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The efficient postcode

A postmaster in Zejtun had to invent an efficient postcode in such a manner that he
would not waste any numbers and that any number could uniquely identify one house
in his district.

His district consisted of 3 locations: Zejtun, Ramla and Bir-id-deheb, Zejtun had 200
houses, Ramla had 100 and Bir-id-deheb had another 100. Each house had 10 people
living in it.

At the post-office he had a pigeonhole shelving for 400 addresses. How could he invent
a code, such that each code would identify uniquely on residence, that from the number
the postman could directly know which location it was for.

So he gave the following addresses within each location:

Zejtun: from 000 to 199; Ramla: from 00 to 99; Bir-id-deheb: from 00 to 99;

Now he had to adapt the code so that each house in the district had a unique address, so
he adopted the following code:

Zejtun: from 0 to 199; Ramla: from 200 to 299; Bir-id-deheb: from 300 to 399

In this manner the postman knew that if the code started with 0 or 1 it was intended for
Zejtun, with 2 it was for Ramla and 3 it was for bir-id-deheb.

So he organised his pigeonhole shelving as follows:


             ADDRESS LOCATION             ADDRESS WITHIN LOCATION
             [000]                        [000]
                         Zejtun           ...
             [199]                        [199]
             [200]                        [00]
                         Ramla            ...
             [299]                        [99]
             [300]                        [00]
                      Bir-id-deheb        ...
             [399]                        [99]
However at Ramla, the houses were small and could host only 5 persons. So he
combined two houses together, with the same address, in such a way that each address
would involve a whole family of 10 persons.


             ADDRESS LOCATION             ADDRESS WITHIN LOCATION
             [000]                        [000]
                         Zejtun           ...
             [199]                        [199]
             [200]                        [00]        [00]
                         Ramla            ...         ...
             [299]                        [99]        [99]
             [300]                        [00]
                      Bir-id-deheb        ...
             [399]                        [99]



He then had the idea to organise a demonstration to the people in each village to explain
the working of the new code. So he organised a minibus service which would pick a
particular family from each location and bring them to the post office for a
demonstration. What should be the capacity of this minibus? Obviously 10 since each
household consisted of 10 persons.




                                     10-BIT
Zejtun
                             [000]
                                                                    000




                                      Zejtun sub post office


 Starting with 0 or 1
                            [199]
                                                                    199


                            [200]        Ramla
                  Starting with 2
                                                                          00




                                     Ramla sub post office

                            [299]                                         99
Starting with 3
                            [300]        Bir-id-Deheb

                                                                     00




                                     Bir-id-Deheb sub post office
                            [399]                                    99
Modern times had this postmaster program an automatic selector to sort the mail to
individual sub-post offices in each location.



              0

              1

              2

              3



                   Full Address                                       Zejtun
              A2                                                        A2

              A1                                                        A1

              A0                                                        A0

                                                                       Ramla

                                                                        A1

                                                                        A0


                                                                       Bir-id-deheb

                                                                        A1

                                                                        A0




For example an address of 223 would mean house number 23 at Ramla, an address of
166 would mean house 166 in Zejtun and an address of 325 would mean house 25 in
Bir-id-deheb.
POSTMASTER GOT TRANSFERED
The big success in the code he had invented, made him famous, and with fame came all
the envy of his colleagues which started inventing rumours which led the post master
general to transfer him to Binland, a place where people could only count up to 1. In fact
these people did all their mathematics in binary (base 2) and to avoid confusion, in
speaking, they adopted a Hexadecimal notation (base 16).

In this land there were only two places, one called ROM and the other called RAM each
had 256 locations and the Central Postal Unit could address all 512 locations in this
land. So he had to adopt the postcode system to this new land.

Not being such a good mathematician, he had to get acquainted with binary numbers.

To do this he adopted a table which gave him the number of digits required to have a
number of unique addresses:


        n = number of bits in address    2n = number of addressable locations
                     1                   2
                     2                   4
                     3                   8
                     4                   16
                     5                   32
                     6                   64
                     7                   128
                     8                   256
                     9                   512
                    10                   1024 ( called 1Kilo)
                    20                   1024x1024 (called 1Mega)
                    30                   1024x1024x1024 (called 1Giga)
                    40                   1024x1024x1024x1024 (called 1Tera)



From this table he deducted that to be able to address all the 512 locations in his land he
had to have an address which is 9 bits long and since each place had 256 locations, each
had to have an address of 8 bits within the place.

Since each location contained 8 bits as the inhabitants of Binland were called, the bus
carrying the bits from each loction had to be 8 bit wide.



Soon he realised that this was an even simpler case than he had before
ROM

                     [000000000B] = [000H]         [00000000B] = [00H]
Starts with 0




                     [011111111B] = [0FFH]        [11111111B] = [FFH]




                                                  RAM
                     [100000000B] = [100H]
                                                   [00000000B] = [00H]




Starts with 1

                     [111111111B] = [1FFH]         [11111111B] = [FFH]




     The automatic machine he programmed was even Simpler:

                        0             ROM
                                             A7     A6    A5    A4       A3   A2   A1   A0
                A8

                A7

                A6

                A5

                A4

                A3

                A2

                A1

                A0
                        1           RAM
                                             A7     A6    A5    A4      A3    A2   A1   A0
Just as he had launched the new system, the postmaster general called him to announce
that a new place called I/O had been added to his district, and that he had to cater for
their mail too.

After a few minutes of rage, the postmaster couldn’t be bothered to redo everything
from scratch. So he decided to keep the I/O on a separate map and introduced a new
signal stamp called M/IO’ which indicated if the mail was addressed to the Main portion
or the I/O section. Thus if this signal was 1 it would indicate that the address is referring
to the Main and if it was 0 the address would be referring to I/O.
  M/IO’    1                                 ROM

                        0                        A7   A6    A5    A4    A3    A2    A1   A0
               A8

               A7

               A6

               A5

               A4

               A3

               A2

               A1

               A0
                            1
       1                                         A7   A6    A5   A4    A3    A2    A1    A0

                                           RAM
       0
                                                 A8   A7   A6    A5    A4    A3    A2    A1     A0
                                           I/O




    M/IO’ A8...A0 Section        A8...A0              M/IO’ A8...A0 Section        A7...A0
    0     [000H]                 [000H]               1     [000H]                 [00H]
                                 ...                                 ROM           ...
                                                            [0FFH]                 [FFH]
                         I/O                                [100H]                 [00H]
                                                                     RAM           ...
               [1FFH]            [1FFH]                     [1FFH]                 [FFH]
This solution could accomodate an extra 512 locations in I/O. Yet being on a separate
map, locations in I/O felt a bit Isolated. Also this added extra complexity to the machine
which consequently made mail distribution slower. The postmaster also noticed that
512 addresses for I/O were superfluous and that actually there were only 256
addresses needed for this place.
                                                    A9...A0 Section           A7...A0
So one fine day he decided it would be              000H]                     [00H]
better to map everything together on a                            ROM         ...
single map. He noticed that to do this the          [0FFH]                    [FFH]
address would need another bit since now            [100H]                    [00H]
the locations had become 768 locations.                           RAM         ...
With the 10th bit in the address added, he          [1FFH]                    [FFH]
would be able to address 1024 locations             [200H]                    [00H]
                                                                   I/O        ...
which would leave 256 locations for further
                                                    [2FFH]                    [FFH]
expansion.
                                                    [300H]                    [00H]
So he redrew the map from scratch.                             Expansion
                                                    [3FFH]                    [FFH]

        CPU
                                       ROM
         A9
                                           A7   A6    A5    A4    A3   A2    A1    A0
         A8                                                                                  I/O

         A7                                                                                  A7

         A6                                                                                  A6

         A5                                                                                  A5

         A4                                                                                  A4

         A3                                                                                  A3

         A2                                                                                  A2

         A1                                                                                  A1

         A0                                                                                  A0


                                          A7    A6    A5    A4   A3    A2    A1    A0
              0
        00
                                          RAM
        01    1

        10    2

        11    3
The Sorting Machine – also called Address Decoder

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The efficient postcode

  • 1. The efficient postcode A postmaster in Zejtun had to invent an efficient postcode in such a manner that he would not waste any numbers and that any number could uniquely identify one house in his district. His district consisted of 3 locations: Zejtun, Ramla and Bir-id-deheb, Zejtun had 200 houses, Ramla had 100 and Bir-id-deheb had another 100. Each house had 10 people living in it. At the post-office he had a pigeonhole shelving for 400 addresses. How could he invent a code, such that each code would identify uniquely on residence, that from the number the postman could directly know which location it was for. So he gave the following addresses within each location: Zejtun: from 000 to 199; Ramla: from 00 to 99; Bir-id-deheb: from 00 to 99; Now he had to adapt the code so that each house in the district had a unique address, so he adopted the following code: Zejtun: from 0 to 199; Ramla: from 200 to 299; Bir-id-deheb: from 300 to 399 In this manner the postman knew that if the code started with 0 or 1 it was intended for Zejtun, with 2 it was for Ramla and 3 it was for bir-id-deheb. So he organised his pigeonhole shelving as follows: ADDRESS LOCATION ADDRESS WITHIN LOCATION [000] [000] Zejtun ... [199] [199] [200] [00] Ramla ... [299] [99] [300] [00] Bir-id-deheb ... [399] [99]
  • 2. However at Ramla, the houses were small and could host only 5 persons. So he combined two houses together, with the same address, in such a way that each address would involve a whole family of 10 persons. ADDRESS LOCATION ADDRESS WITHIN LOCATION [000] [000] Zejtun ... [199] [199] [200] [00] [00] Ramla ... ... [299] [99] [99] [300] [00] Bir-id-deheb ... [399] [99] He then had the idea to organise a demonstration to the people in each village to explain the working of the new code. So he organised a minibus service which would pick a particular family from each location and bring them to the post office for a demonstration. What should be the capacity of this minibus? Obviously 10 since each household consisted of 10 persons. 10-BIT
  • 3. Zejtun [000] 000 Zejtun sub post office Starting with 0 or 1 [199] 199 [200] Ramla Starting with 2 00 Ramla sub post office [299] 99 Starting with 3 [300] Bir-id-Deheb 00 Bir-id-Deheb sub post office [399] 99
  • 4. Modern times had this postmaster program an automatic selector to sort the mail to individual sub-post offices in each location. 0 1 2 3 Full Address Zejtun A2 A2 A1 A1 A0 A0 Ramla A1 A0 Bir-id-deheb A1 A0 For example an address of 223 would mean house number 23 at Ramla, an address of 166 would mean house 166 in Zejtun and an address of 325 would mean house 25 in Bir-id-deheb.
  • 5. POSTMASTER GOT TRANSFERED The big success in the code he had invented, made him famous, and with fame came all the envy of his colleagues which started inventing rumours which led the post master general to transfer him to Binland, a place where people could only count up to 1. In fact these people did all their mathematics in binary (base 2) and to avoid confusion, in speaking, they adopted a Hexadecimal notation (base 16). In this land there were only two places, one called ROM and the other called RAM each had 256 locations and the Central Postal Unit could address all 512 locations in this land. So he had to adopt the postcode system to this new land. Not being such a good mathematician, he had to get acquainted with binary numbers. To do this he adopted a table which gave him the number of digits required to have a number of unique addresses: n = number of bits in address 2n = number of addressable locations 1 2 2 4 3 8 4 16 5 32 6 64 7 128 8 256 9 512 10 1024 ( called 1Kilo) 20 1024x1024 (called 1Mega) 30 1024x1024x1024 (called 1Giga) 40 1024x1024x1024x1024 (called 1Tera) From this table he deducted that to be able to address all the 512 locations in his land he had to have an address which is 9 bits long and since each place had 256 locations, each had to have an address of 8 bits within the place. Since each location contained 8 bits as the inhabitants of Binland were called, the bus carrying the bits from each loction had to be 8 bit wide. Soon he realised that this was an even simpler case than he had before
  • 6. ROM [000000000B] = [000H] [00000000B] = [00H] Starts with 0 [011111111B] = [0FFH] [11111111B] = [FFH] RAM [100000000B] = [100H] [00000000B] = [00H] Starts with 1 [111111111B] = [1FFH] [11111111B] = [FFH] The automatic machine he programmed was even Simpler: 0 ROM A7 A6 A5 A4 A3 A2 A1 A0 A8 A7 A6 A5 A4 A3 A2 A1 A0 1 RAM A7 A6 A5 A4 A3 A2 A1 A0
  • 7. Just as he had launched the new system, the postmaster general called him to announce that a new place called I/O had been added to his district, and that he had to cater for their mail too. After a few minutes of rage, the postmaster couldn’t be bothered to redo everything from scratch. So he decided to keep the I/O on a separate map and introduced a new signal stamp called M/IO’ which indicated if the mail was addressed to the Main portion or the I/O section. Thus if this signal was 1 it would indicate that the address is referring to the Main and if it was 0 the address would be referring to I/O. M/IO’ 1 ROM 0 A7 A6 A5 A4 A3 A2 A1 A0 A8 A7 A6 A5 A4 A3 A2 A1 A0 1 1 A7 A6 A5 A4 A3 A2 A1 A0 RAM 0 A8 A7 A6 A5 A4 A3 A2 A1 A0 I/O M/IO’ A8...A0 Section A8...A0 M/IO’ A8...A0 Section A7...A0 0 [000H] [000H] 1 [000H] [00H] ... ROM ... [0FFH] [FFH] I/O [100H] [00H] RAM ... [1FFH] [1FFH] [1FFH] [FFH]
  • 8. This solution could accomodate an extra 512 locations in I/O. Yet being on a separate map, locations in I/O felt a bit Isolated. Also this added extra complexity to the machine which consequently made mail distribution slower. The postmaster also noticed that 512 addresses for I/O were superfluous and that actually there were only 256 addresses needed for this place. A9...A0 Section A7...A0 So one fine day he decided it would be 000H] [00H] better to map everything together on a ROM ... single map. He noticed that to do this the [0FFH] [FFH] address would need another bit since now [100H] [00H] the locations had become 768 locations. RAM ... With the 10th bit in the address added, he [1FFH] [FFH] would be able to address 1024 locations [200H] [00H] I/O ... which would leave 256 locations for further [2FFH] [FFH] expansion. [300H] [00H] So he redrew the map from scratch. Expansion [3FFH] [FFH] CPU ROM A9 A7 A6 A5 A4 A3 A2 A1 A0 A8 I/O A7 A7 A6 A6 A5 A5 A4 A4 A3 A3 A2 A2 A1 A1 A0 A0 A7 A6 A5 A4 A3 A2 A1 A0 0 00 RAM 01 1 10 2 11 3
  • 9. The Sorting Machine – also called Address Decoder