21. Logic Puzzle: There are three crates, one with apples, one with oranges, and one with both apples and oranges mixed. Each crate is closed and labeled with one of three labels: Apples, Oranges, or Apples and Oranges. The label maker broke and labeled all of the crates incorrectly. How could you pick just one fruit from one crate to figure out what’s in each crate?
Answer: Pick a fruit from the crate marked Apples and Oranges. If that fruit is an apple, you know that the crate should be labeled Apples because all of the labels are incorrect as they are. Therefore, you know the crate marked Apples must be Oranges (if it were labeled Apples and Oranges, the Oranges crate would be labeled correctly, and we know it isn’t), and the one marked Oranges is Apples and Oranges. Alternately, if you picked an orange from the crate marked Apples and Oranges, you know that crate should be marked Oranges, the one marked Oranges must be Apples, and the one marked Apples must be Apples and Oranges.
Hardest Logic Puzzles for Adults
22. Logic Puzzle: A teacher writes six words on a board: “cat dog has max dim tag.” She gives three students, Albert, Bernard and Cheryl each a piece of paper with one letter from one of the words. Then she asks, “Albert, do you know the word?” Albert immediately replies yes. She asks, “Bernard, do you know the word?” He thinks for a moment and replies yes. Then she asks Cheryl the same question. She thinks and then replies yes. What is the word?
Answer: Dog. Albert knows right away because he has one of the unique letters that only appear once in all the words: c o h s x i. So, we know the word is not “tag.” All of these unique letters appear in different words, except for “h” and “s” in “has,” and Bernard can figure out what the word is from the unique letters that are left: t, g, h, s. This eliminates “max” and “dim.” Cheryl can then narrow it down the same way. Because there is only one unique letter left, the letter “d,” the word must be “dog.” (For more on this answer, watch the video below.)
23. Logic Puzzle: You have five boxes in a row numbered 1 to 5, in which a cat is hiding. Every night he jumps to an adjacent box, and every morning you have one chance to open a box to find him. How do you win this game of hide and seek?
Answer: Check boxes 2, 3, and 4 in order until you find him. Here’s why: He’s either in an odd or even-numbered box. If he’s in an even box (box 2 or 4) and you check box 2 and here’s there, great; if not you know he was in box 4, which means the next night he will move to box 3 or 5. The next morning, check box 3; if he’s not there that means he was in box 5 and so the next night he’ll be in box 4, and you’ve got him. If he was in an odd-numbered box to begin with (1, 3, or 5), though, you might not find him in that first round of checking boxes 2, 3 and 4. But if this is the case, you know that on the fourth night he’ll have to be in an even-numbered box (because he switches every night: odd, even, odd, even), so then you can start the process again as described above. This means if you check boxes 2, 3, and 4 in that order, you will find him within two rounds (one round of 2, 3, 4; followed by another round of 2, 3, 4). For more on this answer, watch the video below.
24. Logic Puzzle: The “Monty Hall” problem was made famous when it appeared in Parade magazine’s “Ask Marilyn” column in 1990, and it was so counterintuitive it had everyone from high school students to top mathematical minds questioning the answer—but rest assured, the solution is accurate. Named for the Let’s Make a Deal game show host, the puzzle goes like this: You are given three doors to choose from, one of which contains a car and the other two contain goats. After you’ve chosen one but haven’t opened it, Monty, who knows where everything is, reveals the location of a goat from behind one of the other two doors. Should you stick with your original choice or switch, if you want the car?
Answer: You should switch. At the beginning, your choice starts out as a one in three chance of picking the car; the two doors with goats contain 2/3 of the chance. But since Monty knows and shows you where one of the goats is, that 2/3 chance now rests solely with the third door (your choice retains its original 1/3 chance; you were more likely to pick a goat to begin with). So, the odds are better if you switch.
Near Impossible Logic Puzzle for Adults
25. Logic Puzzle: This conundrum, a variation on a lying/truth problem, has famously been called the hardest logic puzzle ever. You meet three gods on a mountain top. One always tells the truth, one always lies, and one tells the truth or lies randomly. We can call them Truth, False and Random. They understand English but answer in their own language, with ja or da for yes and no—but you don’t know which is which. You can ask three questions to any of the gods (and you can ask the same god more than one question), and they will answer with ja or da. What three questions do you ask to figure out who’s who?
Answer: Before getting to the answer, let’s think of a hypothetical question you know the answer to, such as “Does two plus two equal four?” Then, phrase it so you’re asking it as an embedded question: “If I asked you if two plus two equals four, would you answer ja?” If ja means yes, Truth would answer ja, but so would False (he always lies, so he’d say ja even though he really would answer da). If ja means no, they both would still answer ja—in this case, False would answer the embedded question with ja, but saying da to the overall question would be telling the truth, so he says ja. (Random’s answer would be meaningless because we don’t know whether he lies or tells the truth.)
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