Added all exercises files to the repository.

1_binary/0_add_sub/ex/ex_add_sub.txt

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+Base 2
+======
+
+Binary addition and subtraction exercises
+-----------------------------------------
+
+If you have no previous experience in the binary base, it is crucial that you
+solve all the exercises here. Make sure to solve everything before you move on
+with the course.
+
+Anything marked with *Bonus* is not required for you to understand the following
+material, however It will probably make you smarter.
+
+
+0.  Addition. Solve the following. Write the solutions in base 2.
+    Use a pen and a paper.
+    Note: {100}_2 means 100 in base 2.
+
+  0.0   {100}_2 + {1}_2 = ?
+
+  0.1   {1111}_2 + {1}_2 = ?
+
+  0.2   {1001110}_2 + {111011}_2 = ?
+
+  0.3   {1101}_2 + {1101}_2 = ?
+
+  0.4   {1101}_2 + {1101}_2 + {1101}_2 + {1101}_2 = ?
+
+  0.5   Bonus: How much is {8}_10 * {1101}_2 ? (in base 2)
+        Note: * means multiplication.
+
+1. Subtraction. Solve the following. Use a pen and a paper:
+
+  1.0 {100}_2 - {1}_2 = ?
+
+  1.1 {11000}_2 - {11}_2 = ?
+
+  1.2 {11001101}_2 - {1011}_2 = ?
+
+  1.3 {1000000}_2 - {111011}_2 = ?
+
+  1.4 {10101}_2 - {1010}_2 = ?
+
+  1.5 {1000}_2 - {100}_2 = ?
+
+  1.6 {1100}_2 - {110}_2 = ?  (Bonus: Could you guess it?)
+
+
+Happy thinking :)

1_binary/1_base_conversion/ex/ex_base_conversion.txt

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+Base 2
+======
+
+Base conversion exercises
+--------------------------
+
+Hey, some Base conversion exercises are included here. The first ones are pretty
+usual, and could be dull at times, but please do them if this is your first time
+with base 2. Also, make sure that you do them with a pen and a piece of paper. I
+can't emphasize it more, you got to get the feeling through the hands, at least
+on the first time.
+
+As you move on with the exercises there are some that require more thinking.
+As usual, anything marked with *Bonus* is not required for you to understand the
+following material, however it will probably make you smarter.
+
+
+0.  Binary to Decimal conversion. Convert the following numbers to base 10:
+
+    You may use a calculator if you feel like it, however please don't use the
+    built in conversion function. 
+
+    It is important that you do this yourself, to make sure that you get the
+    feeling of it. Some things could only be learned through the hands.
+
+
+  0.0   {10}_2 = ?
+
+  0.1   {100}_2 = ?
+
+  0.2   {101}_2 = ?
+
+  0.3   {1001110}_2 = ?
+
+  0.4   {100110011}_2 = ?
+
+  0.5   {111111}_2 = ?
+
+  0.6   {1000}_2 = ?
+
+  0.7   {10000}_2 = ? (Bonus: Could you guess this one? How?)
+
+
+1.  Decimal to Binary conversion. Convert the following numbers to base 2:
+
+  1.0   {2}_10 = ?
+
+  1.1   {5}_10 = ?
+
+  1.2   {83}_10 = ?
+
+  1.3   {134}_10 = ?
+
+  1.4   {128}_10 = ?
+
+  1.5   {63}_10 = ?
+
+  1.6   {256}_10 = ? (Bonus: Could you guess this one? How?)
+
+
+2.  Conversion between various bases and 10:
+
+    Make sure that you pick a smart method to do the conversion in the following
+    exercises :)
+
+  2.0   {2312}_5 = {?}_10 ?
+
+  2.1   {10120}_3 = {?}_10 ?
+
+  2.2   {121}_10 = {?}_3 ?
+
+  2.3   {121}_10 = {?}_6 ?
+
+
+3. Conversion between two arbitrary bases.
+
+    Convert the following numbers' representations.
+
+    Hint: Given that you know how to convert to base 10 and from base 10 easily,
+    you could just use base 10 as a mediator base.
+
+    Example: {102}_3 = {?}_5 ?
+
+    Solution:
+    First we convert from base 3 to base 10. 
+    We use Direct Evaluation method:
+
+    {102}_3 = 2*3^0 + 0*3^1 + 1*3^2 = 2 + 0 + 9 = {11}_10
+
+    Now we want to convert the result from base 10 to base 5. We use the
+    Remainder Evaluation method:
+
+    11 |  1    11 % 5 = 1
+     2 |  2     2 % 5 = 2
+
+    We obtain the number {21}_5. Hence {102}_3 = {21}_5
+
+  3.0   {10010}_2 = {?}_5 ?
+
+  3.1   {102}_3 = {?}_2 ?
+
+  3.2   {11011}_2 = {?}_3 ?
+
+  3.3   {1402}_5 = {?}_2 ?
+
+
+4.  Bonus: For the mathematically inclined:
+    (I still recommend you to at least read this. You might find it
+    interesting.)
+
+  4.0 Existence of representation: 
+
+      During the presentation we implicitly assumed that there is a
+      representation in base 2 for every number that could be represented in
+      base 10. How could you prove it?
+
+      Generally, how could you prove that using base b for some b > 1, every
+      integral quantity could be represented in base b?
+
+  4.1 Uniqueness of representation:
+
+      During the representation we mentioned that every quantity could be
+      represented in a unique way in base 2. That means: Every quantity could be
+      represented as a sum of distinct powers of 2 in a unique way. prove it.
+
+      Generally: Let b be some base for b > 1. Show that for any quantity
+      q, there is only one way to represent q in base b.
+
+      Hint for b=2:
+      Assume that there are two different representations as sums of distinct
+      powers of 2 for the same quantity q. Then we get:
+      2^(a_0) + 2^(a_1) + ... = q = 2^(b_0) + 2^(b_1) + ...
+      Cancel the similar terms from the two sides of the equation, and try to
+      understand if equality is possible.
+
+
+5. Bonus: Symmetry of bases:
+
+  5.0 We now know that there is nothing special about base 10. Why then do we
+      use it as a mediator in Question 3, instead of converting directly between
+      two arbitrary bases?
+
+  5.1 Once in a lifetime experience: Try to convert the number
+      {1241}_5 into base 3 directly, without using base 10 as a mediator. (Use
+      your mind, a pen and a paper).
+
+
+6. Bonus: Divisibility by interesting numbers.
+
+  6.0 How could you check if a binary represented number is divisible by 2?
+      And By 4? Could you generalize it?
+
+  *6.1* How could you check if a binary represented number is divisible by 3?
+
+
+7. Bonus: Complexity of conversion algorithms.
+   Required: Basic understanding of what is complexity.
+
+   It might be interesting to find out how fast do the base conversion
+   algorithms work. 
+
+   For example: looking at the execution of the Direct Evaluation method while
+   converting a number n from base b to base c: The algorithm iterates over all
+   the digits of the number n, and invokes one addition operation and one
+   exponentiation for every digit. A total of about O(log{b}(n)) operations.
+
+   In this context we choose basic operations to be addition, subtraction,
+   division and exponentiation.
+
+  7.0 What is the amount of basic operations it takes to run the "Finding
+      largest power" method?
+
+  7.1 What is the amount of basic operations it takes to run the "Remainder
+      evaluation" method?
+
+
+Happy thinking :)

1_binary/2_hexadecimal/ex/ex_hexadecimal.txt

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+Base 2
+======
+
+Base 16 exercises
+------------------
+
+Hello! Some Base 16 related exercises are included here. As usual, the first
+exercises are a bit easy (And might be boring), however it is very important
+that you do those if you want to do the next ones, and especially if you want to
+understand the rest of the course. We are going to use base 16 everywhere in
+this course.
+
+To make sure that you understand things right, Use a calculator from time to
+time to check your answers. The Windows calculator (When in Programmer mode) has
+the feature of changing the base of the current number. It has the built in
+options of Hex, Dec, Oct(Octal - base 8) and Bin. Use those to check your
+answers.
+
+Although you could use a calculator or a computer to solve the simpler
+exercises, make sure that you do them with a pen and a paper.
+
+
+0.  Convert the following numbers to binary:
+
+  0.0   0xD = {?}_2
+
+  0.1   0xF = {?}_2
+
+  0.2   0x3A = {?}_2
+
+  0.3   0x4532 = {?}_2
+
+  0.4   0x9A9B = {?}_?
+
+
+1. Convert the following binary numbers to base 16:
+
+  1.0   {10}_2 = {?}_16
+
+  1.1   {110}_2 = {?}_16
+
+  1.2   {1011}_2 = {?}_16
+
+  1.3   {11110101}_2 = {?}_16
+
+  1.4   {1111100}_2 = {?}_16
+
+  1.5   {1010101111001101}_2 = {?}_16
+
+
+2. Convert the following numbers (Pay attention to the wanted result base):
+
+  2.0   {10110}_2 = {?}_8
+
+  2.1   {1100111}_2 = {?}_8
+
+  2.2   {101001}_2 = {?}_4
+
+  2.3   {3725}_8 = {?}_2
+
+  2.4   {57137}_8 = {?}_4   (Hint: Try to use base 2 as a mediator.)
+
+
+3.  Think of an easy way to convert numbers from base 16 to base 4, and vice
+    versa.
+
+
+4.  Basic divisibility rules:
+
+  4.0   How could you check if a number represented in base 16 is divisible by
+        2?
+
+  4.1   Bonus: How could you check if a number represented in base 16 is
+        divisible by 4? By 8?
+
+
+*5*.Bonus: How could you check if a number represented in base 16 is divisible
+    by 3? And by 15? Could you think of another interesting divisibility rule?
+
+
+6.  Bonus: It is known that a number x has n digits when it is represented in
+    base 4. How many digits will x have when it is represented in base 8?
+
+
+7.  Bonus: 
+    Required: Basic familiarity with the notion of complexity.
+
+    What is the computational complexity of the new hex to binary
+    conversion algorithm we learned about in this lesson? Is it faster than
+    using the Remainder Evaluation method?
+
+
+Happy thinking :)