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How to Solve Logic Puzzles

Last Updated: October 8, 2023

This article was reviewed by Timothy Huynh . Timothy Huynh is a Rubik’s Cube Artist based in Tampa, Florida. Timothy has been solving Rubik's Cubes professionally since 2016 and became a Guinness World Record Holder with advanced Rubik's cube-solving skills. Timothy has a follower base of over 8.6 million YouTube subscribers and 124k followers on Instagram. Internet celebrities including David Dobrik and Simu Liu have sung his praises, and Pixar has deemed his work "incredible." This article has been viewed 296,380 times.

This article includes general advice for logical reasoning problems, as well as thorough instructions for solving the most common type of logic puzzle. This type of puzzle provides a list or paragraph of clues, then asks you a question that requires you to use the clues to answer. Many books and websites that contain these logic puzzles come with a grid to help you solve them, but this article also includes instructions for making your own.

Setting up a Grid

• Here's an example problem: Three friends named Anna, Brad, and Caroline agree to bring one dessert each to a birthday party. Each friend is wearing a different color shirt. Anna wears a blue shirt. The person who brought brownies couldn't find her red shirt today. Brad didn't bring any dessert at all, which made the person wearing a yellow shirt irritated. Which person brought the ice cream?
• The example question, like all logic puzzles of this type, asks you to match two categories together. You start out knowing the names of several people and the names of several desserts, but you don't know who brought which dessert. Using the clues in the description, you need to figure out how to match each person to a dessert until you know who brought the ice cream. There's actually a third category, shirt color, which should help you get to your answer.
• Note : skip to Using a Grid if the puzzle already comes with a grid set up. Skip to Solving Other Logic Puzzles if your puzzle does not fit this description.

• Write each list separately. When the puzzle mentions a name, add it to a list of names. When the puzzle mentions a color, add it to a separate list of colors.
• Each list should have the same number of items once you've finished. If a list is too short, reread the puzzle carefully for more items.
• Some tricky puzzles will give you hints about what someone doesn't have, such as "Brad didn't make a dessert." In this case, you should add "none" to the list of desserts, which should make it the same length as the other lists.

• For instance, let's say you have three lists. Names : Anna, Brad, Caroline; Desserts : brownies, ice cream, none; and Color of Shirts : red; blue, yellow. Write a vertical list in this order: Anna; Brad; Caroline; (draw a thick line here); brownies; ice cream; none; (draw a thick line here); red; blue; yellow.

• Once you're more familiar with this system, you can get away with not writing every list in both places. We will be using this grid to match items in the vertical list (on the left) to items in the horizontal list (at the top), and sometimes you don't need to match every item. If you've never used this method before, stick with these instructions

• If the list to the left of a section and the list above a section are the same, cross it out. You'll never need to compare the list "Anna, Brad, Caroline" to the list "Anna, Brad, Caroline" – you already know that Anna is Anna.
• Cross out duplicate sections. For instance, the section that compares "Anna, Brad, Caroline" on the left and "red, blue yellow" on the top is the same as the section that compares "red, blue, yellow" on the left and "Anna, Brad, Caroline" on the top. Cross off one of these duplicate sections so you only have one to pay attention to. It doesn't matter which you cross off.

Using a Grid to Solve a Logic Puzzle

• Occasionally, a puzzle cannot be fully solved, meaning you won't be able to fill the entire grid. You should still be able to answer the question it asks.

• If you can't find that square, search the other way around. For instance, find the row labeled "blue" and the column labeled "Anna", instead of the other way around.
• Don't start with a clue that tells you something that doesn't apply, such as "Anna doesn't wear a red shirt." While that's a useful clue that should be marked with an "X", this method will assume you started with a clue that gives positive information.

• In our example, the section that has the clue you just circled compares the names of people to the colors of their shirts. The squares we're crossing off are the combinations we've ruled out, which include Brad or Caroline wearing a blue shirt, and Anna wearing a red or yellow shirt. (Typically, the introduction will tell you that each item can only be matched to one item in each other category.)

• If your puzzle gives you clues about what doesn't match, such as "Anna doesn't wear a red shirt", you should put an X in that column. However, since you haven't found a positive match, you should not cross out any other squares.

• Brad did not bring a dessert. Put a circle in the Brad-none square.
• The person wearing a yellow shirt is not Brad. Put an X in the Brad-yellow square.

• If you're solving a puzzle from another country, look up the names to find out whether they are male or female. Puzzle books that are printed more than 20 years ago will sometimes contain names that were once female, but have now become male (or vice versa).

• The green house comes before another house, so it can't be the last one.
• The black house comes after another house, so it can't be the first one.

• Marcus can't be the one who ran the mile in 6 minutes, no one was ahead of him. Cross out the Marcus-6 square.
• Marcus can't be the one who ran in 8 minutes, because that time is less than 5 minutes behind the one before it. Cross out the Marcus-8 square.
• Either the 15 or 25 minute times would work for this clue. You'll have to wait until more squares are crossed off before you can figure out which time was Marcus's.

• Let's say you've discovered that Caroline wears a yellow shirt. Check the yellow shirt column or row for information in other sections.
• Let's say you notice on your chart that the person with a yellow shirt did not bring ice cream. Because you know that person is Caroline, you can also cross out the square that connects Caroline and ice cream.
• Check Caroline's row or column too and transfer information the same way to the yellow shirt column or row.

• If a row or column within a section has every square crossed off, or more than one square with a circle in it, there was probably a mistake made along the way and you may need to start over.

• If an inconsistency occurs, your guess must have been wrong. Go back to what the chart looked like before you made your guess, and make the opposite one. Always keep track of when you made your guess with a new copy or a different color ink so it's easy to reverse if the guess was wrong.

• If you got the answer without filling out your entire chart, you may not be able to check every clue. As long as your chart doesn't contradict the clues you can check, you are probably correct.

Answering Logical Reasoning Problems

• For example: "A cell phone has fallen down a one foot (30cm) hole. How do you retrieve it? You have a wheel of cheese, three chicken feathers, and a flute." The question is designed to get you thinking about how to use bizarre objects in a creative way, but consider each word and you'll notice the hole is shallow enough to reach down and pick up the cell phone.

• For instance, "A wind is blowing from the east, but you are facing the south side of a tree. Which way are the leaves blowing?" If you don't stop to think, you might have heard "east wind" and automatically answer "east". However, the wind is blowing from the east, so the leaves are actually blowing west.

• For timed tests, if you cannot narrow it down to exactly one answer (or however many the instructions request), you may need to take a guess and move on. Make a note on your notepaper to go back to that question at the end if you have time.

• There are many practice tests available online for free for any major standardized school exam. If you can't find your exact exam, search for practice logic tests that match your education level.

• If the question doesn't give you enough information, make an assumption or estimate and state it clearly. For instance, say "Let's say the skyscraper is 100 stories tall and has 20 windows on each story" or "First, I'll assume everyone is following the speed limit, and then I'll consider what changes if some people are traveling faster."

Expert Q&A

Things You'll Need

• Graph paper
• For difficult puzzles, keep track of which clue you used by putting the number of the clue in your grid instead of a circle. You may need to add numbers to each sentence of the puzzle description first if the clues do not come in a numbered list. Thanks Helpful 0 Not Helpful 0
• Some people prefer to keep the duplicate sections when setting up a graph, while others dislike having to keep the same information in two places. Thanks Helpful 0 Not Helpful 0
• If you have a spreadsheet program on your computer, you can set up your grid there using the border tool to outline the cells. Then, if you have to choose between two answers (see Step 13), you can simply copy and paste the entire 'solution so far' to another section of the spreadsheet to prove or disprove your guess. Thanks Helpful 0 Not Helpful 0

You Might Also Like

• ↑ http://www.psychometricinstitute.com.au/Psychometric-Guide/Logical-Reasoning-test.html

About This Article

To solve logic puzzles, start by rewriting the question to eliminate any unnecessary or nonsensical information. Then, make a list of important clues, such as colors, names, and words that indicate a particular order, like “before” and “after.” If the puzzle is a multiple choice question, check each answer to see if it contradicts something in the question, or if the answer can’t be deduced from the given information. Afterwards, if you’re still stuck, reread the puzzle to see if you’ve missed any clues. To learn more, including how to solve logic puzzles using a grid, scroll down. Did this summary help you? Yes No

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Solving Propositional Logic Word Problem

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Propositional logic is a formal language that treats propositions as atomic units. A typical propositional logic word problem is as follows:

A, B, C, D are quarreling quadruplets. If A goes to the party, then B will not go. If C goes to the party, then B will not go. What is the largest possible number that will go to the party?

Logic is the study of valid reasoning. It is applied not only in studies but also in our day-to-day lives. Using simple logical reasoning and deduction, we are able to obtain information from a certain premise. Likewise, we can also verify or disprove statements.

We will be discussing ways of identifying common mistakes and how to avoid them. We will also talk about different proof techniques, such as using Venn diagrams and analogies so that you have a toolkit for solving logic word problems.

What's wrong with this proof?

Proof by venn diagram, proof by analogy.

In this section, we will use familiar notations used in propositional logic. You might want to familiarize yourself with Propositional Logic first.

Like solving any other questions, we should always ask ourselves what we can and can't do when writing out our reasoning. The first step to learn how to solve propositional logic problems is to list out what can't be done or what is not a possibility so we can narrow down what the possible scenarios are. Remember that it is very easy to fall into an erroneous conclusion based on faulty reasoning. Take the statements below as an example, if the first statement is true, is the second statement also true?

\[\begin{array} &\text{"If it's raining, then I can't play soccer".} &\text{"If I can't play soccer, then it's raining."} \end{array}\]

It is pretty clear that the issue here is: there can be other reasons why I can't play soccer, which doesn't necessarily depend on the weather. If we make such simple errors in reasoning when the context is very clear, just imagine what will happen when you are less certain about statements that are more vague. In the next paragraph, we will be introduced to these errors.

Converse and Inverse Errors

As a beginner, the most common mistake you can make is to assume that the converse and/or inverse of the original statement is also true. Take a look at the two sections below:

Introduction to Converse Error with erroneous reasoning:

Premise : If it's raining, then I can't play soccer. Conclusion : If I can't play soccer, then it's raining. Explanation : From the first statement, we are given a condition and a result: "raining" as a condition and "I can't play soccer" as a result. The entire premise is phrased in such a way that if the condition is fulfilled, then the result will occur. However, the conclusion shows that if the result is fulfilled, then the condition will occur. This does not make sense because it is not necessary for the condition to take place if the result occurs first. This is known as a converse error. In a general form, the argument for a converse error is as follows: If P occurs, then Q occurs. Q occurs. Therefore, P also occurs.

Introduction to Inverse Error with erroneous reasoning:

Premise : If it's raining, then I can't play soccer. Conclusion : If it's not raining, then I can play soccer. Explanation : From the first statement, we are given a condition and a result: "raining" as a condition and "I can't play soccer" as a result. The entire premise is phrased in such a way that if the condition is fulfilled, then the result will occur. However, the conclusion shows that if the condition does not occur, then the result does not occur either. This does not make sense because there can be other reasons/factors such that the result does occur. This is known as an inverse error. In a general form, the argument for an inverse error is as follows: If P occurs, then Q occurs. P does not occur. Therefore, Q also does not occur.

It may now be abundantly clear that it is easy to identify we've made an erroneous reasoning. However, what if the statements given appear more vague? This is the reason why we introduce the two errors above (converse error and inverse error) to show that not all wrong statements are easily identifiable. Simply put, the relationship between two events do not necessarily imply that one causes the other. In short, we are pointing out the common fact that "correlation does not imply causation".

Now that we have seen these mistakes first hand, let's do another example to remind ourselves that they are mistakes and we can hopefully avoid them in the future. Keep in mind that some of the converse/inverse statements can appear ridiculous but some do not.

We are given the following statement: If today is Sunday, then the weather is sunny. \[\] \(\qquad \text{ (i)}\) Write the inverse and converse of this statement. \(\qquad \text{(ii)}\) Identify which of these statements you have made is not logical and explain why. \(\text{(i)}\) Inverse and Converse Inverse: If today is not Sunday, then the weather is not sunny. Converse: If the weather is sunny, then today is Sunday. \(\text{(ii)}\) Logical or Illogical Though they are the inverse and converse of the original statement, we must keep in mind that they might not necessarily be an error. However, there is no harm in checking whether they are correct or not. The inverse statement implies that the day has a direct relation on the weather being sunny or not, which is ludicrous because there can also be non-sunny days which do not fall on a Sunday. The converse statement implies that only if the weather is sunny then the day is Sunday, which is also ludicrous because they can also have a sunny weather on days not falling on a Sunday. \(_\square\)

Pinpoint the exact error

Now that we can identify how the errors occur, let's take a step further and apply these techniques so that we can pinpoint exactly where the error occurs. Note that the easiest way to identify where the error arose is to convert logical statements into symbolic forms (like P implies Q). Let's try the following example.

Taking the long view on your education, you go to the Prestige Corporation and ask what you should do in college to be hired when you graduate. The personnel director replies that you will be hired only if you major in mathematics or computer science, get a \(\text{B}^\text{+}\) average or better, and take accounting. You do, in fact, become a math major, get a \(\text{B}\) average and take accounting. You return to Prestige Corporation, make a formal application, and are turned down. Did the personnel lie to you? Let's list down the requirements to be hired: \(\begin{array}{r r l} & \text{(i)} & \text{Major in mathematics or computer science}\\ & \text{(ii)} & \text{Get a } \text{B}^\text{+} \text{ average or better}\\ & \text{(iii)} & \text{Take accounting}\\ \end{array}\) Since you became a math major, criteria \(\text{(i)}\) is satisfied. Since you got a \(\text{B}\) average instead of a \(\text{B}^\text{+}\) average, criteria \(\text{(ii)}\) is not satisfied. Since you took accounting, criteria \(\text{(iii)}\) is satisfied. Since you did not satisfy all the criteria and were turned down, the personnel didn't lie to you. \(_\square\)

Now that you're familiar with writing out these statements and identifying possible errors, let's try another example that uses such a property!

A store has been raided by looter/s, who drove away in a car. Three well-known criminals Satvik, Krishna and Sharky are brought to the police station for questioning. Inspector Aditya of the police extracts the following facts:

\((1)\) None other than Satvik, Krishna and Sharky was involved in the robbery. \((2)\) Sharky never does a job without using Satvik (and possibly others) as accomplices. \((3)\) Krishna doesn't know how to drive.

Find the person who, in any case, is guilty.

This problem is the part of my set "Is This What You Call Logic?"

Formal terminologies.

In the previous sections, we have learned the two most common errors that students will make when solving a logical reasoning problem. However, we did not formally touch on the terminologies for those terms: converse error and inverse error. Let's begin!

Contrapositive : A statement is logically equivalent to its contrapositive. The contrapositive negates both terms in an implication and switches their positions. For example, the contrapositive of "P implies Q" is the negation of Q implies the negation of P.

Converse : The converse switches the positions of the terms. The converse of "P implies Q" is "Q implies P".

"If and only if", sometimes written as iff and known as equivalence, is implication that works in both directions. "P if and only if Q" means that both "P implies Q" and "Q implies P".

Let's try a few examples that cover this area!

\(\text{ (i)}\) Write down the contrapositive statement for \[\text{"If you are human, then you have DNA."}\] \(\text{(ii)}\) Write down the two if-then statements for \[\text{"A polygon is a quadrilateral if and only if the polygon has 4 sides."}\] \(\text{ (i)}\) contrapositive \(\qquad\) If you do not have DNA, then you are not human. \(\text{(ii)}\) if-then statements \(\qquad\) If a polygon is a quadrilateral, then it has 4 sides. \(\qquad\) If a polygon has 4 sides, then it is a quadrilateral. \(_\square\)

Simple, isn't it? Let's try some problems that apply the techniques we have learned above.

If Jeff spends 5 hours playing video games, then he cannot finish his math homework.

If Jeff finishes his math homework, then he will do well on his next math test.

Based on this information, which of the following is logically correct?

Selena, Jennifer and Miley wear a blue dress, yellow dress, and green dress in an unknown order. It is known that:

1) If Selena wears blue, then Jennifer wears green. 2) If Selena wears yellow, then Miley wears green. 3) If Jennifer does not wear yellow, then Miley wears blue.

What is the color of the dress Selena is wearing?

Now that we have mastered these techniques, let's move on to the following section for other cool proof techniques!

In this section, we will be applying some basic rules of set notations. You might want to familiarize yourself with sets and Venn diagram first.

In the previous section, we have learned the most common ways in identifying and pinpointing errors. In this section, we will apply the use of Venn Diagram as an alternative proof in solving logical reasoning problems. But what's the benefit of this? Well, it's simple: We do not need to verbalize these statements and we can use the visual aids to guide us to solve these problems.

Recap of set notations and Venn diagram \[\]

Let's do a brief recap for the application of Venn Diagrams by taking the following as an explicit example:

Consider \(W,X,Y,Z\) as sets, each with their own elements in them. Then by interpreting the Venn Diagram, we can obtain information like:

• All elements in set \(W\) is in set \(Y\).
• All elements in set \(X\) is in set \(Z\).
• Not all elements in set \(Y\) is in set \(W\).

How to use Venn diagrams to solve a logical word problem \[\]

Let us consider the following statements and deduce whether the conclusion is true or false by Venn diagram:

True or false? \[\] It is given that all birds have wings. All chickens are birds. Therefore, all chickens have wings. Explanation : By Venn diagram, the statement "A chicken is a bird." implies that the set "all chickens" is a subset of "all birds." Thus we can say that all chickens have the same characteristics as a bird. Because it is given that all birds have wings (a characteristic), all chickens have wings too. Thus the conclusion is correct. Note : We should keep in mind that this only works if the premise is true. For example, if we replace the word "wings" by "forearms" in the first statement, then the conclusion of "All chickens have forearms." will inevitably be true despite its ridiculous claim.

Food for thought : If all phones have batteries and I have a phone, does it mean that my phone has a battery?

Careful! There are other ways of drawing out Venn diagrams!

Though it may appear very simple to set up a Venn diagram, the setup may not necessarily be unique. Let us consider a revised version of the statements above and deduce whether the conclusion is true or false by Venn diagram.

True or false? \[\] It is given that all birds have wings. All chickens are birds. Therefore, all birds are chickens. Explanation : By Venn diagram, the statement "A chicken is a bird" implies that the set "all chickens" is a subset of "all birds." Thus we can say that all chickens have the same characteristics as a bird. However, it is not necessarily true that all birds share the same characteristics of a chicken. (Sounds familiar? It's converse error.) So the claim "All birds are chicken" must be false. Note : To fix the conclusion, you should say " Some birds are chickens" instead of " All birds are chickens."

Now that we know the fundamental applications of proof by Venn diagram, let's apply these knowledge we learned on the following examples:

True or False?

\(\quad\) All pangs are pings. \(\quad\) Some pings are pongs. \(\quad\) Therefore, some pangs are pongs.

Johannes has several written publications on his bookshelf. Albert notices that all of the comics are paperback books, and that some of the paperback books are manga. Are all comics manga?

Image Credit : Wikimedia Johannes Jansson

Isn't the proof by Venn diagram fun? You don't need to use actual words to formalize these statements. Looks very unusual, right? But it works. Speaking of unusual, is it possible to solve these logical statements if we were to spice things up by dramatizing out the statements? Yes, we can! Proof by analogy is another proof technique to solve logical problems. See the following section:

How do we solve texts that are seemingly hard to decipher?

All pangs are pings. Some pings are pongs. Therefore some pangs are pongs.

Consider the logical statements given above. Since we can't relate to or identify what pangs, pings, or pongs are, it will appear that these terms are vague or all too similar. How are we supposed to solve problems like this if we have little to no clue to what is going on? Well, proof by analogy will be useful here: this is when we dramatize or caricaturize the terms used.

For example, we may call pangs as humans, pings as apes, and pongs as gorillas. With these new terms, we are able to visualize what they are. Rewriting them into the original 3 statements shows that

\[\text{All humans are apes. Some apes are gorillas. Therefore some humans are gorillas.}\]

So the given conclusion is wrong because of the ridiculousness of the conclusion "Some humans are gorillas."

However, an important question to ask is why this works. This is too good to be true, right? Or, are we running into some wrong argument? Why does this work?

Explanation of how this works

The reason why proof by analogy works is because we make an inference that if the objects have multiple similar characteristics, and it is given that you know one of them have an extra characteristics (call it X), then it is not a bad inference to conclude that the other object shares that same characteristic X.

To put it short, the generalized/structured form for proof by analogy is:

P and Q has similar properties \(x_1, x_2, x_3, \ldots, x_n\). We know that P has a further property \(y\). Therefore, Q probably has property \(y\) too.

Now let's try a modified version of the ping-pang-pong question from earlier!

True or false? \[\] \(\qquad\) All yangs are yengs and yings. \(\qquad\) Some yengs are yings. \(\qquad\) Then, all yengs are yangs. This is false. Let "yangs" be defined as "pets", "yengs" as "tigers", and "yings" as "cats". So it is true (or at least still reasonable) that all pets are cats and tigers and that some tigers are cats. But it is not true that all tigers are pets. \(_\square\) Note: The reason why proof by analogy works best here is because we couldn't label or identify any characteristics for yangs, yengs, and yings. Therefore, a sensible approach is to prove by analogy.

Now that you're ready to solve logical problems by analogy, let's try to solve the following problem again, but this time by analogy!

Suitability of analogy

Notice from the previous section that we've mentioned that "Q probably has property \(y\) too." instead of "Q definitely has a property \(y\) too." This is because the argument may provide what appears to be the right evidence, but the conclusion does not always follow. This subsection explains why this proof (arguemnt) might not always work.

Though it is true that we highlight or amplify parallel characteristics, the differences between things can often overwhelm their similiarities. One might note that it is always possible to extend an analogy to the point of absurdity. For illustration, take the following famous "Information Argument":

DNA is a code. A code requires an intelligence. Therefore, DNA comes from an intelligence.

Yes, this will sound completely logical if we apply the implications approach. That is, it's P implies Q and Q implies R, so P implies R. However, the argument here is not valid because the statement "DNA is a code." is purely an analogy and thus it is not an entirely accurate statement to begin with. Thus, we have started with a wrong premise. So the merits of analogy do not hold. This is further explained in Analyzing arguments from analogy .

We can see that proof by analogy is very useful and can also be used to make incorrect conclusions. Thus, one must be careful in labeling certain characteristics when using this method. Let's see the following examples to see how the proof by analogy backfires:

True or false? \[\] All squares and rectangless are convex, have four sides and form right angles at their vertices. All squares have sides of the same length. Therefore, all rectangles have sides of the same length. This is obviously false because by definition, all rectangles do not have sides of the same length, but only squares have sides of the same length. We make the wrong conclusion that rectangles also have this characteristic because it is known previously that both share a number of characteristics. \(_\square\)

True or false?

It is given that Amy, Bernadette and Penny are good friends of Sheldon and Leonard.

Leslie is a good friend of Leonard.

Therefore, Leslie is a friend of Sheldon as well.

Image Credit: Wikimedia TBBT. No copyright infringement intended.

• Propositional Logic Using Algebra .
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Math and Logic Puzzles

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Whosoever shall solve these puzzles shall Rule The Universe!

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20 Tough Riddles for Adults That Will Have You Scratching Your Head

Put your logic and math skills to the test. No cheating!

So go grab a pencil and a piece of scratch paper and prepare to rip your hair out (and we really do mean that in the best way possible). When you think you’ve got the right answer, click the link at the bottom of each riddle to find the solution. Got it wrong? No worries, you have 19 other riddles to test out.

Navigate Through Our Riddles:

Puzzmo / The King’s Orders / How Many Eggs? / The Gold Chain / Pickleball / Circuit Breaker / Two Trains, Two Grandmas / Ant Math / Peppermint Patty / Great American Rail Trail / A Cruel SAT Problem / Movie Stars Cross a River / Tribute to a Math Genius / One Belt, One Earth / Elbow Tapping / Whiskey Problem / Doodle Problem / Stumping Scientists / What ’ s On Her Forehead? / Keanu for President / Who Opened the Lockers?

Riddle No. 1: The King’s Orders Make for One Hell of a Brain Teaser

Difficulty: easy.

King Nupe of the kingdom Catan dotes on his two daughters so much that he decides the kingdom would be better off with more girls than boys, and he makes the following decree: All child-bearing couples must continue to bear children until they have a daughter!

But to avoid overpopulation, he makes an additional decree: All child-bearing couples will stop having children once they have a daughter! His subjects immediately begin following his orders.

After many years, what’s the expected ratio of girls to boys in Catan?

The likelihood of each baby born being a girl is, of course, 50 percent.

Ready for the solution? Click here to see if you’re right .

Riddle No. 2: How Many Eggs Does This Hen Lay?

This problem is in honor of my dad, Harold Feiveson. It’s due to him that I love math puzzles, and this is one of the first problems (of many) that he gave me when I was growing up.

A hen and a half lays an egg and a half in a day and a half. How many eggs does one hen lay in one day?

Riddle No. 3: The Gold Chain Math Problem Is Deceptively Simple

Difficulty: moderate.

You’re rummaging around your great grandmother’s attic when you find five short chains each made of four gold links. It occurs to you that if you combined them all into one big loop of 20 links, you’d have an incredible necklace. So you bring it into a jeweler, who tells you the cost of making the necklace will be $10 for each gold link that she has to break and then reseal.

How much will it cost?

Riddle No. 4: Try to Solve This Pickleball Puzzle

Difficulty: 🚨hard🚨.

Kenny, Abby, and Ned got together for a round-robin pickleball tournament, where, as usual, the winner stays on after each game to play the person who sat out that game. At the end of their pickleball afternoon, Abby is exhausted, having played the last seven straight games. Kenny, who is less winded, tallies up the games played:

Kenny played eight games

Abby played 12 games

Ned played 14 games

Who won the fourth game against whom?

How many total games were played?

Riddle No. 5: Our Circuit Breaker Riddle Is Pure Evil. Sorry.

The circuit breaker box in your new house is in an inconvenient corner of your basement. To your chagrin, you discover none of the 100 circuit breakers is labeled, and you face the daunting prospect of matching each circuit breaker to its respective light. (Suppose each circuit breaker maps to only one light.)

To start with, you switch all 100 lights in the house to “on,” and then you head down to your basement to begin the onerous mapping process. On every trip to your basement, you can switch any number of circuit breakers on or off. You can then roam the hallways of your house to discover which lights are on and which are off.

What is the minimum number of trips you need to make to the basement to map every circuit breaker to every light?

The solution does not involve either switching on or off the light switches in your house or feeling how hot the lightbulbs are. You might want to try solving for the case of 10 unlabeled circuit breakers first.

Riddle No. 6: Two Trains. Two Grandmas. Can You Solve This Tricky Math Riddle?

Jesse’s two grandmothers want to see him every weekend, but they live on opposite sides of town. As a compromise, he tells them that every Sunday, he’ll head to the subway station nearest to his apartment at a random time of the day and will hop on the next train that arrives.

If it happens to be the train traveling north, he’ll visit his Grandma Erica uptown, and if it happens to be the train traveling south, he’ll visit his Grandma Cara downtown. Both of his grandmothers are okay with this plan, since they know both the northbound and southbound trains run every 20 minutes.

But after a few months of doing this, Grandma Cara complains that she sees him only one out of five Sundays. Jesse promises he’s indeed heading to the station at a random time each day. How can this be?

The trains always arrive at their scheduled times.

Riddle No. 7: Here’s a Really F@*#ing Hard Math Problem About Ants

Max and Rose are ant siblings. They love to race each other, but always tie, since they actually crawl at the exact same speed. So they decide to create a race where one of them (hopefully) will win.

For this race, each of them will start at the bottom corner of a cuboid, and then crawl as fast as they can to reach a crumb at the opposite corner. The measurements of their cuboids are as pictured:

If they both take the shortest possible route to reach their crumb, who will reach their crumb first? (Don’t forget they’re ants, so of course they can climb anywhere on the edges or surface of the cuboid.)

Remember: Think outside the box.

Riddle No. 8: This Peppermint Patty Riddle Is Practically Impossible

You’re facing your friend, Caryn, in a “candy-off,” which works as follows: There’s a pile of 100 caramels and one peppermint patty. You and Caryn will go back and forth taking at least one and no more than five caramels from the candy pile in each turn. The person who removes the last caramel will also get the peppermint patty. And you love peppermint patties.

Suppose Caryn lets you decide who goes first. Who should you choose in order to make sure you win the peppermint patty?

First, solve for a pile of 10 caramels.

Riddle No. 9: Can You Solve the Great American Rail-Trail Riddle?

This problem was suggested by the physicist P. Jeffrey Ungar.

Finally, the Great American Rail-Trail across the whole country is complete! Go ahead, pat yourself on the back—you’ve just installed the longest handrail in the history of the world, with 4,000 miles from beginning to end. But just after the opening ceremony, your assistant reminds you that the metal you used for the handrail expands slightly in summer, so that its length will increase by one inch in total.

“Ha!” you say, “One inch in a 4,000 mile handrail? That’s nothing!” But … are you right?

Let’s suppose when the handrail expands, it buckles upward at its weakest point, which is in the center. How much higher will pedestrians in the middle of the country have to reach in summer to grab the handrail? That is, in the figure below, what is h ? (For the purposes of this question, ignore the curvature of the Earth and assume the trail is a straight line.)

Pythagoras is a fascinating historical figure.

Riddle No. 10: This Riddle Is Like an Especially Cruel SAT Problem. Can You Find the Answer?

Amanda lives with her teenage son, Matt, in the countryside—a car ride away from Matt’s school. Every afternoon, Amanda leaves the house at the same time, drives to the school at a constant speed, picks Matt up exactly when his chess club ends at 5 p.m., and then they immediately return home together at the same constant speed. But one day, Matt isn’t feeling well, so he leaves chess practice early and starts to head home on his portable scooter.

After Matt has been scooting for an hour, Amanda comes across him in her car (on her usual route to pick him up), and they return together, arriving home 40 minutes earlier than they usually do. How much chess practice did Matt miss?

Consider the case where Amanda meets Matt exactly as she’s leaving their house.

Riddle No. 11: Can You Get These 3 Movie Stars Across the River?

Three movie stars, Chloe, Lexa, and Jon, are filming a movie in the Amazon. They’re very famous and very high-maintenance, so their agents are always with them. One day, after filming a scene deep in the rainforest, the three actors and their agents decide to head back to home base by foot. Suddenly, they come to a large river.

On the riverbank, they find a small rowboat, but it’s only big enough to hold two of them at one time. The catch? None of the agents are comfortable leaving their movie star with any other agents if they’re not there as well. They don’t trust that the other agents won’t try to poach their star.

For example, Chloe’s agent is okay if Chloe and Lexa are alone in the boat or on one of the riverbanks, but definitely not okay if Lexa’s agent is also with them. So how can they all get across the river?

There isn’t just one way to solve this problem.

Riddle No. 12: This Ludicrously Hard Riddle Is Our Tribute to a Late Math Genius. Can You Figure It Out?

On April 11, John Horton Conway , a brilliant mathematician who had an intense and playful love of puzzles and games, died of complications from COVID-19. Conway is the inventor of one of my favorite legendary problems (not for the faint of heart) and, famously, the Game of Life . I created this problem in his honor.

Carol was creating a family tree, but had trouble tracking down her mother’s birthdate. The only clue she found was a letter written from her grandfather to her grandmother on the day her mother was born. Unfortunately, some of the characters were smudged out, represented here with a “___” . (The length of the line does not reflect the number of smudged characters.)

“Dear Virginia,

Little did I know when I headed to work this Monday morning, that by evening we would have a beautiful baby girl. And on our wedding anniversary, no less! It makes me think back to that incredible weekend day, J___ 27th, 19___ , when we first shared our vow to create a family together, and, well, here we are! Happy eighth anniversary, my love.

Love, Edwin”

The question: When was Carol’s mother born?

This problem is inspired by Conway’s Doomsday Rule .

Riddle No. 13: To Solve This Twisty Math Riddle, You Just Need One Belt and One Earth

Imagine you have a very long belt. Well, extremely long, really … in fact, it’s just long enough that it can wrap snugly around the circumference of our entire planet. (For the sake of simplicity, let’s suppose Earth is perfectly round, with no mountains, oceans, or other barriers in the way of the belt.)

Naturally, you’re very proud of your belt. But then your brother, Peter, shows up—and to your disgruntlement, he produces a belt that’s just a bit longer than yours. He brags his belt is longer by exactly his height: 6 feet.

If Peter were also to wrap his belt around the circumference of Earth, how far above the surface could he suspend the belt if he pulled it tautly and uniformly?

Earth’s circumference is about 25,000 miles, or 130 million feet … but you don’t need to know that to solve this problem.

Riddle No. 14: This Elbow Tapping Riddle Is Diabolical. Good Luck Solving It.

In some future time, when the shelter-in-place bans are lifted, a married couple, Florian and Julia, head over to a bar to celebrate their newfound freedom.

They find four other couples there who had the same idea.

Eager for social contact, every person in the five couples enthusiastically taps elbows (the new handshake) with each person they haven’t yet met .

It actually turns out many of the people had known each other prior, so when Julia asks everyone how many elbows they each tapped, she remarkably gets nine different answers!

The question: How many elbows did Florian tap?

What nine answers did Julia hear?

Riddle No. 15: You’ll Need a Drink After Trying to Solve This Whisky Riddle

Alan and Claire live by the old Scottish saying, “Never have whisky without water, nor water without whisky!” So one day, when Alan has in front of him a glass of whisky, and Claire has in front of her a same-sized glass of water, Alan takes a spoonful of his whisky and puts it in Claire’s water. Claire stirs her whisky-tinted water, and then puts a spoonful of this mixture back into Alan’s whisky to make sure they have exactly the same amount to drink.

So: Is there more water in Alan’s whisky, or more whisky in Claire’s water? And does it matter how well Claire stirred?

The size of the spoon does not matter.

Riddle No. 16: The Doodle Problem Is a Lot Harder Than It Looks. Can You Solve It?

This week’s riddle is relatively simple—but sinister all the same.

The question: Can you make 100 by interspersing any number of pluses and minuses within the string of digits 9 8 7 6 5 4 3 2 1? You can’t change the order of the digits! So what’s the least number of pluses and minuses needed to make 100?

For instance, 98 - 7 - 6 + 54 - 32 shows one way of interspersing pluses and minuses, but since it equals 107, it’s not a solution.

I call this a “doodle problem”: one that’s best worked on during meetings where you might be doodling otherwise.

You might want to start looking for solutions that use a total of seven pluses and minuses (although there are ways to use fewer).

Ready for the solution? Click here to see if you’re right.

Riddle No. 17: This Math Puzzle Stumped Every Scientist but One. Think You Can Crack It?

Difficulty: hard.

In honor of Freeman Dyson, the renowned physicist who died last month , here’s a legendary tale demonstrating his quick wit and incredible brain power.

One day, in a gathering of top scientists, one of them wondered out loud whether there exists an integer that you could exactly double by moving its last digit to its front. For instance, 265 would satisfy this if 526 were its exact double—which it isn’t.

After apparently just five seconds , Dyson responded, “Of course there is, but the smallest such number has 18 digits.”

This left some of the smartest scientists in the world puzzling over how he could have figured this out so quickly.

So given Dyson’s hint, what is the smallest such number?

My second grader has recently learned how to add a 3-digit number to itself using the classic vertical method:

18-digit numbers, of course, can be added in the same way.

Riddle No. 18: Figure Out What’s on Her Forehead

Cecilia loves testing the logic of her very logical friends Jaya, Julian, and Levi, so she announces:

“I’ll write a positive number on each of your foreheads. None of the numbers are the same, and two of the numbers add up to the third.”

She scribbles the numbers on their heads, then turns to Jaya and asks her what her number is. Jaya sees Julian has 20 on his forehead, and Levi has 30 on his. She thinks for a moment and then says, “I don’t know what my number is.” Julian pipes in, “I also don’t know my number,” and then Levi exclaims, “Me neither!” Cecilia gleefully says, “I’ve finally stumped you guys!”

“Not so fast!” Jaya says. “Now I know my number!”

What is Jaya’s number?

Jaya could be one of two numbers, but only one of those numbers would lead to Julian and Levi both not knowing their numbers. Why?

Riddle No. 19: Can You Get Keanu Reeves Elected As President?

It’s 2024, and there are five candidates running in the democratic primary: Taylor Swift, Oprah Winfrey, Mark Cuban, Keanu Reeves, and Dwayne Johnson. (Hey, it could happen.) As usual, the first primary is in Iowa.

In an effort to overcome its embarrassment after the 2020 caucus debacle , the Iowa Democratic Party has just announced a new, foolproof way of finding the best candidate: there will be four consecutive elections.

First, candidate 1 will run against candidate 2. Next, the winner of that will run against candidate 3, then that winner will run against candidate 4, and finally the winner of that election will run against the final candidate. By the transitive property, the winner of this last election must be the best candidate ... so says the Iowa Democratic Party.

Candidate Keanu has been feeling pretty low, as he knows he is ranked near the bottom by most voters, and at the top by none. In fact, he knows the Iowa population is divided into five equal groups, and that their preferences are as follows:

Keanu is childhood friends with Bill S. Preston, Esq., the new head of the Iowa Democratic Party. Preston, confident that the order of the candidates doesn’t matter for the outcome, tells Keanu he can choose the voting order of the candidates.

So what order should Keanu choose?

How would Keanu fare in one-to-one races against each candidate?

Riddle No. 20: Who Opened All These Damn Lockers?

There are 100 lockers that line the main hallway of Chelm High School. Every night, the school principal makes sure all the lockers are closed so that there will be an orderly start to the next day. One day, 100 mischievous students decide that they will play a prank.

The students all meet before school starts and line up. The first student then walks down the hallway, and opens every locker. The next student follows by closing every other locker (starting at the second locker). Student 3 then goes to every third locker (starting with the third) and opens it if it’s closed, and closes it if it’s open. Student 4 follows by opening every fourth locker if it’s closed and closing it if it’s open. This goes on and on until Student 100 finally goes to the hundredth locker. When the principal arrives later in the morning, which lockers does she find open?

Make sure you pay attention to all of the factors.

Laura Feiveson is an economist for the government, a storyteller, and a lifelong enthusiast of math puzzles.  She lives in Washington, DC with her husband and two daughters. 

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Making Sense of Logic Puzzles

While the world of logic puzzles is vast, finding your niche can make you a stronger solver.

By Isaac Aronow

The rich and complex philosophical concept of logic is the engine that drives an uncountable number of puzzles and games. Sudoku, at least in part, uses the same type of deductive reasoning that drove Aristotle to write, “All men are mortal. Socrates is a man. Therefore, Socrates is mortal.”

For our purposes, let’s loosely define a logic puzzle as one that offers all the information a solver needs to deduce a solution that must be true. A traditional logic puzzle won’t require any outside knowledge, such as trivia or words in a foreign language, for someone to arrive at the solution. As Wyna Liu, a puzzle editor at The New York Times, said, “You don’t need to have any experience as a detective to know how to play Clue.”

While many other games have elements of logic, such as crossword puzzles and Connections , you might not be able to solve them without knowing, for example, the ingredients of a mojito or that Ottawa is the capital of Canada. Ms. Liu, who creates the Connections boards, said that, to her, those games are “word puzzles that incorporate logic elements,” like pattern recognition and sorting.

For many, logic puzzles are just a way to pass the time on their commutes, but for others, they are a lifelong pursuit. The global community of dedicated puzzlers is small but tightknit.

India, for example, has a very devoted community of puzzle experts. Prasanna Seshadri, an Indian Puzzle Championship winner who helps run Logic Masters India, said, “We have a thriving community and a website where we hold regular competitions,” though the group is still small and has struggled to break into India’s larger puzzle culture, he said.

Logic puzzles are also used as a benchmark in standardized testing. The Law School Admission Test, or LSAT, for example, uses logic to determine a student’s potential for success in legal education and practice. Since 1982, the test has contained an analytical reasoning section, colloquially known as logic games.

Mark Murray, a spokesman for the Law School Admission Council, which administers the LSAT, said that the test is so focused on logic because “making a logical argument is a fundamental part of the legal system.”

Look for Specific Information First

To start solving logic puzzles, it’s important to remember that all the information you need is inside the puzzle itself. Sometimes it will be specific information, such as a number in a Sudoku square or a hint like “Knights always tell the truth.”

Ms. Liu said, “There is something very comforting about having all the information encapsulated in front of you,” adding, “It’s not a ‘You know it or you don’t’.”

It also helps to know what a piece of information eliminates. If Professor Plum has to have committed the murder, then none of the other characters would have been able to do it. If there’s a number already placed in a Sudoku grid, then another of that number can’t be placed in the same row or column.

Because of the complete information within a logic puzzle, there’s never a need to guess randomly. While the difficulty level of uncovering that information changes with the amount given, there’s nearly always something that a solver can hook into to make progress.

It can also be helpful to take notes. Each type of puzzle has its own best practices for notation, but ultimately the method you choose is deeply personal. What works best for you may not work best for someone else, so try a few different ways for each puzzle.

Every Puzzle Type is Different

There are many different types of logic puzzles, all with their own strategies, styles and variants. Sudoku rises above many others in terms of popularity among puzzle fans, but the number of puzzles and variants out there seems endless. Fortunately, for every puzzle type examined for this article, there are at least a few resources, online or in print, to help you find a strategy that works for you.

If you’re having a hard time with a specific type of puzzle, don’t let that sour you on others. Mr. Seshadri said in an email that, because there are so many different types of logic puzzles, “I would tell anyone to give a bunch of different genres a go before deciding if it’s for them.”

Mr. Seshadri is known in the Indian puzzle community as a Sudoku specialist, but, he said in an email, “Truthfully, Sudoku is not one of my favorite puzzle genres, I’m more fond of the ones that involve drawing loops or shading cells.” He added that if solvers don’t like Sudoku, they should try other puzzle types, such as Slitherlink, a puzzle where a solver forms a loop by connecting dots, or Nurikabe, where the solver shades cells to form “islands” in a grid.

Trying different types of puzzles can also help you improve your overall solving skills. Tammy McLeod, a U.S. Sudoku championship winner, said, “To get better at puzzles in general, there’s no substitute for the experience one would get from trying various types.”

Just Jump In

Many puzzles have consistent community support. The World Puzzle Federation runs online contests for solvers all over the world. Logic Masters India, a subsidiary of the World Puzzle Federation that helped with the Puzzle Mania spread in The New York Times this year, regularly hosts beginner contests that anyone is allowed to join, in addition to their more advanced offerings.

If you want information on a specific type of puzzle, seeking out that puzzle’s community can be a great way to gain knowledge, though not always necessary. Ms. McLeod said that after trying many different types of puzzles, “I have discovered strategies on my own that let me finish puzzles a lot faster,” adding, “Certainly I could have learned them from someone else, but I haven’t actually researched to see if anyone is sharing techniques for those puzzle types.”

You may not immediately recognize exactly how much you’re improving at solving puzzles, but if you keep practicing, you will be rewarded for your efforts. Ms. Liu said that she had been getting back into Sudoku lately and that after solving consistently for a few months, “I couldn’t tell you what the patterns are, but I can tell you that I’m better at it.”

Isaac Aronow is an editor on the Games team. More about Isaac Aronow

It’s Game Time!

Take your puzzling skills in new directions..

WordleBot , our daily Wordle companion that tells you how skillful or lucky you are, is getting an upgrade. Here’s what to know .

The editor of Connections , our new game about finding common threads between words, talks about how she makes this daily puzzle feel fun .

We asked some of the best Sudoku  solvers in the world for their tips and tricks. Try them to  tackle even the most challenging puzzles.

Read today’s Wordle Review , and get insights on the game from our columnists.

We asked Times readers how they play Spelling Bee. The hive mind weighed in with their favorite tips and tricks .

Ready to play? Try Wordle , Spelling Bee  or The Crossword .

LOGIC PUZZLES

Test your ingenuity and knowledge with these logic puzzles. Deduction is the key so get your thinking cap dusted off and start logically working through our puzzling content. Sharpen Your Mind and Put Your Problem Solving Skills to Work.

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You'll like these riddle puzzles on days that end with Y. You'll like these logic puzzles are available on days that end with Y. How many days can you try out our logic puzzles? Answer: Every day!

You Walk Up To A Mountain That Has Two Paths

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Little tommy tittletat riddle.

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Math equation riddle.

Which One Does Not Belong?

If you have two coins which total 35 cents and one of the coins is not a dim, i am a 5 letter word take away the first letter and i am a place's name, open me, and you can me without a mirror riddle, redo what has been undone, most feathers, what only works the first time you use it, bus stop riddle, number of chairs, what's 3/7 chicken, 2/3 cat, and 2/4 goat, i can hold you prisoner, the wise daughter riddle, if you see me, i see you. if you move, then i'll move too, search riddles, follow riddles, share riddles.

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Digtial Nonogram: color logic 4+

Solve pixel puzzle and jigsaw, designed for ipad.

• Offers In-App Purchases

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Description.

Digital Nonogram is a fun logical number game that contains a huge number of crossword puzzles. Solve the picture puzzles by following the number clues! Digital Nonogram offers thousands of puzzles of varying difficulty and categories, and is updated regularly to ensure that the game never gets boring. The game is also fun for young and old and is very easy to play. Whether you are new to number painting or already an experienced player, you will have fun with this game. Challenge your mind while solving puzzles and become a true Nonogram master! Gameplay: - Use numbers to solve puzzles. - Numbers above the columns are read from top to bottom. - Numbers to the left of the row are read from left to right. - If you find a square that should not be shaded, mark it with an “X”. - Color the squares according to the numbers and fill in the color correctly to get the final pattern. Game Features: - Graphic puzzles with multiple themes to color - Multiple difficulty puzzles - Complete the picture puzzles to get puzzle pieces and have fun with the puzzle at the same time - Daily Challenges: Solve puzzles every day and earn special trophies every month! - Hints are available if you get stuck while solving non-graphics. Number puzzle games are easy to learn and addictive once you start playing. Enjoy the endless fun of Nonogram!

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The developer, BigJoy , indicated that the app’s privacy practices may include handling of data as described below. For more information, see the developer’s privacy policy .

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The following data may be used to track you across apps and websites owned by other companies:

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The following data may be collected but it is not linked to your identity:

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Privacy practices may vary, for example, based on the features you use or your age. Learn More

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English, Simplified Chinese

• 150 Diamonds $1.99
• 6800 Golds $9.99
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• 48000 Golds $49.99
• 2800 Golds $4.99
• 15000 Golds $19.99
• 7200 Diamonds $49.99
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• 2200 Diamonds $19.99
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About this game.

• All islands must be separated by water
• The water must form one connected river
• 2x2 water squares, called "pools" , are not allowed

• Hand-crafted puzzle campaigns
• Different environments, seasons and times of day
• Relaxing visuals and music
• Infinitely generated puzzles
• Intuitive level editor
• Daily challenges
• City building
• Multiplayer

System Requirements

• Requires a 64-bit processor and operating system
• Memory: 1 GB RAM
• DirectX: Version 11
• Additional Notes: Final requirements pending
• Processor: Apple Silicon or Intel

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