Ex. 5: Mr. C, of the Eccentrics’ Chess Club, has a system of play that rigidly pre-
Scribes, for every possible position, both for White and slack (except for
Those positions in which the player is already mated) what is the player’s best
Next move. The theory thus defines a transformation from position to posi-
Tion. On being assured that the transformation was a closed one, and that C
Always plays by this system, Mr. D. at once offered to play C for a large
Stake. Was D wise?
C:
↓
A transformation may have an infinite number of discrete
Operands; such would be the transformation
1 2 3 4 …
4 5 6 7 …
Where the dots simply mean that the list goes on similarly without
End. Infinite sets can lead to difficulties, but in this book we shall
Consider only the simple and clear. Whether such a transformation
Is closed or not is determined by whether one cannot, or can
(respectively) find some particular, namable, transform that does
Not occur among the operands. In the example given above, each
Particular transform, 142857 for instance, will obviously be found
Among the operands. So that particular infinite transformation is
Closed.
Ex. 1: In A the operands are the even numbers from 2 onwards, and the trans-
Forms are their squares:
2 4 6 …A: ↓ 4 16 36 …
Is A closed?
Ex. 2: In transformation B the operands are all the positive integers 1, 2, 3, …and
each one’s transform is its right-hand digit, so that, for instance, 127 → 7,
and 6493 → 3. Is B closed?
↓
Notation. Many transformations become inconveniently
Lengthy if written out in extenso. Already, in S.2/3, we have been
Forced to use dots ... to represent operands that were not given
Individually. For merely practical reasons we shall have to
Develop a more compact method for writing down our transforma-
Tions though it is to be understood that, whatever abbreviation is
Used, the transformation is basically specified as in S.2/3. Several
Abbreviations will now be described. It is to be understood that
They are a mere shorthand, and that they imply nothing more than
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Has already been stated explicitly in the last few sections.
Often the specification of a transformation is made simple by
Some simple relation that links all the operands to their respective
Transforms. Thus the transformation of Ex. 2/4/1 can be replaced
By the single line
Operand → operand plus three.
The whole transformation can thus be specified by the general
Rule, written more compactly,
Op. → Op. + 3,
Together with a statement that the operands are the numbers 1, 2 3
And 4. And commonly the representation can be made even
Briefer, the two letters being reduced to one:
n → n + 3 (n = 1, 2, 3, 4)
The word “operand” above, or the letter n (which means exactly
The same thing), may seem somewhat ambiguous. If we are think-
Ing of how, say, 2 is transformed, then “n” means the number 2
And nothing else, and the expression tells us that it will change to
The same expression, however, can also be used with n not
Given any particular value. It then represents the whole transfor-
Mation. It will be found that this ambiguity leads to no confusion
In practice, for the context will always indicate which meaning is
Intended.
Ex. 1: Condense into one line the transformation
1 2 3A: ↓ 11 12 13
Ex. 2: Condense similarly the transformations:
1 → 71 → 1
2 → 14 b:2 → 4c:a:
3 → 213 → 9
{
{
{
{
{
1 → 1
2 → 1/2
3 → 1/3
1 → 1
2 → 2
3 → 3
D:
{
1 → 10
2 → 9
3 → 8
E:
1 → 1
2 → 1
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3 → 1
F:
12
13
A N I N T R O D UC T I O N T O C Y B E R NE T I C S
C H A NG E
We shall often require a symbol to represent the transform of
Such a symbol as n. It can be obtained conveniently by adding a
prime to the operand, so that, whatever n may be, n → n'. Thus, if
The operands of Ex. 1 are n, then the transformation can be written
as n' = n + 10 (n = 1, 2, 3).
Ex. 3: Write out in full the transformation in which the operands are the three
numbers 5, 6 and 7, and in which n' = n – 3. Is it closed?
Ex. 4: Write out in full the transformations in which:
(i) n'
(ii) n'
= 5n (n = 5, 6, 7);
= 2n2 (n = – 1, 0,1).
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