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You stand before a panel with two buttons on it. One of them will dispense your favorite beverage, and you're dying for a sip. The other will destroy all life throughout the universe, so you'd really rather not press that one. You don't know which button is which, but fortunately the buttons have three alien attendants--each a different race--to help you. You know a little bit about the alien languages, but unfortunately you don't know which alien uses which language. The aliens are identified with signs reading "1", "3", and "6".

The aliens' languages are based on gestures, and there are four polite ways to ask a question: point with the left index finger, the left index finger and middle finger, the right index finger, or the right index finger and middle finger, for 50 seconds. Any other form of gesture would be considered rude, and would cause the aliens to immediately shoot you (which would be bad).

One of the aliens is an Altairian. If you ask a question by pointing with your left hand (one or two fingers), it will (after 49 seconds) nod its head if the left button dispenses a beverage and do nothing if it does not. Likewise if you ask a question by pointing with your right hand, it will not its head if the right button dispenses a beverage and do nothing if it does not.

One of the other aliens is a Denebrian. If you ask a question by pointing with one finger (either hand), it will (after 49 seconds) nod its head if the alien with the closer number is an Altairian or else do nothing. If you point with two fingers (either hand), it will nod its head if the further number is an Altairian or else do nothing.

The remaining alien is a Snoozorian. It's going to sleep for the next five years no matter what you do, and can thus be counted upon to simply do nothing.

You've been called to battle, and need to leave within the next two minutes, but you really want your beverage before then. This wouldn't be enough time, had you not learned in your latest linguistics class that it's possible to politely address two aliens simultaneously subject to the following constraints:

  1. One must simultaneously use both arms to address the aliens, with one arm pointed at each; the arms must not be crossed, and one must continue to point with both arms for the full 50 seconds.

  2. If you ask questions which both merit affirmative responses, the aliens will get annoyed and shoot you (again, not good).

Note: This is my first attempt at formulating a variation of a "Lady or the tiger" puzzle; rather than make some of the guardians deceitful, I decided to instead make them vary in the questions they answer. Since only one alien knows where the right button is, and since the first alien one talks to might not know anything, it's necessary to be able to talk to all three aliens, but being able to simply ask a question of each without restriction doesn't seem very satisfying. If people like the concept, but not the particulars, I'd be open to advice on how to make the puzzle more interesting and/or satisfying.

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  • $\begingroup$ Can we visibly tell which alien is sleeping? $\endgroup$ – EFrog Jan 17 '15 at 20:32
  • $\begingroup$ The intention in the present formulation is that the (non) response from the sleeping alien is indistinguishable from a negative response by one of the others. In an earlier formulation of the puzzle, I had the two non-sleeping aliens teleport away after two question-answering opportunities, leaving the sleeping one behind (but only after one has asked the questions); I don't think that's necessary, however. $\endgroup$ – supercat Jan 17 '15 at 20:39
  • $\begingroup$ @EFrog: You just gave me an idea for a possible improved formulation: replace the snoozer with a Metoo; if asked a question simultaneously with someone else, it will nod if that other alien does so. The aliens can distinguish a Metoo's echo from an affirmative answer, and do not consider it rude, but you can't tell the difference. If you ask a question of two aliens and both nod, one of them will be a Metoo and the other will be answering affirmatively, but you won't be able to tell which is which. I'll have to see if that formulation allows the proper button to be chosen in all cases. $\endgroup$ – supercat Jan 17 '15 at 20:51
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    $\begingroup$ After spending one minute fifty seconds thinking about what questions to ask, I realize I have run out of time and sadly head off to battle with no beverage. There I am surprised by a Centaurian because I am too busy thinking about interesting logic puzzles. Please continue. $\endgroup$ – Callidus Jan 18 '15 at 4:09
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Solution:

Point two fingers with your left hand at alien 1, and two fingers with your right at alien 3.

  • If alien 1 nods, then point one finger with your left hand at alien 1, and one finger with your right at alien 6. If alien 6 nods, choose the right button, otherwise choose the left button.
  • If alien 3 nods, then point 1 finger with your left hand at alien 1, and 1 finger with your right at alien 6 (as above). If one of the alien nods, choose the right button, otherwise choose the left button.
  • If neither alien 1 or alien 3 nodded, then point two fingers with your left at 3 and two fingers with your right at 6. If alien 6 nods, choose the right button, otherwise choose the left button.

Explanation:

Firstly, point two fingers with your left hand at alien 1, and two fingers with your right at alien 3. You won't get shot because:

  • If alien 1 is an Altairian, then alien 3 will do nothing (either because he is the Snoozorian, or because the alien with the farther number (6) is not an Altairian).
  • If alien 1 is a Denebrian and nods, then alien 6 is the Altairian and alien 3 is the Snoozorian and will do nothing. If he doesn't nod, alien 3 is the Altairian.
  • If alien 1 is a Snoozorian, then he won't nod.

If alien 1 nods, then point one finger with your left hand at alien 1, and one finger with your right at alien 6. If alien 6 nods, choose the right button, otherwise choose the left button.

  • If alien 1 nods, he is the Altairian (the Denebrian wouldn't nod when pointed at with one and two fingers), so choose the left button. If alien 6 is the Denebrian, then he won't nod because the alien with the close number (alien 3) isn't the Altairian.
  • If alien 6 nods, he must be the Altarian, so we choose right. Alien 1 wouldn't nod, because he would be the Denebrian, and the alien with the number closer to him was not the Altairian. If he were the Denebrian, then alien 3 would be the Altairian, and alien 1 the Snoozorian - but alien 1 nodded the first time, so this is a contradiction.
  • If neither alien nods, then alien 1 must be the Denebrian (if he was the Altairian, he would have nodded both times), and so alien 3 is the Altairian. He didn't nod when we pointed our right hand at him the first time, so we choose the left button.

If alien 3 nods, then point 1 finger with your left hand at alien 1, and 1 finger with your right at alien 6 (as above). If one of the alien nods, choose the right button, otherwise choose the left button.

  • If alien 1 nods, then alien 3 is the Altairian (if alien 1 was the Altairian, then alien 3 wouldn't have nodded, because we pointed two fingers at him), and the correct button is that on the right.
  • If alien 6 nods, then the correct button is the one on the right (either 3 or 6 is the Altairian, and they both nodded when we pointed with our right hand to them).
  • If neither alien nods, then alien 3 is the Denebrian (if he was the Altairian, then one of the others would be the Denebrian and would nod). He nodded when we pointed 2 fingers at him, so 6 is the Altairian. He didn't nod when we pointed with our right hand, so choose the left button.

If neither alien 1 or alien 3 nodded, then point two fingers with your left at 3 and two fingers with your right at 6. If alien 6 nods, choose the right button, otherwise choose the left button.

  • If alien 3 nods, then he is the Altairian (if he was the Denebrian, he would have nodded the first time), so choose the left button. You won't get shot because either 6 is the Snoozorian, or the farther alien from him (alien 1) is not the Altairian.
  • If alien 6 nods, then he is either an Altairian so we choose the button on the right, or the Denebrian (meaning alien 1 was the Altairian). When we pointed with our left to alien 1, he didn't nod, so we also choose the button on the right. Again, we won't get shot because if alien 6 was the Altairian, alien 3 would have nodded the first time, and if alien 6 was the Denebrian, 1 was the Altairian and therefore 3 was the Snoozorian.
  • If neither alien nods, then 3 is not the Altairian (we have pointed both hands at him). 3 cannot be the Snoozorian either, because if 6 was the Altairian, 1 (being the Denebrian) would have nodded the first time, and if 6 was the Denebrian, he would have nodded because 1 was the Altairian. Therefore 3 is the Denebrian, and 1 is the Altairian, so we choose left.
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  • $\begingroup$ How did you like the puzzle? Did it seem original, or are there others of similar vein (where guardians answer different questions, and part of the challenge is to identify what question each guardian was answering)? I would have liked to have had the third guardian be able to arbitrarily answer yes or no (without the person knowing what the guardian was going to do) but I wasn't able to solve the puzzle under those conditions, though it might be possible to formulate rules that would make it possible (e.g. perhaps if the third guardian could only answer yes if the other guardian... $\endgroup$ – supercat Jan 18 '15 at 23:37
  • $\begingroup$ ...wouldn't do so). Do you have any advice for how to best formulate such puzzles so as to be interesting, and avoid being either too easy nor frustratingly difficult? $\endgroup$ – supercat Jan 18 '15 at 23:38
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    $\begingroup$ @supercat Having just started doing this sort of thing, I'm afraid I can't give you any meaningful advice. But here are some cursory ideas (I don't know whether they've been done before): perhaps have more than two possible answers, or maybe a systematic variation in how each guardian answers? You'd have to do some experimenting to figure out how to balance the difficulty though. In regards to having a guardian answer randomly, I don't think any puzzle with a random element can be solved (you wouldn't be able to tell the guardian apart from the others). $\endgroup$ – Volatility Jan 19 '15 at 0:28
  • $\begingroup$ The intention with the guardian that answers arbitrarily is that a proper solution should work no matter how the guardian answers. There could be a few variations--in some the player may need to figure out which guardians to ignore based on the other guardians' answers; in others, the arbitrary-answer guardian may be revealed after the player's last opportunity to question other guardians. In no case would there be "randomness" involved. The presumption would be that if there was a way the guardian could answer that would confound the player, the guardian would answer that way. $\endgroup$ – supercat Jan 19 '15 at 2:29

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