0 1 2 3 4 5 6 7 8 9 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001
Let ABCD represent a 4-bit binary-coded decimal number as described above. For example, the decimal digit 7 is represented in binary-coded decimal as 0111, so in this case, A = 0, B = 1, C = 1, and D = 1.
A seven-degment display can be used to display each of the ten decimal digits as shown below.
We can define a circuit abstractly below that requires four Boolean inputs A, B, C, and D represent a decimal digit encoded in binary-coded decimal format, and seven Boolean outputs s1, s2, ..., s7 to control each segment of the display:
A segment si is lit if si is 1 (true) and not lit if si is 0 (false).
-
Derive a Boolean formula for s2 that is
true (1) if and only if segment s2 is lit.
Your answer should be of the following form:
s2 = (________) ∨ (________) ∨ (________) ∨ ...
where each missing section is a boolean expression with all 4 input variables that is true when ABCD represents a decimal digit that results in the segment s2 of the display being lit. To help you get started, the digit 0 (binary 0000) will light segment s2, so the first expression in the formula above would be
(¬A ∧ ¬B ∧ ¬C ∧ ¬D)
since s2 would be true (i.e. 1) if A = 0, B = 0, C = 0, and D = 0. You will need to find all of the other missing expressions for s2.
- Derive a simple Boolean expression for s6 that is true (i.e. 1) if and only if the segment s6 is lit. (HINT: Segment s6 is lit for all digits except when the digit is the number 2.)
X Y | C S --------------- 0 0 | 0 0 0 1 | 0 1 1 0 | 0 1 1 1 | 1 0
This adder can be represented abstractly using the following diagram:
-
Give Boolean formulas for C and S as functions of X and Y.
- Using two half-adders represented abstractly as boxes and one OR gate, show how to build a full-adder as defined in class. Your picture should have three inputs, X, Y, and Carry_In and two outputs, S and Carry_Out.
0000: DAT #0 #6 0001: DAT #0 #1 0002: ADD -1 -1 0003: SUB #1 -3 0004: JMN -2 -4 0005: DAT #0 #0
-
Explain clearly what the instructions at address 2 is computing.
-
Explain clearly what the instructions at address 3 is computing.
-
Explain clearly what the instructions at address 4 is computing.
- What is this program computing overall?
-
A linear congruential random number generator is created
with a = 5, c = 9, m = 16. What is the period of this
random number generator if it starts with a seed of 0?
What is the sequence of values
generated by this random number generator in one period?
Show your work for full credit.
- Consider a pseudo random number generator based on the linear congruential method with a = 81, c = 337 and m = 1000. Using irb and RandomLab, generate a set of 20 random numbers from this generator. Try this with several different seeds. Look at each sequence of numbers you get. What is the pattern in each of the sequences that would make you question whether you want to use this generator? (HINT: For each sequence, look at the last digit of each number in the sequence.)
- Write a Ruby function spin1() that returns an
integer and simulates
the wheel spinner from the Milton Bradley game of Life. The wheel
contains the numbers from 1 to 10.
- Write a Ruby function spin2() that returns an
integer and simulates
the Showcase Showdown wheel from the Price Is Right game show.
The wheel contains the multiples of 5 from 5 to 100.
- Write a Ruby function roll1() that returns an integer
and simulates
a backgammon doubling cube. The 6-sided cube contains the values 2, 4,
8, 16, 32, and 64.
- Write a Ruby function roll2() that returns a string
and simulates
a poker die. The 6-sided cube contains the values "A", "K",
"Q", "J", "10", and "9". (You should simulate just one die.)
def roll1()
return rand(6) + 1 # simulate a roll of a six-sided die
end
def roll2()
return rand(6) + 1 # simulate a roll of a six-sided die
end
def simple_game()
total = 0
for i in 1..10 do
if roll1() != roll2() then
total = total + roll1() + roll2()
else
total = 0
end
end
return total
end