Smullyan's logical machines

In several of his wonderful books (most notably in The lady or the tiger? and other logical puzzles ) the logician Raymond Smullyan introduces several "logical machines" that operate on numbers and strings according to specified rules.

Numerical machine from The lady or the tiger?

Rules:
  • For any number X, the number 2X is accepted by the machine, and 2X produces X (if you input "233" to the machine, it will produce "33")
  • For any numbers X and Y, if X produces Y then 3X produces Y2Y (called "the associate of Y"): if you input 3233 to the machine, it will output 33233, since 233 produces 33!
  • For any numbers X and Y, if X produces Y then 4X produces the inverse of Y
  • For any numbers X and Y, if X produces Y then 5X produces YY (the repetition of Y)
Input a number in the area below and press 'Evaluate' to see the machine result:

Some problems for you to try on this machine

  • Find a number X which produces 2
  • Find a number X which produces X
  • Find a number X which produces its associate (X2X)
  • Find a number X which produces its own inverse
  • Find a number X which produces the repetition of the inverse of its own associate
If you read Smullyan's The lady or the tiger? you will understand how to solve all these problems (and many more!), and you will learn interesting properties of the machine.

Montecarlo's safebox machine from The lady or the tiger?

This logical machine is supposed to drive the lock of an important safebox. This machine will accept strings; some strings will block the safebox, some strings will be neutral (these will not produce any effect on the lock), and some strings will open the safebox. There is a certain relationship between some pairs of strings, and we will say that some string is "especially related" to another string; the meaning of this relationship is not clarified further! These are the properties of the machine:
  • For any string x, QxQ is especially related to x (for example, QGBQ is especially related to GB).
  • If the string x is especially related to the string y, then Lx is especially related to Qy (for example, since QGBQ is especially related to GB, then LQGBQ is especially related to QGB).
  • If the string x is especially related to y, then Vx is especially related to the reverse of y (for example, since LQGBQ is especially related to QGB, we see that VLQGBQ is especially related to BGQ, the reverse of QGB).
  • If the string x is especially related to y, then Rx is especially related to the repetition of y (for example, since LQGBQ is especially related to QGB, then RLQGBQ is especially related to QGBQGB, the repetition of QGB).
  • If x is especially related to y then, if x blocks the lock, y will be neutral, and if x is neutral, y will block the lock.
With these conditions, it is possible to find a string that will open the safe. Can you find one? (Hint: what happens if we enter a string which is especially related to itself?)
Input a string in the area below and press 'Check' to see what other string it is "especially related" to:


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(c) 2004 Manuel Heras