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Generating Formulas

The previous linear ordering games we studied were static and finite. We were given a fixed number of elements and were asked questions about their possible orderings. Generating formulas, however, tend to be dynamic, in the sense that a basic sequence is given that is used to “generate” other sequences by repeated applications of the formulas. Because the formulas can be applied indefinitely, the sequences often have no end—though typically we are interested in only the beginning of the sequence.

A particular computer code uses only the letters A, B, C, and D. A “word” is formed in the code according to the following rules:


ABC is the basic word from which all other words are constructed.


D must appear in a word more than once, if at all.


Interchanging the first and last letters in a word creates a new word.


Adding a pair of Ds to the end of a word creates another word.

Notice that the third and fourth conditions are permissive. That is, they could be applied but don’t have to be.


Note: With permissive conditions, the contrapositive rule of logic does not apply.

The second condition, on the other hand, is mandatory: if D occurs in a word, it must occur at least once more.


Note: With mandatory conditions, the contrapositive does apply.

There are only two basic types of questions to these games:
  1. Those that ask you to derive a new sequence from a basic sentence. In the game above, for example, you may be given the word ABC and then asked to derive a new word by applying the fourth and third rules, in that order.
  2. Those that ask you to “discover” from where a sequence was derived. In the game above, for example, you may be asked “From which word was the word DBCDA derived?”
The latter type of question tends to be more difficult since there are many paths you can retrace, only one of which will lead to the correct answer.
Because working backwards is often difficult, look for opportunities to reverse the direction by using the contrapositive. But apply the contrapositive only to mandatory conditions.

Generating-formula games are one of the few types of games for which it is not advisable to draw a diagram. In fact, typically you cannot draw a diagram. Nevertheless, you may want to symbolize the “rules” for easy reference.

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