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Aladdin and the Lethal Liquid -- SOLUTION (The "Binary Winery")
Posted By: Nack Ballard In Response To: Aladdin and the Lethal Liquid (Nack Ballard)
Date: Friday, 10 May 2019, at 7:18 a.m.
Congratulations to Casper, John O'Hagan, Karol Szczerek, Brian Lonergan, Bob Koca, Ian Terry, Mike Clapsadle, G L Harvie, Joe Russell, and Sam Pottle, for solving the puzzle!
With so few subjects, Aladdin's first thought was to administer the wines a few seconds or few minutes apart, relying on the literal interpretation that illness suddenly strikes only and always after precisely 24 hours, give or take a very small increment of time. However, when people say "exactly 24 hours" (the wording that the queen overheard), it might easily be plus or minus 10–15 minutes (or conceivably 30) or so, meaning that deaths might overlap in an uncertain fashion. With the stakes this high, it was a risk Aladdin dared not take.
There is an episode of Three Stooges called Malice in the Palace (not one of their best, IMO). Sam Pottle referred to "the Chalice in the Palace." Either of these could have been an alternate puzzle title to "(Aladdin and) the Lethal Liquid." (I saved "...the Binary Winery" for the solution post as it would otherwise be too big a hint.)
This is actually an old puzzle, and I imagine you can find some variation of it on the internet. I had never heard it until someone e-mailed it to me couple of months ago. He posed it with a thousand bottles and ten prisoners, which made it slightly more difficult because "a thousand" is not an exact power of 2, but it was close enough that it set off an alarm bell in my head and I quickly hit on the right idea.
After some pondering, a similar bell went off in Aladdin's head, as he realized that the number of prisoners (six) matches the number of sides on a backgammon cube, on which 64 is the highest number. This mathematical nicety was not intended by King Shahryar; it was just a happy coincidence. Simply put, 2^6 = 64.
Aladdin cut up 64 pieces of paper, and numbered them from 0 to 63. Underneath, he translated each of these numbers to its six-digit binary (base 2) equivalent:
.....0 = 000000
.....1 = 000001
.....2 = 000010
.....3 = 000011
.....4 = 000100...and so on.
Aladdin attached the labels (randomly) to the bottles and laid out drinking glasses numbered 1 to 6 (and scribbled the initials of a different prisoner's name next to each number, to avoid later confusion). The glass on his far left corresponded to the first digit (of the six-digit binary), and the glass next to that the second digit, etc. The glass on his far right corresponded to the sixth digit.
Whenever a label had "1" in a given position, he poured a few drops of that wine into the appropriate glass. For example, if a bottle's label was 100110, he poured drops into glass #1, #4 and #5.
Another way to look at it is that glass #1 received drops from bottles 32 through 63 (100000 through 111111). Glass #2 received drops from bottles 16–31 (010000-011111) and 48–63 (110000-111111). Glass #3 received drops from bottles 8–15, 24–31, 40–47 and 56–63... At the other end of the spectrum, glass #6 received drops from all the odd-numbered bottles (any with a 1 in the sixth position).
With this judicious scheme, having 64 possible outcomes, each prisoner receives (a different) 32 samples and has a 50% chance to survive. In many cases, more or fewer than the (average of) three die. An example outcome is that if only prisoners 2 and 3 die, the bottle labeled 011000 can be pegged as the poisoned one. If all six prisoners die, 111111 is the poisoned bottle.
As it happened, none of the prisoners suffered any ill effects. Aladdin even waited a 25th hour to be sure. Then he reported definitively to the emperor that 000000 was the poisoned bottle.
"Excellent!" his majesty trumpeted. "So, I'm sure you won't mind if I pour you a few drops from each of the other 63 bottles into your glass, and you can kick off our ceremony with a public toast?"
"Absolutely," Aladdin smiled, his eyes shining. In the strength of that reaction, the sovereign shrugged and decided not to bother. Aladdin was greatly relieved (though he dared not show it) that his bluff had worked, as after all he had based his logic on hearsay. He would rather take his chances on drinking from a few random bottles, like everyone else. Vintage Aladdin.
He marked the (unopened) 000000 bottle "Deadly Poison" and had it locked in the royal safe. Then he removed the labels of the other 63, recorked them, and added a decorative bow to the top of each bottle. The party, which then started without further delay, was magnificent; and the many different wines were delicious, having excellent scent, balance, depth and finish.
Nevertheless, Aladdin stayed on the job. Observing the guests carefully, he was able to deduce the identity of the criminal. [To readers: What do you suppose was his clue?]
"My leige: if you send investigators to his winery, chances are that you will find corroborating evidence of his evildoing." Indeed that was the case, and the next day the proud potentate arrested the would-be assassin. He would have his public hanging after all!
True to his word, the dynamic dynast freed the six lucky prisoners. However, he first extracted a blood oath from each, on the lives of their families, that they would never cheat at backgammon again.
Nack
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