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• Number Theory
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Take a guided, problem-solving based approach to learning Logic. These compilations provide unique perspectives and applications you won't find anywhere else.

What's inside

• Introduction
• Puzzles and Riddles
• Multi-Level Thinking
• The Rational Detective
• Syllogisms and Sets
• Logic Machines
• Arithmetic With Logic Gates
• Propositional Logic
• First-Order Logic

Joy of Problem Solving

• Intro to Problem Solving
• Coin Rearrangements
• Truth Tellers and Liars
• Operator Searches
• Matchstick Puzzles

Community Wiki

Browse through thousands of Logic wikis written by our community of experts.

• Truth-Tellers and Liars
• Cryptogram - Problem Solving
• Solving Propositional Logic Word Problem
• Mind Reading with Math
• Information Compression
• K-level thinking
• Chess Puzzles
• Arithmetic Puzzles - Operator Search
• Arithmetic Puzzles - Fill in the Blanks
• Elimination Grids
• Grid Puzzles
• Combinatorial Games - Definition
• Combinatorial Games - Winning Positions
• Tic Tac Toe
• Sprague Grundy Theorem
• Chess Puzzles - Reduced Games
• Chess Puzzles - Opening Strategies
• Chess Puzzles - Rook Strategies
• Rook Polynomial
• Game Theory
• Nash Equilibrium
• Zero-Sum Games
• Prisoner's Dilemma
• Braess' Paradox
• Utility Functions
• Cognitive Bias
• Monty Hall Problem
• Birthday Problem
• Two-Envelope Paradox
• Simpson's Paradox
• Berkson's Paradox
• Newcomb's Paradox
• Benford's Law
• Mathematics of Voting
• Survivorship Bias
• Russell's Paradox
• Zeno's Paradox
• Gabriel's Horn
• Truth Tables
• Proof by Contradiction
• Mathematical Logic and Computability
• Mathematical Logic and Computability II (continuation)
• Propositional Logic Using Algebra
• Venn Diagram
• Predicate Logic

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Gizmodo Monday Puzzle: Can You Solve This Brutally Hard Logic Puzzle?

A magic store sells truth-telling machines. alas, one of them is broken..

One of the most famous logic puzzles of all time involves a fork in the road, where one path leads to freedom and the other leads to your death. Each path has a guard, and one of them always tells the truth and the other always lies, but you don’t know which is which. You get to ask one yes/no question to one guard to guarantee your road to freedom. What do you ask? It won’t work if you simply ask one of them, “Does your road lead to freedom?” because if they reply, “Yes,” then it could be a truth-teller setting you free or a liar condemning you to death.

This puzzle featured in the 1986 acid trip, Labyrinth , and the protagonist solved it by asking one of them, “Would the other guard tell me that your path leads to freedom?” Combining the two guards in this way cleverly ensures that you get bad advice (i.e. if the guard says, “No” then choose his path, and if the guard says, “Yes” then choose the other path). The idea is that a lie about the truth or the truth about a lie both yield reliably bad intel that you should reject. A solution I like even better is to ask a guard, “What would you say if I asked you if your path leads to freedom?” This subtle rephrasing forces the liar into a double negative. I invite you to work through the details.

Legendary logician Raymond Smullyan dubbed these “Knights and Knaves” puzzles and produced dozens of variants on them in his writings, including this week’s Monday puzzle. The so-called “ Hardest Logic Puzzle Ever ” stars knights and knaves and requires solvers to ask three questions to crack. While that one is an absolute mind-melter, I’m tempted to assert that the puzzle I present below is the hardest puzzle of this type in which you only ask a single question. Let me know in the comments if you have other contenders.

Did you miss last week’s puzzle? Check it out here , and find its solution at the bottom of today’s article. Be careful not to read too far ahead if you haven’t solved last week’s yet!

Puzzle #11: Smullyan’s Truth Machines

A magic store sells truth-telling machines. The machines each have a yellow light and a blue light that illuminate in response to true-or-false questions, with one light corresponding to “true” and the other to “false.” The store owner has three machines. She tells you that two of them work and tell the truth, but one of them is broken and outputs random answers. She doesn’t know which one is broken. Furthermore, she doesn’t know which light colors correspond to true and false, and the meaning of the lights might differ between machines (i.e., in one machine a yellow light might mean “true,” while in another machine a yellow light might mean “false”). Ask a single true-or-false question to one machine to guarantee that you walk away with a functioning truth-telling machine.

Machines that answer randomly are considerably harder to wrangle than knaves who always lie. Consistent liars mislead in predictable ways! Not knowing what the lights mean and that they can differ across machines poses even more obstacles. While this puzzle is not for the faint of heart, determined mortals can solve it.

Discuss your thoughts in the comments, and I’ll be back next week with the solution and a new puzzle. Do you know a cool puzzle that I should cover here? Send it to me at [email protected]

Solution to Puzzle #10: Physics Stumpers

Last week, I left you with two puzz les from the physical world. Shout-out to Spessartine for giving good, concise explanations for both puzzles.

The first challenge asked:

• You have a piping hot cup of coffee that’s too hot to drink. You can either pour a splash of cold milk into it and then let it sit for 10 minutes, or first let it sit for 10 minutes and then add the milk. In which scenario will the coffee end up cooler, or are they equivalent? In both scenarios, assume you pour the same amount of milk and it is the same temperature.

The coffee will be cooler if you first let it sit for 10 minutes and only then pour the milk in. There are two ways that the coffee loses heat. One is by dissipating into the environment, and the other is by transferring heat to the cold milk until they mix and become one (cooler) temperature. Critically, things dissipate heat into the environment faster the hotter they are. So when the coffee is at its hottest, it’s losing heat very quickly. Adding milk right away slows this dissipation, and instead that early heat gets transferred to the milk and stays inside the mug. For cooler coffee, it’s better to let it lose as much heat as it can to the air before introducing milk.

The second question:

• You’re in a canoe in the middle of a pond and you brought a rock with you. You pick up the rock, drop it into the water, and watch it sink to the bottom. Does the water level of the pond rise or fall (however imperceptibly) when you do this, or does it stay the same?

The water level falls when you drop the rock into the pond. A submerged object displaces its volume. In other words, imagine the rock were made of water and we poured that water into the pond. That would raise the level of the pond exactly the same amount as submerging the rock does. A floating object, on the other hand, displaces its weight in water. This means that when the rock is in the canoe, if the rock weighs 1 pound, then it contributes a pound’s worth of water to the height of the pond. So the question becomes: which is more—an amount of water equal to the volume of the rock or an amount of water equal to the weight of the rock? Weight wins this competition. In fact, we know this because the rock sinks. Ordinary objects sink in water precisely when it would take more water to equal their weight than to equal their volume. If the rock had been something much less dense, like a big airy pumice stone, then its volume in water could exceed its weight in water, and such a stone would have floated.

Ice floats in water. For a neat bonus teaser, what happens to the water level in a glass when the ice melts? Does it rise, fall, or stay the same? Let me know what you think in the comments.

Last week, I posed a bonus poker question in the solution section, courtesy of reader Joshua Lehrer: In Texas hold ‘em against one opponent, imagine you get to pick which two cards you get dealt and which two cards they get dealt. Then the dealer flips the five community cards, with no betting between them (maybe you’re all in). Which cards should you choose for yourself and for your opponent to maximize your chances of winning? My gut instinct was to pick the best hand for myself, (two aces) and the worst hand for my opponent (a 7 and a 2 that don’t share a suit). It turns out that you can do better when you get to pick both hands. Shout-out to reader Donnie from Las Vegas, who emailed me the correct answer. The answer is to give yourself two kings and give your opponent a king and a 2, where the 2 shares a suit with one of your kings. Awfully unexpected! Apparently, this gives the side with the two kings a 94.16% chance of winning, whereas two aces versus the 7 and the 2 only gives an 88.74% chance of winning.

5 Logical Puzzles That Will Bend Your Brain

March 16, 2021 by Anthony Persico

5 Logical Puzzles That Will Bend Your Brain (Answers Included!)

A Post By: Anthony Persico

Are you looking to take on some fun and challenging logical puzzles and put your brain power to the test?

Attempting to solve logic puzzles is one of the best ways to improve your problem-solving and logical thinking skills while having lots of fun at the same time.

Today’s post shares a collection of 5 logical puzzles for kids and adults that gradually progress in difficulty—and there’s also a special bonus puzzle that has been stumping people for over 100 years!

Note that the solution to each logic puzzle is included at the bottom of the post (we recommend trying all of the problems on your own first and then checking to see if your answers are correct afterwards).

Are you ready to get started?

Some of these puzzles can be solved in under one minute, while others are tricky enough to stump you all day long, putting your brain to the test!

Today, you’ll have a chance to give your brain a workout by figuring out a few of these famous logic and number puzzles.

So, are you up for the challenge? And don’t forget to try the bonus puzzle—a simple yet tricky riddle that has been around for over 100 years.

And if you need more detailed explanations of how to solve each logic puzzle, check out our 5 Logical Puzzles That Will Bend Your Brain video below, and be sure to click the like button and leave a comment!

Watch the 5 Logic Puzzles Video:

Logical puzzle #1.

How many total blocks are in the diagram below?

Keep reading to the bottom of this page to see the answer key AND click here to see the video that explains the solution to this problem.

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Logical Puzzle #2

How far can a bear walk into the woods?

Logical Puzzle #3

How can you flip the gummy bear pyramid upside down by moving only 3 gummy bears?

Are you looking for more super fun Math Riddles, Puzzles, and Brain Teasers to share with your kids?

The best-selling workbook 101 Math Riddles, Puzzles, and Brain Teasers for Kids Ages 10+!  is now available as a PDF download. You can get yours today by clicking here .

Logical Puzzle #4

On Bert’s 14th Birthday, his younger brother Chip was half his age. If today is Bert’s 31st birthday, how old is Chip?

Logical Puzzle #5

If there are 7 bears in a room and they each hug each other once and only once, how many total bear hugs were there altogether?

Bonus Logical Puzzle!

An explorer who walks one mile south, one mile east, and one mile north and ends up right back where he started. While he is walking, he sees a bear. What color is the bear?

Ready for the answers?

Remember that there’s no rush to find the answers to today’s logic puzzles. It’s ok to wait to until you’ve found an answer to each problem before scrolling down to see the answers below.

ANSWER KEY:

1.) 11 Blocks

2.) Half-Way (after that, the bear is walking out of the woods)

3.) Diagram:

4.) Chip is 24 years old

5.) 21 Hugs

Bonus: The Bear is White (Polar Bear at the North Pole)

Are you looking for more super fun Math Puzzles to share with your kids?

My best-selling workbook 101 Daily Math Challenges for Engaging Students in Grades 3-8  is now available as a PDF download. You can get yours today by clicking here .

Did I miss your favorite math riddle for kids? Share your thoughts, questions, and suggestions in the comments section below!

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By Anthony Persico

Anthony is the content crafter and head educator for YouTube's  MashUp Math  and an advisor to Amazon Education's ' With Math I Can ' Campaign. You can often find me happily developing animated math lessons to share on my YouTube channel  . Or spending way too much time at the gym or playing on my phone.

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