Adder

THE LOOLOOLO
ADDING MACHINE

There are 10 types of people in this world�
those who know binary, and those who don't.

A thousand years ago, the Looloolos, like many other primitive tribes, knew only two numbers: 0 for none, and 1 for all. Over the centuries, as the Looloolos grew into a civilization, they, like most civilizations, discovered other numbers between zero and all. Unlike other civilizations, however, they did not create any new digits for their new numbers. We use ten digits, 0 to 9, to notate any and all numbers. Looloolos, even today, use only 0's and 1's. Yet, they add, subtract, multiply, and divide numbers just as big, or small, as our numbers. How do they do it?

In the beginning, addition was not much of a challenge for Looloolos. Adding nothing to nothing always results in nothing: 0+0=0. Adding nothing to one is still one: 1+0=1. Adding one to nothing is also one : 0+1=1. But the Looloolos ran into trouble when they wanted to add 1 plus 1. We know that 1+1=2, but what is simple for us was a challenge for them. The Looloolos have no digit for the number two. But they discovered a neat trick. They learned to carry. We know what carry means. When we add 9 plus 1, we carry a 1, and the sum is 10.

When Looloolos add 1 plus 1, they do not write the sum as 2 because they have no digit for the number two. For Looloolos, the sum of 1 plus 1 looks like 10; but, for them, 10 is the number two. They write the sum like this:

Just as we carry a 1 when adding 9+1, so Looloolos carry a 1 when adding 1+1.

Here is how Looloolos would add 3+5=8:

Here is what 3+5=8 looks like showing the carries:

1110 <� carries 
 011
 101
����
1000

Notice the similarity to how we add 91+909=1000:

 110 <� carries  
  91
 909
����
1000

ASSIGNMENT
Not too many centuries past, after the discovery of Looloolo addition, before a young Looloolo couple discovered how to use the switches in Looloolo water machines to create the first Looloolo adding machine. Here is how Loo and LooLoo did it. Loo and LooLoo noticed a pattern in the way older Looloolos added:

0 plus 0 output a 0
0 plus 1 output a 1
1 plus 0 output a 1
1 plus 1 output a 0 with a carry.

So Loo, or LooLoo, made a table of these 4 possibilities that looked like this:

 
A       0   0   1   1
B       0   1   0   1
����������������������
SUM     0   1   1   0

Turn this table sideways and what do you see?


A B | SUM
����������
0 0 |  0
0 1 |  1
1 0 |  1
1 1 |  0
����������

LLLLLLLLLL! It's the Truth Table for XOR, the eXclusive OR! Notice the table ignores carrying. Of course, all young Looloolos are taught in elementary school how to implement truth tables with AND OR NOT blocks. So this is easy for them. Is it easy for you too? Four points if you can design a system of blocks and pipes that implement the XOR truth table above. Three extra points if you can do it using only 4 blocks. Hint: Look over others' pipe diagrams, set matchings, and truth tables for clues. Review set properties and their equalities. If you can reduce a logical expression to ANDs ORs and NOTs ( & + ~), then you can pipe it. Remember:

when the A and B sources are both turned off, there is no flow out the output pipe.
when A is off and B is on, there is a flow out the output pipe.
when A is on and B is off, there is also a flow out the output pipe.
when the A and B sources are both turned on, there is no flow out the output pipe.

Design your system using the pipes applet. Save your pipes image as a gif file. Then send it to me privately as an attachment to a message. Warning! Once again do not confuse the + that means OR with the + that means SUM.

Click here for a Adder Demo.

Comments:

From Bravo - 12/5/06 7:01 PM

(A+B)~A&B=output

A&B|(A~B)|(B~A)|(A+B)|output

OFF OFF OFF OFF OFF

OFF ON OFF ON ON

OFF OFF ON ON ON

ON OFF OFF OFF OFF

A X B IS TRUE IF EITHER A OR B, BUT NOT BOTH.

a |b| a x b The X represents or and T for true and F false

_________________

T T F

T F T

F T T

From wHolt - 10/20/06 1:17 PM

Use the examples on the Water Logic page to design your adder.
Your adder must implement the truth table for the XOR table.
It's a puzzle. You can do it.

From GolfGirl - 10/19/06 2:18 PM

I agree.. I really need an example to follow.. I am confused!

From 7Iron - 10/19/06 10:09 AM

Are there any examples to follow? Im not sure I understand how to show the 0's and 1's in the Pipe diagram.

Last Modified 10/5/06 10:48 PM

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