projecteuler.net/problem=35>`_ (circular primes)
* `Euler problem 36 <https://projecteuler.net/problem=36>`_ (integers palindromic in base 2 and 10)
* `Euler problem 37 <https://projecteuler.net/problem=37>`_ (truncatable primes)
* `Euler problem 38 <https://projecteuler.net/problem=39>`_ (integer right triangles, aka pythagorean triples)
* `Euler problem 39 <https://projecteuler.net/problem=53>`_ (binomials greater than a milion)
* `Euler problem 40 <https://projecteuler.net/problem=57>`_ (continued fractions)