Instructions:
You are encouraged to work on the problem set in groups and turn in one problem set for the entire group. Remember to put all your names on the solution sheet. Also, remember to put the name of the TA and the time for the discussion section you would like the problem set turned back to you. Show your work.

Please remember to turn in the Student Information Sheet with a recognizable photograph on the same day you turn in this problem set.

1. (Adapted from problem 1.5 in the textbook)
   Say we had a "black box," which takes two numbers as input and outputs their sum. See Figure 1.7a in the Textbook. Say we had another box capable of multiplying two numbers together. See figure 1.7b. We can connect these boxes together to calculate p * (m + n). See Figure 1.7c. Assume we have an unlimited number of these boxes. Show how to connect them together to calculate:
   1. ax+b
   2. The average of the four input numbers w, x, y, and z
   3. a² + 2ab + b² (can you do it with one add box and one multiply box?)
   4. a⁶ (can you do it using only 3 multiply boxes?)


2. (1.14)
   Suppose we wish to put a set of names in alphabetical order. We call the act of doing so sorting. One algorithm that can accomplish that is called the bubble sort. We could then program our bubble sort algorithm in C, and compile the C program to execute on an x86 ISA. The x86 ISA can be implemented with an Intel Core microarchitecture. Let us call the sequence "Bubble Sort, C program, x86 ISA, Core microarchitecture" one transformation process.
   Assume we have available four sorting algorithms and can program in C, C++, Pascal, Fortran, and COBOL. We have available compilers that can translate from each of theses to either x86 or SPARC, and we have available three different microarchitectures for x86 and three different microarchitectures for SPARC.
   1. How many transformation processes are possible?
   2. Write three examples of transformation processes.
   3. How many transformation processes are possible if instead of three different microarchitectures for x86 and three different microarchitectures for SPARC, there were two for x86 and four for SPARC.


3. (2.3)
   1. Assume that there are about 400 students in your class. If every student is to be assigned a unique bit pattern, what is the minimum number of bits required to do this?
   2. How many more students can be admitted to the class without requiring additional bits for each student's unique bit pattern?
   
   
4. (Adapted from 2.13)
   Without changing their values, convert the following 2's complement binary numbers into 8-bit 2's complement numbers.
   1. 010110
   2. 1101
   3. 1111111000
   4. 01


5. (2.22)
   Create two 16-bit 2's complement integers such that their sum causes an overflow.
   


6. (Adapted from 2.37) (updated on 09/08/08)
   If n and m are both 4-bit 2's complement numbers, and s is the 4-bit result of adding them together, write a logical equation, using only the logical operations described in Section 2.6, that assigns the value 1 to the logical variable V if the addition causes an overflow, and the value 0 to the logical variable V if the addition does not cause an overflow.
   


7. (2.42)
   A computer programmer wrote a program that adds two numbers. The programmer ran the program and observed that when 5 is added to 8, the result is the character m. Explain why this program is behaving erroneously.
   


8. (Adapted from 2.50)
   Perform the following logical operations. Express your answers in hexadecimal notation.
   • xABCD OR x9876
   • x1234 XOR x1234
   • xFEED AND (NOT(xBEEF))


9. (2.54)
   Fill in the truth table for the equations given. The first line is done as an example.
   Q1 = NOT (NOT(X) OR (X AND Y AND Z))
   Q2 = NOT ((Y OR Z) AND (X AND Y AND Z))



X Y Z	Q1 Q2
0 0 0	0 1



10. (2.55 a-d)
    We have represented numbers in base-2 (binary) and in base-16 (hex). We are now ready for unsigned base-4, which we will call quad numbers. A quad digit can be 0, 1, 2, or 3.
    
    1. What is the maximum unsigned decimal value that one can represent with 3 quad digits?
    2. What is the maximum unsigned decimal value that one can represent with n quad digits (Hint: your answer should be a function of n)?
    3. Add the two unsigned quad numbers: 123 and 121.
    4. What is the quad representation of the decimal number 42?


11. (2.40) (added on 09/05/08)
    Write the decimal equivalents for these IEEE foating point numbers.
    
    1. 0 10000000 00000000000000000000000
    2. 1 10000011 00010000000000000000000
    3. 0 11111111 00000000000000000000000
    4. 1 10000000 10010000000000000000000


12. (2.56) (added on 09/05/08)
    Define a new 8-bit floating point format with 1 sign bit, 4 bits of exponent, using an excess-7 code (that is, the bias is 7), and 3 bits of fraction. If xE5 is the bit pattern for a number in this 8-bit floating point format, what value does it have? (Express as a decimal number)