# the 3 robot heads

in the robot room in castle of dr. brain there are three robot heads. you need to program each of them to fetch some items in a small maze. the problem is that not all of the robot heads work correct all the time.

there is more information one can get by spending one hint coin or by trial and error: one robot always does as told, one robot always does the opposite of what it is told, one robot head alternates between correct and opposite.

here is what the three robot heads say about themselves:

propeller head: "I am programmed to always follow instructions precisely. Iron Face also follows instructions precisely"

saucer head: "I am programmed to always follow instructions precisely. I am the only precise order-follower here."

iron face: "I am programmed to always follow instructions precisely. Saucer Head never follows orders correctly."

i figured this out when i played this game as a child, and i figured it out again now. but somehow i don't feel satisfied because my method of solving the puzzle is by some combination of backtracking and trial and error. i know from my logic class (years ago) that one can deduce the solution somehow, but i fail to do so.

can someone provide me with a formal logical approach to solve this puzzle.

note: i am not interested in only the solution. i know that already.

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## 2 Answers

Never played this game, but I know logic!

All 3 heads say "I am programmed to always follow instructions precisely", but only one of them actually does. That means that the only one who is telling the truth with the first statement is the only one who follows orders precisely. The other two are lying with the first statement, so one has to have a true second statement (the one that alternates) and the other has to have a false second statement (the one who is always opposite).

1. Propeller Head always follows orders - Impossible, because the second statement would be false, violating the necessity for the first statement to be true. Only one of them always follows orders.
2. Saucer Head always follows orders - Impossible, because otherwise both Propeller Head and Iron Face must be lying in their second statements for this to be correct, yet one of them must have a true second statement.
3. Iron Face always follows orders - Possible, because this makes Propeller Head's second statement true and Saucer Head's second statement false, which fits our requirements.

The derivation of this is that Propeller Head alternates between following precisely and following opposite, Saucer Head always follows the opposite, and Iron Face always follows precisely.

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i understand your arguments why PH is impossible and why IF is possible, but i don't understand your argument why SH is impossible. i worked out following argumentation: suppose SH always follows orders (both of SHs statements are true). that means PHs and IFs first statement are false. that means either PHs or IFs second statement must be true. IFs second statement can't be true (SH is supposed to be the correct order follower). PHs second statement also can't be true (IF is already established as always not following orders). thus the proposition can't be true (SH impossible). – lesmana Aug 15 '10 at 23:27
i just read your text again and now i understand your argument for saucer head. it is very very ... concise. – lesmana Aug 15 '10 at 23:35
Funny how I never realized that, excepting we discuss the inferred semantics of it, all of them can be said to follow orders precisely, in that they will have a precise response to your command, whether they follow it or the opposite; even the goofy, confused looking one has a simple precision in the switching from one behavior to the other. If "Iron Face /also/ always follows orders precisely", then for that to be true, mustn't he "also"? – user42191 Feb 10 '13 at 3:00

It's been so long since I played this: is the constraint that "one robot always does as told, one robot always does the opposite of what it is told, one robot head alternates between correct and opposite" really only given if you ask for a hint? (As in, it isn't stated at the outset?)

I just realised that if you remove the constraint that there is exactly one of each type of robot (i.e., you allow for multiple robots to have the same behaviour), then there are actually quite a number of consistent solutions. Here I enumerate all of the possibilities. First, I'll reword the problem more simply and formally. I use "Truth", "Lies" and "Alternate" to describe the three robot behaviours.

• Propellor: Propellor is Truth. Iron is Truth.
• Saucer: Saucer is Truth. Propellor is not Truth. Iron is not Truth.
• Iron: Iron is Truth. Saucer is Lies.

Now for the full decision tree:

• Propellor is Truth. Therefore, Iron is Truth.
• Iron is Truth. Therefore, Saucer is Lies. (Consistent)
• Propellor is Lies. Therefore, Iron is not Truth.
• Iron is Lies. Therefore, Saucer is not Lies.
• Saucer is Truth. (Consistent)
• Saucer is Alternate. His first statement is false, second is true. (Consistent)
• Iron is Alternate. His first statement is false, second is true. Therefore, Saucer is Lies. (Consistent)
• Propellor is Alternate. His first statement is false, second is true. Therefore, Ison is Truth.
• Iron is Truth. Therefore, Saucer is Lies. (Consistent)

Therefore, there are five consistent possibilities:

• Propellor is Truth. Saucer is Lies. Iron is Truth.
• Propellor is Lies. Saucer is Truth. Iron is Lies.
• Propellor is Lies. Saucer is Alternate. Iron is Lies.
• Propellor is Lies. Saucer is Lies. Iron is Alternate.
• Propellor is Alternate. Saucer is Lies. Iron is Truth.

Only the last possibility has all three robots with different behaviours, so it is the correct answer. But it is interesting to note that if you allow two robots to have the same behaviour, then there are other ways to interpret their statements. I like the one where Saucer Head is telling the truth all along and the other clowns are lying through their teeth -- finally some redemption for poor misunderstood Saucer Head!

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