The MY Missile

Copyright ©1997 by Paul Niquette.  All rights reserved.
 

t was convenient for our analysis in this puzzle that the Soviets chose names of towns near their launch complexes such that we may substitute A, B, C, D for their abbreviations. Also, the use of 0 and 1 may have mnemonic value, representing OUT and IN for the instantaneous placement of missiles. Thus...
 
Agejevo
Belousovo
Chanino
Detcino
A
B
C
D
OUT
OUT
OUT
OUT
=
0
0
0
0
OUT
OUT
OUT
IN
=
0
0
0
1
OUT
OUT
IN
IN
=
0
0
1
1
OUT
OUT
IN
OUT
=
0
0
1
0
OUT
IN
IN
OUT
=
0
1
1
0
OUT
IN
IN
IN
=
0
1
1
1
OUT
IN
OUT
IN
=
0
1
0
1
OUT
IN
OUT
OUT
=
0
1
0
0
IN
IN
OUT
OUT
=
1
1
0
0
IN
IN
OUT
IN
=
1
1
0
1
IN
IN
IN
IN
=
1
1
1
1
IN
IN
IN
OUT
=
1
1
1
0
IN
OUT
IN
OUT
=
1
0
1
0
IN
OUT
IN
IN
=
1
0
1
1
IN
OUT
OUT
IN
=
1
0
0
1
IN
OUT
OUT
OUT
=
1
0
0
0

Sophisticated solvers will recognize that particular sequence for postulating Soviet missile movements as "reflected binary." The right-most non-zero columns reflect vertically the binary values in the table above the (2n)th entry.  Huh?  Reflected binary is but one of many families of coding sequences by which only a single bit is changed at each step. One of them meets the conditions of the problem..
 


A
B
C
D
0
0
0
0
0
1
0
0
0
1
2
1
0
0
1
3
1
0
0
0
4
1
1
0
0
5
1
1
0
1
6
1
1
1
1
7
1
0
1
1
8
0
0
1
1
9
0
1
1
1
10
0
1
0
1
11
0
1
0
0
12
0
1
1
0
13
1
1
1
0
14
1
0
1
0
15
0
0
1
0

Thus, assuming that the Soviets wanted to distribute the moves uniformly among their missiles, they would have chosen a sequence as indicated in the table. {HyperNote}

HyperNote

In the early sixties, this "state assignment" solution was discovered during the logical design of a peripheral controller and came to be called a "Niquette Counter."  It exploited the observation that for a finite automaton with 2n states, uniform state transitions can be made only when 2n is divisible by n. That does not happen as often as one might think.

    The design of the earliest automata will be forever credited to heavyweights in the logic arena dating back to Aristotle, with a raft of logicians showing up in the 19-Century to help out: George Bentham, George Boole, D.F. Gregory, Sir William Hamilton, Georg von Holland, Henry Longueville Mansel, O.H. Mitchell, Augustus De Morgan, George Peacock, Charles Sanders Peirce, Thomas Solly, J. Veitch, and John Venn.
Despite all of their contributions, the sequence of states in the first Niquette Counter was not produced by any of their venerated methods. It was some 20th-Century foozling that did the trick.  Oh well, a simple adaptation of the "Veitch diagram" came in handy.  It is one of the logical tools long familiar to designers of computers, whether maxi, mini, or micro.  Here is how the diagram looked after the solution was found, with the numbers in the squares corresponding to the sequence numbers in the rows of the table above.
 

A = 1
A = 1
A = 0
A = 0
B = 1
4
5
10
11
C = 0
B = 1
13
6
9
12
C = 1
B = 0
14
7
8
15
C = 1
B = 0
3
2
1
0
C = 0

D = 0
D = 1
D = 1
D = 0

Now that you have seen the solution, you might try starting with a blank table and creating your own Niquette Counter. Care to guess how many there are?

Finally, suppose that U.S. spy satellites discovered extra roadways being built by the Russians, such that squadrons of MY missiles could be moved into and out of their respective launch sites up to three at a time. Sophisticated solvers might enjoy strategizing retrospcctively on behalf of the Kremlin, with the objective of maximizing the security of the MY missiles while still assuring that all weapons are moved in equal amounts. {Return}

 


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