Let me reveal more of the secrets in this puzzle.

In my high-school Math there was a chapter “Logic”. This puzzle bases on nothing more complicated than that simple Logic lesson.

Principles

- All “cheating points” come from a statement contradicting earlier statements.
- Rules of the game are also kinds of statements (told before the game), so that a statement against the rules of the game is a contradiction.

What is Contradiction

(from my example given above)

11 too low

12 too high

After the first statement, the valid range becomes [12…100]

Statement 2 implies the valid range is [1…11]. The combination of two statements (use AND to find the overlapping range of two sets) will result in a null range. Null range is not allowed in the game, so that statement 2 makes a contradiction.

Valid range can shrink after new statements. It will never expand.

11 too low

21 too low (New valid statement. Valid range becomes [22…100])

12 too high (contradiction)

Extreme cases

(from the first sample in puzzle)

1 too high

contradiction to the rule of the game (numbers below 1 are not allowed)

Legal tricks 1 - noise

adding valid statements to the above cases. Add to the front, end, and/or middle.

11 too low

21 too low

12 too high

can become:

11 too low

21 too low

20 too low (noise, does not change the valid range)

12 too high (contradiction)

Legal tricks 2 - repetition

11 too low

12 too high

can become

11 too low

11 too low

12 too high

or become

11 too low

21 too low (added valid statement)

11 too low (repetition)

12 too high

Remember, noise can be added at the front, end, and/or middle, in multiple places.

Legal tricks 3 - different direction

All the above examples can be completely or partially reversed (reverse the sequence or reverse “high” vs “low”). The contradiction point will be different. Or there can be no contradiction at all.

reversed sequence

12 too high

11 too low

21 too low

11 too low

(After all these explanation, you should know where the contradiction is.)

All these tricks and principles are illustrated in the existing test cases with some possible combinations. Why not giving the full combination of all possibilities? Normally no puzzle can. However, you are now armed with the knowledge to take care of the rest of other unlisted combinations.