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Pipe Game

Pipe Game

THE GAME OF PIPES



In the exercise today, you will become familiar with a new set of symbols for AND, OR, NOT.

The SQUARE will represent an AND:   Square

The CIRCLE will represent an OR:        Circle

The TRIANGLE will represent a NOT:   Triangle


Imagine water, or one of your favorite beverages, flowing through pipes,
and obeying the following rules:

Rules for The Game of PIPES

 Equivalence

  1. Pipes on the far left are input source pipes, and are tagged A, B, C, etc.
  2. The one pipe on the far right is the output pipe, and shows the results of all the plumbing to its left.
  3. Liquid flowing from the input pipes is controlled by the AND, OR, and NOT blocks. These blocks are switches that control the flow of water through the pipes. They function the same as the & + – operators you have previously mastered:
    • Liquid flows out of an AND block only if both its input pipes are turned ON;
      else the output from an AND is turned OFF.
    • Liquid flows out of an OR block if one or both input pipes are turned ON;
      else the output from an OR is turned OFF.
    • Liquid flows out of a NOT block if its input pipe is turned OFF; else,
      the opposite, if its input pipe is turned ON, its output is turned OFF.
  4. Many pipes may output from one pipe, but no pipe may input into another pipe.
    All pipes must flow into a block, except the final output pipe.
  5. All blocks require one and only one output pipe.
  6. The AND block requires exactly two input pipes.
  7. The OR block requires exactly two input pipes.
  8. The NOT block requires exactly one input pipe.
  9. All blocks must be connected by pipes.

ASSIGNMENT

 Within the Pipes applet, create a system of pipes that obey the game rules. Start with two input pipes A and B. Use each of the 3 blocks at least once. Be creative as you were with the CSG doodle. The 3 blocks in the Pipe Game, OR AND NOT, correspond to Union, Intersection, and Negation in your CSG doodle. So that we can follow the flow of water through your diagram, do what a good plumber does, label your input and output pipes with a tag. When you are satisfied with your diagram, copy it to a wiki comment using the HoverSnap utility.

Next make a truth table that lists all the possible ON/OFF positions for pipes A and B. Let 1=ON=water flows. Let 0=OFF=no water flows. In columns A and B, list all possible inputs to pipes A and B. Copy the blank table below to your comment. Add or delete columns as needed. This should speed up your table making. Trace the flow of water thru your pipes for each pair of inputs A and B. The OUT column is the column that corresponds to the final block in your diagram. At the bottom of the output column, label the column OUT . Use Brian's Truth Table Constructor to make this assignment, and future assignments, easier! 
   A B |
   —————————
   0 0 |  ?
   0 1 |  ?
   1 0 |  ?
   1 1 |  ?
   —————————
         OUT
Summary of Output Rules:
 Output from an AND block is 1, only if both inputs are 1; else it is 0.
Output from an OR block is 1, if one or both inputs is 1; else it is 0.
Output from a NOT block is 1, if input is 0; else 0, if input is 1.

Write the ouput in symbolic notation as you have done before with your other truth tables. Post your pipes diagram and table in a wiki comment. Be sure no one else has already done yours, or you will need to do it over again. Fit your pipes diagram within the comment window, or I will delete it; and you will receive no points. You earn 7 points for this assignment, if you:
  1. Use AND block at least once
  2. Use OR block at least once
  3. Use NOT block at least once
  4. Connect all blocks by pipes correctly
  5. Tag Input and Output Pipes correctly
  6. Make Corresponding Truth Table for Input and Output Pipes
  7. Label Table with Equivalent Table from 16 Table assignment
Also, you lose 1 point for each rule that you break—until you correct it.

EXAMPLE PIPE GAMES:
Imagine turning on and off your A and B tap water. How would the water flow thru this system? Regardless of the inputs A and B, the truth table for this system always outputs 0. Therefore, it is equivalent to the Contradiction table in the 16 Tables assignment.
? 

   A B | ((A&B) + A) & –A
   ————————————————————————————
   0 0 |    0   0    0 1
   0 1 |    0   0    0 1
   1 0 |    0   1    0 0
   1 1 |    1   1    0 0
   ————————————————————————————
  steps     1   2    4 3
                    OUT = CONTRADICTION TABLE
Which of the 16 truth tables is this one equal to?



Here is another example:
? 

   A B | OUT = –(A+B)+(A&B)
   —————————————————————————
   0 0 |  1
   0 1 |  0
   1 0 |  0
   1 1 |  1
   —————————————————————————
Which of the 16 truth tables is this one equal to?

Notice that –(A+B)+(A&B) is also equal to (A+B)–>(A&B)
because they have the same truth table.
You can verify this with a truth table or the set matcher.


HOW TO USE THE PIPES APPLET

 The Java applet for the Pipes Game contains a text box for writing pipe tags, and 4 buttons labelled AND, OR, NOT, and TAG. To place a block or tag , first click one of the buttons AND, OR, NOT, or TAG. Then click on the board where you want to place the upper left corner of the block or tag. If you don't like where you put it, click inside the block or tag, and it will vanish. If you select the TAG button, and then click on the board, you will copy the tag in the text box to the board. To draw a pipe, press the mouse button where you want the pipe to go, and hold it down while you drag the mouse to where you want the pipe to end. Be sure to give the pipe some diameter, or you will not see it. If it is not to your liking, click inside it to remove it. Of course, if you really foul things up, you can always hit the refresh button at the top of this browser. As usual, go forth and play. Have fun!

Click here for a Pipe Game Demo.

Comments:

From wHolt - 11/23/06 10:15 AM

BassLady - which table did you diagram?
Label your table.

From BassLady - 11/22/06 10:07 PM

pipes

 AB - (A & B) + (A + B) 
 00  10  10 
 0 1 1 0 1 1
 10  1 0 1 1
 11 0  11  1

Try # 87

From DirtyBird - 10/25/06 3:20 PM

Oh, contradiction. I see what you are saying now.

From wHolt - 10/24/06 11:05 AM

Which of the 16 tables always outputs 0's ?

From DirtyBird - 10/23/06 7:53 PM

What about a converse negation?

 

From wHolt - 10/23/06 1:15 PM

DirtyBird - not a NAND either. Look on 16 Tables page for answer.

From DirtyBird - 10/22/06 7:56 PM

NAND Table

From wHolt - 10/22/06 12:53 PM

Tiger - which Table is the table for your diagram. I see two tables.

DirtyBird - Your diagram label does not match your table - which is not a Converse table.

From Tiger - 10/21/06 8:42 PM

A B  (A&~B)+A 
 0 0    0  1   0
 0 1    0   0   0
 1 0    0   1   1
 1 1    1   0   1

 

A B  Out=(A&~B)+A 
 0 0  0
 0 1  0
 1 0  1
 1 1  1

This is truth table A.

From wHolt - 10/19/06 1:05 PM

THE FOLLOWING FUTURE PLUMBERS OF AMERICA EARNED 7 POINTS:

7Iron
BassLady
Boki
Capricorn
CatsEyes
Centerfield
Cheana
Harkar
Houdini
JooJoo
Lexiowen
Melewen
Pac
Phoenix
Poovey
Pringle
Soller
Sunshine
SuperDuke
Taurus
Zonino

Those who did not make the list yet, perhaps forgot to label their tables...

From wHolt - 10/19/06 1:04 PM

Sunshine finished!

DirtyBird-
your table may be a converse table,
but it does not match your diagram.
do a table for your diagram.
~(A+B)&A

BassLady-
your diagram is good now, 
but the output pipe is labeled wrong.
it should match your table.
way to go on negating those ~()

From BassLady - 10/18/06 9:17 PM

New Pipes

 AB - (A & B) & - (A+B) 
 0 0 1 0 11 0 
 0 1 1 0 0 0 1
 1 0 1 0 0 0 1
 11  0100  1

From Sunshine - 10/18/06 2:25 PM

NAND table

From wHolt - 10/18/06 11:53 AM

THE FOLLOWING FUTURE PLUMBERS OF AMERICA EARNED 7 POINTS:

7Iron
Boki
Capricorn
CatsEyes
Centerfield
Cheana
Harkar
Houdini
JooJoo
Lexiowen
Melewen
Pac
Phoenix
Poovey
Pringle
Soller
SuperDuke
Taurus
Zonino

Those who did not make the list yet, perhaps forgot to label their tables...

From wHolt - 10/18/06 11:51 AM

Cheana, Poovey, Soller- Good ones!

Dirty Bird-
your table does not match you diagram.
you diagram ANDs ~(A+B) with A
Fix.

BassLady - you have 2 outputs from your AND. Fix

From Soller - 10/18/06 1:30 AM

This is a Tau table.

From BassLady - 10/17/06 11:56 PM

New Pipes

 AB ((A + B & - A)  + B  
 00  0 1 0 
 01  0 1 1 
 10  0 0 0 
 11  1 0 1 
      

This is an "OR" table

From Poovey - 10/17/06 7:21 PM

From Cheana - 10/17/06 4:23 PM

This is equi. to a TAUTOLOGY (TAU)

From wHolt - 10/17/06 2:45 PM

Sunshine - yours is not a negation table. Look again.

Kathi - your Ps and Qs are not relevant.

DirtyBird - you notted (A+B), not (A&B)
fix table.

Soller - you have 3 inputs into your AND block. Fix.

Harkar - yours is a goody!

From Draco - 10/17/06 2:42 PM

A B | (A+B) & -A
   —————————
   0 0 |  0
   0 1 |  1
   1 0 |  0
   1 1 |  0
   —————————
         OUT=converse negation

From Harkar - 10/16/06 10:26 PM

pipegame3

pipegame4

This is a Converse table.

From Soller - 10/16/06 9:48 PM

From Kathi - 10/16/06 5:49 PM

 

 

  P

Q

 | 

 P--> Q

 0

 0

 |

 1 

 0

 1

 |

 1

 1

 0

 |

 1

 1

 1

 |

 1

 P->Q = ~(A+B)+~(A&B)

A 

B

|

 ~(A + B) + ~(A & B)

 0

0 

 

 1     0     1  1    0

 0

 1

 

 0     1     1  1   0

 1

 0

 

 0     1     1  1   0

 1

 1

 

 0     1     0  0   1

NAND table

 

 

From Sunshine - 10/16/06 2:42 PM


This is a negation table

From wHolt - 10/16/06 1:56 PM

THIS PAGE IS CLOSING DOWN TO NEWCOMERS

PIPES THAT HAVE LEAKS, CONTINUE FIXING

From wHolt - 10/16/06 1:53 PM

DirtyBird -
you diagrammed ~(A+B)&A
fix table.

From wHolt - 10/16/06 1:52 PM

THE FOLLOWING FUTURE PLUMBERS OF AMERICA EARNED 7 POINTS:

7Iron
Boki
Capricorn
CatsEyes
Centerfield
Harkar
Houdini
JooJoo
Lexiowen
Melewen
Pac
Phoenix
Pringle
SuperDuke
Taurus
Zonino

Those who did not make the list yet, perhaps forgot to label their tables...

From Taurus - 10/16/06 1:02 PM

 A B (-A +B)  & -B
 0 0 1 1 0 1 1
 0 1 1 1 1 0 0
 1 0 0 0 0 0 1
 1 1 0 1 1 0 0

This is equivalent to a NOR table.

From Sunshine - 10/16/06 10:33 AM

From TBird - 10/16/06 8:41 AM

A B      (A & B)   +  (-A)
0 0         0      0   1
0 1         0      0   1
1 0         0      0   0
1 1         1      1   0

From CatsEyes - 10/16/06 8:41 AM

This is a IMPLICATION table

From Capricorn - 10/16/06 12:04 AM

 A B (-A +(-B )) &A 
 0 0 1 1 1 0 0
 0 1 1 1 0 0 0
 1 0 0 1 1 1 1
 1 1 0 0 0 0 1

This is a NEGATION table

From Pringle - 10/15/06 11:02 PM

 (-A + -B) & -A

 A B | -A
   —————————
   0 0 |  1
   0 1 |  1
   1 0 |  0
   1 1 |  0
   —————————

 this is the same as the negation of A table.

From Lexiowen - 10/15/06 10:35 PM

pipegame

pipetable

This is a converse negation table

From Zonino - 10/15/06 10:05 PM

This is the AND table.

From Harkar - 10/15/06 8:16 PM

Reconfigured Problem

Pipegame

pipegame chart

This is a Converse table.

From Bubba - 10/15/06 5:39 PM

A  B (-B & (A + B)) 
 0 0 1        0    0
 1 0 1         1    0
 0 1 0          0    1
 1 1 0          1    1

From CenterField - 10/15/06 4:18 PM

Pipe Game

From wHolt - 10/15/06 1:29 PM

Poovey - diagram OK, but it does not fit with table.

Soller - you have 2 pipes running into 1 pipe. Fix.

Kathi - your table is not the NOR. fix it.

GolfGirl - give us a table.

Tiger - you have 3 pipes going into your AND. Fix.

Draco - name your table (one of the 16)

From wHolt - 10/15/06 1:20 PM

THE FOLLOWING FUTURE PLUMBERS OF AMERICA EARNED 7 POINTS:

7Iron
Boki
Houdini
JooJoo
Melewen
Pac
Phoenix
SuperDuke

From Draco - 10/15/06 1:20 PM

A B | (A+B) & -A
   —————————
   0 0 |  0
   0 1 |  1
   1 0 |  0
   1 1 |  0
   —————————
         OUT=converse negation

From JooJoo - 10/15/06 1:09 PM

(A +B) & (-A)

 Here is the truth table that is equal to my pipe game: (A +B) & (-A)

 

My pipe game (A +B) & (-A) is equal to a converse negation truth table (NCV)

Converse negation to equal to the negation of converse and (A +B) & (-A)

Therefore my pipe game (A +B) & (-A)= -(B ->A)

From Tiger - 10/15/06 1:09 PM

   A B |  (A& -B) + A
   —————————
   0 0 |     0  1   0
   0 1 |     0  0   0
   1 0 |     0  1   1
   1 1 |     1  0   1
   —————————
A B   OUT= (A  & -B) + A
 0 0  0
 0 1  0
 1 0  1
 1 1  1

This is truth table A.


        

From GolfGirl - 10/15/06 1:07 PM

This is a nor table

From Kathi - 10/15/06 5:10 AM

  P

Q
 | 
 P--> Q
 0
 0
 |
 1 
 0
 1
 |
 1
 1
 0
 |
 1
 1
 1
 |
 1
 P->Q = ~(A+B)+~(A&B)

A 

B
|
 ~(A + B) + ~(A & B)
 0
0 
 
 1     0     1  1    0
 0
 1
 
 0     1     1  1   0
 1
 0
 
 0     1     1  1   0
 1
 1
 
 0     1     0  0   1

NOR table


From Phoenix - 10/15/06 4:24 AM

pipes

pipetable

This is an OR table

From Soller - 10/15/06 12:16 AM

 

From Pac - 10/14/06 6:47 PM

Output = (A&(A+B))&(-(A+B)) 

This is a CON (contradiction table) - always false regardless of input

ABOutput
000
010
100
110

From Poovey - 10/14/06 4:30 PM

From wHolt - 10/14/06 2:55 PM

BassLady ~A&B comes out of the AND
also your OR has only one input. Fix.

Sunshine - your truth table does not match your diagram.
Put the () where your diagram says to. Redo table.

From Sunshine - 10/14/06 12:09 PM

From BassLady - 10/13/06 10:23 PM

Pipes

A B -A &  (A + B) & (A & B)
 00  1 00  00 
 0 1 1 00  01 
 10  0 0 0 01 
 1 1 00  1 01 

Contradiction Table

From wHolt - 10/13/06 2:19 PM

Fro - your table is not an AND table. Check the 16 table page.

Dirty Bird - you have 4 pipes attached to your OR
and only 2 pipes to your AND.
Also your notation is not that of a NAND.

From Fro - 10/12/06 8:13 PM

From the "AND" table

From wHolt - 10/12/06 6:20 PM

THE FOLLOWING FUTURE PLUMBERS OF AMERICA EARNED 7 POINTS:

7Iron
Boki
Houdini
Melewen
SuperDuke

From wHolt - 10/12/06 6:15 PM

DONT FORGET TO TELL US WHICH OF THE 16 TABLES YOU DIAGRAMMED!

Poovey - there is only one input in that left AND. Fix.

Fro - which table?

Harkar - what happened to the A&B?

Kathy - which table?

From Boki - 10/12/06 9:24 AM

P -> Q = (~A & ~B) -> ~(A v B)

It is equal to the tautology table.


 

A

B
TAU
0
0
1  
0
1  
1  
1  
0
1  
1  
1  
1 

 

 

 

 

 

 

 

 

From Poovey - 10/11/06 9:53 PM


From Fro - 10/11/06 9:12 PM

From Harkar - 10/11/06 7:51 PM

pipegame3

pipegame4

This is a negation of B table.

From Kathi - 10/11/06 12:02 PM

 

 

  P

Q

 | 

 P--> Q

 0

 0

 |

 1 

 0

 1

 |

 1

 1

 0

 |

 1

 1

 1

 |

 1

 P->Q = ~(A+B)+~(A&B)

A 

B

|

 ~(A + B) + ~(A & B)

 0

0 

 

 1     0     1  1    0

 0

 1

 

 0     1     1  1   0

 1

 0

 

 0     1     1  1   0

 1

 1

 

 0     1     0  0   1

 

 

From wHolt - 10/11/06 11:46 AM

REMEMBER TO TELL US WHICH TABLE YOUR PIPES ARE EQUAL TO !!!

Boki Slick DirtyBird - which table on the 16 Table page is equal to your table?

DirtyBird - Your OR has 3 inputs. No-no

From Slick - 10/10/06 8:16 PM

 

 

Pipes 

 

 A B |  A+(A&-B)
   —————————
   0 0 |  0
   0 1 |  0
   1 0 |  1
   1 1 |  1
   —————————
        
This is equal to an "or" table.

From Boki - 10/10/06 11:30 AM

 

 

P -> Q = (~A & ~B) -> ~(A v B)

 

It is equal to the tautology table.

 

A

B
TAU
0
0
1  
0
1  
1  
1  
0
1  
1  
1  
1  

 

 

From wHolt - 10/9/06 1:57 PM

Here are our apprentice plumbers thus far:
Everyone else, continue piping!

Houdini
Melewen
SuperDuke

From wHolt - 10/9/06 1:55 PM

Kathi - you left off the pipe from A to the first OR.
It is also confusing what comes out of that top NOT.
Can you fix it so that one pipe does not go under another?

From Kathi - 10/9/06 8:45 AM

 **Redone 

 

 P
Q
 |  P--> Q
 0 0 | 1 
 0 1 | 1
 1 0 | 1
 1 1 | 1

 P->Q = ~(A+B)+~(A&B)

A B
|
 ~(A + B) + ~(A & B)
 00   1     0     1  1    0
 0 1  0     1     1  1   0
 1 0  0     1     1  1   0
 1 1  0     1     0  0   1

 

From Melewen - 10/8/06 10:24 PM

This is an "or" table 

From SuperDuke - 10/8/06 7:06 PM

pipes

   A B |(A&~B)+(B&~A)
   —————————---
   0 0 |  0    1    0    1 
   0 1 |  0    0    1    1
   1 0 |  1    1    0    0
   1 1 |  1    0    1    0
   —————————---

   A B |OUTPUT
   —————————
   0 0 |  0
   0 1 |  1
   1 0 |  1
   1 1 |  0
   —————————

This is an "exclusive or" table

From wHolt - 10/8/06 12:33 PM

Slick - your AND block breaks a rule. Which one?

From Slick - 10/7/06 9:22 AM

 

 Pipes

 

Table

   A B | Out= (A&B)+-(A+B)
   —————————
   0 0 |  1
   0 1 |  0
   1 0 |  0
   1 1 |  1
   —————————

From wHolt - 10/6/06 1:07 PM

Good one from Boki
Notice that Boki negated ~A&~B before sending it into the OR block.
~(~A&~B) = A+B. Why? Test it in the set matcher applet.

From Boki - 10/5/06 1:12 PM

P -> Q = (~A & ~B) -> ~(A v B)

 

From wHolt - 10/5/06 12:40 AM

DON'T FORGET TO NEGATE P 

Kathi - if P=(A + B) and Q= ~(A & B)
then P->Q = ~(A+B)+~(A&B)

Boki - negation of ~A&~B = ~(~A&~B) = A+B

ALSO - you only need to post one truth table for this assignment.

From Kathi - 10/4/06 5:30 PM

 

 

 

A B
|
 A + B + ~(A & B)
 00       0    1  1    0
 0 1      1    1  1   0
 1 0      1    1  1   0
 1 1      1    1  0   1

 

From wHolt - 10/4/06 11:29 AM

Nothing says you cannot move to the next one before you finish the previous one.

You are all encouraged to move onto the new problems before you finish the old ones.

From SuperDuke - 10/4/06 1:40 AM

Houdini should go into a math field, dude rules ;)

I am very impressed but I am still hung up on the others so I can't jump on this fun one here :(

From wHolt - 10/3/06 11:19 AM

Good example from Houdini

From Houdini - 10/2/06 2:03 PM

AB◊OUT=-(A+B)+-(A&B)
00◊  1
01◊  1
10◊  1
11◊  0

This pipegame is the same as NAND (not and):

AB♥NAND
00♥1
01♥1
10♥1
11♥0

 

From wHolt - 10/1/06 12:31 AM

THE GAME OF PIPES

Today begins a new module
where you will put your new learned logical skills to the test.
Can you apply your logic skills to build a machine that "thinks"?
Or at least thinks it thinks? Probably not.
But if you have been paying attention to your logic exercises,
and everyone else's,
you will, after a little trial and error, be able to design an adding machine,
one that lies at the "heart" of every computer's "brain".

Start by playing around with the pipe applet. See how it works.
Then meet the challenges in the next four assignments.
Can you use what you have learned this far?
Can you find in past problems what you need for future problems?
If so, you can build your own computer with nothing but logic.
Go plumb!


Last Modified 9/30/06 8:03 PM

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