The Poisoned Wine Bottle Puzzle
How will you find the one bottle out of 1000?
King Bob and his loyal subjects have been planning King Bob Day for years. Finally the day is upon us! The Royal Vintner has set aside 1,000 bottles of the finest aged wine in honour of the occasion.
Tragedy and heartbreak! King Bob learns through his Royal Intelligence Agency that his nemesis, Duke Ralph, has had exactly one of the bottles poisoned. Not one, but ten traitors have been bought by Duke Ralph to sabotage the festivities.
King Bob rounds up the ten traitors and has them thrown into the dungeon. He then personally questions them.
“Where is the poison?” the monarch demands.
“We don’t know,” says Royal Traitor #1. “All we know is that it’s in one of the bottles. They all look the same.”
“Fine,” says King Bob. “Then you each taste 100 bottles until one of you drops dead.”
“That won’t work,” says Royal Traitor #2. “We used a special derivative of Iocaine Powder. One drop of wine will be enough to kill anyone….”
“But,” adds Royal Traitor #3, “it needs about 24 hours to have any effect.”
“It is both tasteless and odourless!” laughs Royal Traitor #1. “You simply will not find the one bottle in 1,000 by testing it on the ten of us!”
“You cannot use any of the wine!” says Royal Traitor #2. “Your party tomorrow is destroyed!”
“Long live Duke Ralph!” they all cheer.
King Ralph consults his Royal Clever Person.
“I despair that the festivities are ruined!” he wails. “Even you, in all your Royal Cleverness, cannot find a way to test 1,000 bottles on only 10 subjects!”
A Mona-Lisa smile grows on the The Royal Clever Person’s face.
What advice does he give the King?
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The Royal Clever Person’s Solution
The Royal Clever Person advises the King to have the bottles all numbered, from 1 to 1,000. “But you must not use decimal numbers. You must use binary numbers,” he says.
“A binary system is like the decimal system we all know. It use a place system. But, instead of the familiar units place, 10’s place, 100’s place and so forth, it relies on powers of 2.”
The Royal Clever Person explains that any number from 1 to 1,000 can be expressed as a 10-digit binary, using only 0’s and 1's.
For example, #742 would be 1011100110:
Because there are ten traitors, each can represent one of the ten powers of two. The wine is distributed according the binary number of the bottle, and the place value of each traitor. In the case of #742, only those in the 0 slots — 256, 16, 8 and 1 — will receive the wine.
After 24 hours, the poison will have taken its toll. Each combination will produce a unique pattern of living/dead traitors. The place values assigned to the living traitors can be summed to determine which bottle has the poison.
King Bob is so impressed with the Royal Clever Person, that he immediately promotes him to Royal Extremely Clever Person. The system works, the poison is found, and the other 999 bottles are salvaged.
The Kingdom rejoices. Mathematics has saved King Bob Day!
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When I published this puzzle elsewhere, an Extremely Clever Person commented with an alternative solution. Place all the bottles of wine in a 5 × 5 grid: 40 bottles per square. Traitors 1 to 5 sample from the rows; traitors 6 to 10 from the columns. Two dead traitors will signify one square on the grid.
This method uses 40 bottles, but it's quick.
Whether the Royal Clever Person thought of this idea is unknown. However, he is paid by the hour, so would have preferred the more tedious solution given.