how to solve cross logic puzzles

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How to Solve Logic Puzzles

Last Updated: October 8, 2023

This article was reviewed by Joseph Meyer . Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. This article has been viewed 295,216 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."

Community 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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Logical Puzzles

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• Andrew Hayes

A logical puzzle is a problem that can be solved through deductive reasoning. This page gives a summary of the types of logical puzzles one might come across and the problem-solving techniques used to solve them.

Elimination Grids

Truth tellers and liars, cryptograms, arithmetic puzzles, river crossing puzzle, tour puzzles, battleship puzzles, chess puzzles, k-level thinking, other puzzles.

Main Article: Propositional Logic See Also: Predicate Logic

One of the simplest types of logical puzzles is a syllogism . In this type of puzzle, you are given a set of statements, and you are required to determine some truth from those statements. These types of puzzles can often be solved by applying principles from propositional logic and predicate logic . The following syllogism is from Charles Lutwidge Dodgson, better known under his pen name, Lewis Carroll.

I have a dish of potatoes. The following statements are true: No potatoes of mine, that are new, have been boiled. All my potatoes in this dish are fit to eat. No unboiled potatoes of mine are fit to eat. Are there any new potatoes in this dish? The first and third statements can be connected by a transitive argument. All of the new potatoes are unboiled, and unboiled potatoes aren't fit to eat, so no new potatoes are fit to eat. The second statement can be expressed as the equivalent contrapositive. All of the potatoes in the dish are fit to eat; if there is a potato that is not fit to eat, it isn't in the dish. Then, once again, a transitive argument is applied. New potatoes aren't fit to eat, and inedible potatoes aren't in the dish. Thus, there are no new potatoes in the dish. \(_\square\)

Given below are three statements followed by three conclusions. Take the three statements to be true even if they vary from commonly known facts. Read the statements and decide which conclusions follow logically from the statements.

Statements: 1. All actors are musicians. 2. No musician is a singer. 3. Some singers are dancers.

Conclusions: 1. Some actors are singers. 2. Some dancers are actors. 3. No actor is a singer.

Answer Choices: a) Only conclusion 1 follows. b) Only conclusion 2 follows. c) Only conclusion 3 follows. d) At least 2 of the conclusions follows.

Main Article: Elimination Grids

Some logical puzzles require you to determine the correct pairings for sets of objects. These puzzles can often be solved with the process of elimination, and an elimination grid is an effective tool to apply this process.

An example of an elimination grid

Elimination grids are aligned such that each row represents an object within a set, and each column represents an object to be paired with an object from that set. Check marks and X marks are used to show which objects pair, and which objects do not pair.

Mr. and Mrs. Tan have four children--three boys and a girl-- who each like one of the colors--blue, green, red, yellow-- and one of the letters--P, Q, R, S.

• The oldest child likes the letter Q.
• The youngest child likes green.
• Alfred likes the letter S.
• Brenda has an older brother who likes R.
• The one who likes blue isn't the oldest.
• The one who likes red likes the letter P.
• Charles likes yellow.

Based on the above facts, Darius is the \(\text{__________}.\)

Main Article: Truth-Tellers and Liars

A variation on elimination puzzles is a truth-teller and liar puzzle , also known as a knights and knaves puzzle . In this type of puzzle, you are given a set of people and their respective statements, and you are also told that some of the people always tell the truth and some always lie. The goal of the puzzle is to deduce the truth from the given statements.

20\(^\text{th}\) century mathematician Raymond Smullyan popularized these types of puzzles.

You are in a room with three chests. You know at least one has treasure, and if a chest has no treasure, it contains deadly poison.

Each chest has a message on it, but all the messages are lying .

• Left chest: "The middle chest has treasure."
• Middle chest: "All these chests have treasure."
• Right chest: "Only one of these chests has treasure."

Which chests have treasure?

There are two people, A and B , each of whom is either a knight or a knave.

A says, "At least one of us is a knave."

What are A and B ?

\(\) Details and Assumptions:

• A knight always tells the truth.
• A knave always lies.

Main Article: Cryptograms

A cryptogram is a puzzle in which numerical digits in a number sentence are replaced with characters, and the goal of the puzzle is to determine the values of these characters.

\[ \begin{array} { l l l l l } & &P & P & Q \\ & &P & Q & Q \\ + && Q & Q & Q \\ \hline & & 8 & 7 & 6 \\ \end{array} \]

In the sum shown above, \(P\) and \(Q\) each represent a digit. What is the value of \(P+Q\)?

\[ \overline{EVE} \div \overline{DID} = 0. \overline{TALKTALKTALKTALK\ldots} \]

Given that \(E,V,D,I,T,A,L\) and \(K \) are distinct single digits, let \(\overline{EVE} \) and \( \overline{DID} \) be two coprime 3-digit positive integers and \(\overline{TALK} \) be a 4-digit integer, such that the equation above holds true, where the right hand side is a repeating decimal number.

Find the value of the sum \( \overline{EVE} + \overline{DID} + \overline{TALK} \).

Main Articles: Fill in the Blanks and Operator Search

Arithmetic puzzles contain a series of numbers, operations, and blanks in order, and the object of the puzzle is to fill in the blanks to obtain a desired result.

\[\huge{\Box \times \Box \Box = \Box \Box \Box}\]

Fill the boxes above with the digits \(1,2,3,4,5,6\), with no digit repeated, such that the equation is true.

Enter your answer by concatenating all digits in the order they appear. For example, if the answer is \(1 \times 23 = 456\), enter \(123456\) as your final answer.

Also try its sister problem.

\[ \LARGE{\begin{eqnarray} \boxed{\phantom0} \; + \; \boxed{\phantom0} \; &=& \; \boxed{\phantom0} \\ \boxed{\phantom0} \; - \; \boxed{\phantom0} \; &=& \; \boxed{\phantom0} \\ \boxed{\phantom0} \; \times \; \boxed{\phantom0}\; &=& \; \boxed{\phantom0} \\ \boxed{\phantom0} \; \div \; \boxed{\phantom0} \; &=& \; \boxed{\phantom0} \\ \end{eqnarray}} \]

Put one of the integers \(1, 2, \ldots , 13\) into each of the boxes, such that twelve of these numbers are used once for each (and one number is not used at all) and all four equations are true.

What is the sum of all possible values of the missing (not used) number?

Main Article: River Crossing Puzzles

In a river crossing puzzle , the goal is to find a way to move a group of people or objects across a river (or some other kind of obstacle), and to do it in the fewest amount of steps or least amount of time.

A famous river crossing problem is Richard Hovasse's bridge and torch problem , written below.

Four people come to a river in the night. There is a narrow bridge, but it can only hold two people at a time. They have one torch and, because it's night, the torch has to be used when crossing the bridge. Person A can cross the bridge in one minute, B in two minutes, C in five minutes, and D in eight minutes. When two people cross the bridge together, they must move at the slower person's pace. The question is, can they all get across the bridge in 15 minutes or less? Assume that a solution minimizes the total number of crosses. This gives a total of five crosses--three pair crosses and two solo crosses. Also, assume we always choose the fastest for the solo cross. First, we show that if the two slowest persons (C and D) cross separately, they accumulate a total crossing time of 15. This is done by taking persons A, C, D: D+A+C+A = 8+1+5+1=15. (Here we use A because we know that using A to cross both C and D separately is the most efficient.) But, the time has elapsed and persons A and B are still on the starting side of the bridge and must cross. So it is not possible for the two slowest (C and D) to cross separately. Second, we show that in order for C and D to cross together that they need to cross on the second pair cross: i.e. not C or D, so A and B, must cross together first. Remember our assumption at the beginning states that we should minimize crosses, so we have five crosses--3 pair crossings and 2 single crossings. Assume that C and D cross first. But then C or D must cross back to bring the torch to the other side, so whoever solo-crossed must cross again. Hence, they will cross separately. Also, it is impossible for them to cross together last, since this implies that one of them must have crossed previously, otherwise there would be three persons total on the start side. So, since there are only three choices for the pair crossings and C and D cannot cross first or last, they must cross together on the second, or middle, pair crossing. Putting all this together, A and B must cross first, since we know C and D cannot and we are minimizing crossings. Then, A must cross next, since we assume we should choose the fastest to make the solo cross. Then we are at the second, or middle, pair crossing, so C and D must go. Then we choose to send the fastest back, which is B. A and B are now on the start side and must cross for the last pair crossing. This gives us, B+A+D+B+B = 2+1+8+2+2 = 15. It is possible for all four people to cross in 15 minutes. \(_\square\)

Main Article: Tour Puzzles See Also: Eulerian Path

In a tour puzzle , the goal is to determine the correct path for an object to traverse a graph. These kinds of puzzles can take several forms: chess tours, maze traversals, eulerian paths , and others.

Find the path that leads from the star in the center back to the star in the center. Paths can only go in the direction of an arrow. Image Credit: Eric Fisk Show Solution The solution path is outlined in red below.

Determine a path through the below graph such that each edge is traversed exactly once . Show Solution There are several possible solutions. One possible solution is shown below, with the edges marked in the order they are traversed.

A chess tour is an interesting type of puzzle in its own right, and is explained in detail further down the page.

Main Article: Nonograms

A nonogram is a grid-based puzzle in which a series of numerical clues are given beside a rectangular grid. When the puzzle is completed, a picture is formed in the grid.

The puzzle begins with a series of numbers on the left and above the grid. Each of these numbers represents a consecutive run of shaded spaces in the corresponding row or column. Each consecutive run is separated from other runs by at least one empty space. The puzzle is complete when all of the numbers have been satisfied. The primary technique to solve these puzzles is the process of elimination. If the puzzle is designed correctly, there should be no guesswork required.

Complete the nonogram: Show Solution

One of the many logical puzzles is the Battleship puzzle (sometimes called Bimaru, Yubotu, Solitaire Battleships or Battleship Solitaire). The puzzle is based on the Battleship game.

Solitaire Battleships was invented by Jaime Poniachik in Argentina and was first featured in the magazine Humor & Juegos.

This is an example of a solved Battleship puzzle. The puzzle consists of a 10 × 10 small squares, which contain the following:

• 1 battleship 4 squares long
• 2 cruisers 3 squares long each
• 3 destroyers 2 squares long each
• 4 submarines 1 square long each.

They can be put horizontally or vertically, but never diagonally. The boats are placed so that no boats touch each other, not even vertically. The numbers beside the row/column indicate the numbers of squares occupied in the row/column, respectively. ⬤ indicates a submarine and ⬛ indicates the body of a ship, while the half circles indicate the beginning/end of a ship.

The goal of the game is to fill in the grid with water or ships.

Main Article: Sudoku

A sudoku is a puzzle on a \(9\times 9\) grid in which each row, column, and smaller square portion contains each of the digits 1 through 9, each no more than once. Each puzzle begins with some of the spaces on the grid filled in. The goal is to fill in the remaining spaces on the puzzle. The puzzle is solved primarily through the process of elimination. No guesswork should be required to solve, and there should be only one solution for any given puzzle.

Solve the sudoku puzzle: Puzzle generated by Open Sky Sudoku Generator Each row should contain the each of the digits 1 through 9 exactly once. The same is true for columns and the smaller \(3\times 3\) squares. Show Solution

Main Article: Chess Puzzles See Also: Reduced Games , Opening Strategies , and Rook Strategies

Chess puzzles take the rules of chess and challenge you to perform certain actions or deduce board states.

One kind of chess puzzle is a chess tour , related to the tour puzzles mentioned in the section above. This kind of puzzle challenges you to develop a tour of a chess piece around the board, applying the rules of how that piece moves.

Dan and Sam play a game on a \(5\times3\) board. Dan places a White Knight on a corner and Sam places a Black Knight on the nearest corner. Each one moves his Knight in his turn to squares that have not been already visited by any of the Knights at any moment of the match.

For example, Dan moves, then Sam, and Dan wants to go to Black Knight's initial square, but he can't, because this square has been occupied earlier.

When someone can't move, he loses. If Dan begins, who will win, assuming both players play optimally?

This is the seventeenth problem of the set Winning Strategies.

Due to its well-defined ruleset, the game of chess affords many different types of puzzles. The problem below shows that you can even deduce whose turn it is from a certain boardstate (or perhaps you cannot).

Whose move is it now?

Main Article: K-Level Thinking See Also: Induction - Introduction

K-level thinking is the name of a kind of assumption in certain logic puzzles. In these types of puzzles, there are a number of actors in a situation, and each of them is perfectly logical in their decision-making. Furthermore, each of these actors is aware that all other actors in the situation are perfectly logical in their decision-making.

Calvin, Zandra, and Eli are students in Mr. Silverman's math class. Mr. Silverman hands each of them a sealed envelope with a number written inside.

He tells them that they each have a positive integer and the sum of the three numbers is 14. They each open their envelope and inspect their own number without seeing the other numbers.

Calvin says,"I know that Zandra and Eli each have a different number." Zandra replies, "I already knew that all three of our numbers were different." After a brief pause Eli finally says, "Ah, now I know what number everyone has!"

What number did each student get?

Format your answer by writing Calvin's number first, then Zandra's number, and finally Eli's number. For example, if Calvin has 8, Zandra has 12, and Eli has 8, the answer would be 8128.

Two logicians must find two distinct integers \(A\) and \(B\) such that they are both between 2 and 100 inclusive, and \(A\) divides \(B\). The first logician knows the sum \( A + B \) and the second logician knows the difference \(B-A\).

Then the following discussion takes place:

Logician 1: I don't know them. Logician 2: I already knew that.

Logician 1: I already know that you are supposed to know that. Logician 2: I think that... I know... that you were about to say that!

Logician 1: I still can't figure out what the two numbers are. Logician 2: Oops! My bad... my previous conclusion was unwarranted. I didn't know that yet!

What are the two numbers?

Enter your answer as a decimal number \(A.B\). \((\)For example, if \(A=23\) and \(B=92\), write \(23.92.)\)

Note: In this problem, the participants are not in a contest on who finds numbers first. If one of them has sufficient information to determine the numbers, he may keep this quiet. Therefore nothing may be inferred from silence. The only information to be used are the explicit declarations in the dialogue.

Of course, the puzzles outlined above aren't the only types of puzzles one might encounter. Below are a few more logical puzzles that are unrelated to the types outlined above.

You are asked to guess an integer between \(1\) and \(N\) inclusive.

Each time you make a guess, you are told either

(a) you are too high, (b) you are too low, or (c) you got it!

You are allowed to guess too high twice and too low twice, but if you have a \(3^\text{rd}\) guess that is too high or a \(3^\text{rd}\) guess that is too low, you are out.

What is the maximum \(N\) for which you are guaranteed to accomplish this?

\(\) Clarification : For example, if you were allowed to guess too high once and too low once, you could guarantee to guess the right answer if \(N=5\), but not for \(N>5\). So, in this case, the answer would be \(5\).

You play a game with a pile of \(N\) gold coins.

You and a friend take turns removing 1, 3, or 6 coins from the pile. The winner is the one who takes the last coin.

For the person that goes first, how many winning strategies are there for \(N < 1000?\)

\(\) Clarification: For \(1 \leq N \leq 999\), for how many values of \(N\) can the first player develop a winning strategy?

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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

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Solve and Create a Logic Puzzle

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Logic puzzles come in a few different flavors, the most common being logic grid puzzles, spatial acuity puzzles, and lateral thinking puzzles.

Logic grid puzzles (which include Sudoku puzzles and KenKen puzzles) are probably mostly encountered as "who-owns-the-zebra" type puzzles. The challenge of these popular puzzles is to ascertain certain facts and reach certain conclusions using deductive reasoning to process several statements.

These deduction puzzles are the most basic logic puzzles and can be easily constructed with varying degrees of complexity to appeal to all levels of solving ability . By differing the number of elements, the amount of information given as well as the inter-connectivity of the defining statements (e.g., direct/loose association), these puzzles can be easy as pie or tough as nails.

Example: A Day at the Races

As an employee, Jack arrives at the racetrack to find three jockeys, Willie, Eddie, and Fidel, sitting in the media room waiting for a press conference to start. Jack's boss (who loves puzzles) has left him a note instructing him to bring each jockey his racing silks and announce to the press which race each jockey has won and which horse he was riding.

The challenge for Jack is that his boss' note contains only six statements:

• Willie has never won the Kentucky Derby.
• The jockey that rode Flash did not wear blue silks.
• The winner of the Belmont Stakes didn't ride Zipper.
• The winner of the Preakness is to the right of Willie.
• The jockey that rode Flash is seated to the left of the Derby winner.
• The jockey that rode Zipper is seated to the left of the jockey who wore yellow.

Is it possible for Jack to know which jockey wore which silks, which race he won and which horse he was riding just by using deductive reasoning?

Create a Logic Grid

To begin, Jack draws a grid with four columns and four rows and then subdivides the lower three rows into three rows each as illustrated above (Fig 1).

In the top row, he writes in the names of the jockeys in order (Willie, Eddie, Fidel) and puts the nine variables in the left-hand column.

He then writes the six statements out below the grid.

Note: The statements that refer to left and right are from the perspective of the viewer when looking at the jockeys at the press conference (in the grid).

Now Jack reads each statement to see what conclusions he can come to for each one.

• Willie has never won the Kentucky Derby .

This is a straightforward statement and allows Jack to eliminate one race option for Willie.

This statement doesn't help until Jack knows either who rode Flash or who wore blue. He puts this aside for now.

Once again Jack needs to know who won Belmont or rode Zipper before he can deduce anything from this statement.

Now Jack can make some progress. He previously eliminated the Derby as an option for Willie and now he can eliminate the Preakness which means he can pencil in his first conclusion. Willie won the Belmont Stakes. Tada! Furthermore, he can eliminate Zipper as an option for Willie (see statement #3).

From this statement, Jack eliminates Flash as an option for Fidel since he is seated on the far right. He now knows that either Willie or Eddie rode Flash.

This statement allows Jack to conclude that Eddie rode Zipper because the only other jockey who is to the left of anyone is Willie who Jack has previously concluded did not ride Zipper. Also, it follows that Fidel must have worn yellow and Willie must have ridden Flash.

Now Jack returns to statement #2. From this statement, he can deduce that Willie must have worn red which means Eddie must have worn blue.

And that completes the puzzle. Way to go, Jack!

How to Create a Logic Puzzle

Draw a simple grid. Use the one above as a guide.

Make Up a Scenario

Three fishermen (fish, technique, locale), three musicians (instrument, song, music genre), three people in a bar (occupation, drink, snack), etc.

Assign Values

After you pencil in the headings, assign three different values to each of the three characters.

Work Backward

Once you have filled the grid, work backward and create statements that will allow the solver to deduce the values for each character.

As well as stating what value the characters have, mix in statements that say what a character does not have/do ("the sax player does not write ballads") and/or that refer to the interrelationship of values and do not refer directly to the characters themselves (the R&B player sings jazz classics). Try to limit yourself to five or six statements.

Test solve the puzzle to make sure it works. Also, make sure you don't have unnecessary or redundant clues. In other words, two clues or statements which lead to the same conclusion.

It's a good idea to have someone test solve your puzzle for accuracy and fairness.

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Can You Solve This River Crossing Puzzle?

By chris higgins | nov 5, 2016.

River crossing puzzles are a classic form of logic puzzle. In them, you're provided with a scenario—some number of entities trying to cross a river using a raft or boat—and a set of constraints (typically, some of the entities might eat each other under certain circumstances).

In the TED-Ed video below , we tackle a variant of the puzzle in which a group totaling six, three lions and three wildebeest, need to cross a river using a raft. Only two animals can go at once. The problem is, if the lions ever outnumber the wildebeest, they'll eat them. How can they all cross the river?

The larger question of this puzzle is how should we solve such puzzles? In the video, the narrator walks through this solution, but explains how it can be generalized by drawing up decision trees. At each step of the puzzle, you lay out all the possible options, then cross out any that don't work. As you proceed, the set of possibilities dwindle until you're left with only a few viable paths.

Here are the conditions for this puzzle (also listed in the video):

1. The raft needs at least one animal to paddle it across the river, and it can hold at most two animals. 2. If the lions ever outnumber the wildebeest on either side of the river (including the animals in the boat if it's on that side), they'll eat the wildebeest. 3. The animals can't just swim across, and there are no tricks; the animals have to use the boat as described.

Tune in to see how it's done:

For more on this puzzle, check out this TED-Ed page which explains its relationship to the The Missionaries and Cannibals Problem .

• Tips & Tricks |

How To Solve A Criss Cross

A Criss Cross is a logic puzzle where you fit all the words in a word list into a grid of interlocking squares. The meaning of the words isn’t relevant to solving the puzzle. There is only one way to fit the words in, so if you complete the puzzle using all the words once, you know that you’ve solved it correctly.

Getting started

To start solving a Criss Cross, look at the squares that cross the given word. In the example, the six-letter words ‘_ E _ _ _ _’ and ‘ _ A _ _ _ _’ cross the given word ‘ETA’.

In the 6 LETTERS list, there’s only one six-letter word with ‘E’ in second letter position: RELENT. Fill this in.

There are four possible six-letter words for ‘_ A _ _ _ _’ so leave this blank for now. Try to progress through the puzzle based on certainties, and by testing the fewest possible options each move you make.

Making progress

Turning now to the first Down word in the top left of the grid, ‘_ _ _ _ _ R’, there are two possible options: PINCER and RICHER.

If you fill in PINCER, you then need a four-letter word, beginning with ‘C’, to start from its fourth letter. However, there isn’t a four-letter ‘C’ word in the list, so you can rule out PINCER and fill in RICHER.

RICHER is confirmed by HURL, which crosses it, as HURL is the only four-letter word beginning with ‘H’. And so it continues…

• Score off the words in the word list as you go along. • Work in pencil so you can test possibilities and erase them if they don’t work. • You often have to work through several moves in your mind and use a process of elimination. • If you get stuck, retrace your moves until the last point you had more than one option to choose from.

And here’s the completed puzzle:

How To Solve A Logic Grid Puzzle

This is a quick start guide on how to use a logic puzzle grid to complete logic puzzles using our free online interactive logic problems.

Getting Started

The first step is to read the scenario, or the story, introducing the puzzle. Then, it is your preference how you approach the clues – you might choose to work through them in order, or to quickly scan them for the easiest clues that you can knock out straight away.

Each value from each category of facts has one unique match to each of the other categories. The best way to demonstrate this is with an example of a completed grid:

Reading through the clues, you will need to find the position where the relevant information intersects in the grid, and cross out (X) the square for information you know to be false, or tick it for information you know to be true.

Using our interactive puzzle grids, you do this by clicking once for X , and a second click to tick the box .

When you tick a box, the other options in that row and column only within the group of intersecting facts you are working in will be crossed out automatically, as they can not be true. For example, once you have determined that Sally has the Chocolate cake, no other Flavors can relate to Sally , and no other Names can relate to the flavor Chocolate , so these options are automatically crossed out for you.

Correcting Mistakes

Clicking a ticked box one more time will clear it and the X’s around it that were added automatically. It will not clear any of those X’s if they were not added automatically – i.e. if you had previously crossed out a square in that row or column, it will remain as an X.

If you have crossed out (X) a box by mistake, you will need to click it twice to clear it . The first click will tick the box, the second click will clear.

Working With Clues

Some clues will be simple: giving you one or more facts that can be immediately marked in the grid, and nothing more. If you are satisfied that there is no further information you can gain from a clue, click it once to mark it as solved . It will be moved to a “Solved Clues” list beneath the Clues section.

Other clues will be more complex. They may give you some initial information that can be marked in the grid, but you will need to return to them later when you have more information.

In difficult puzzles, you may need to take notes. For example if you have a number of inconclusive facts, like “Mary won $100 less than Tom” and “Tom won less than Simon” and “Joseph won less than the person with the purple hat but more than the person with the orange hat”, taking notes is the easiest way to test out which arrangement of facts works, or at least to eliminate some further possibilities.

Working In The Grid

Logic puzzles require you to cross reference information in different parts of the grid in order to fill out more information in other parts, leading you to eliminate further possibilities. You can then revisit your more complex clues, and you may find that they are now more helpful.

While advanced puzzle solving techniques are beyond the scope of this article, remember to check for information in a column that has been ticked, and ensure that same information has been marked in the corresponding row. Also check that information in a row that has a tick has been applied in the relevant column below.

As a quick example, in the below puzzle, because Emma has the Carrot flavor ticked, we can cross out Farewell Party for Emma’s row, because if we look down the Carrot column we can see that Farewell Party is crossed out.

Logic puzzles additionally come with a table to enter your answers in a simplified easy to read form. However if you are using an app or a website like ours, this information will be filled in for you automatically.

Ready to have a go at solving a logic grid puzzle? Try our ‘Very Easy’ rated puzzle: Daycare Dropoffs .

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Woodpuzzle - Number Match Game 4+

Number puzzle and logic game, viviana baeza, designed for ipad.

• Offers In-App Purchases

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Description.

Wood block number puzzle "Woodpuzzle" is an addictive relaxing number game. Jump into the new number match gameplay, solve tons of new challenging number puzzles and train your brain! Challenge your mind and solve the puzzles, then you will find them easy and exciting! This innovative number match game is the best brain teaser for lovers of classic number puzzle games also known as Numberama, Ten Pair, Make Ten, Take Ten, Match Ten, Digits, 10 Seeds. Whenever you want to relax, play number match puzzles. Solving logic puzzles and matching numbers will bring your brain great pleasure. Solving a puzzle a day will help you with logic, memory, and maths skills training! So if you like classic board games, try Woodpuzzle. How to play? - You need to find and match the pairs of same numbers(3-3, 5-5) or pairs that add up to 10(2-8, 4-6). Simply tap on the two numbers one by one to eliminate them from the board. - The pairs of numbers must be located side by side. You can cross them out vertically, horizontally or diagonally and also you can make a pair when one number stands in the last cell in the line and another stands in the first cell in the line below. - There can also be empty cells between the 2 matching numbers. - Try to clear the numbers on the board to achieve the highest score. - When there are no more numbers to remove, you can add more numbers. What you get: - Beautifully easy and simple, no pressure and no time limit. - Daily Challenges. Play every day, complete daily challenges for a given month, and win unique and beautiful gems. - Hints to help you reach the goal faster. - Auto-save: If you get distracted and quit your Woodpuzzle game unfinished, we will save it for you so you can continue anytime. - Challenging to break your highest score. - Easy to play. Classic puzzle game and number game for all ages! - More than 1000 levels! With thousand of levels, Woodpuzzle offers endless hours of brain-teasing entertainment. Moreover, the game supports multiple languages, so you can play it in your preferred language. Whether you're a fan of math puzzles or just looking for a fun and challenging game to pass the time and train your brain, Woodpuzzle is the perfect choice. Download now and start matching!

Version 1.5

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The developer, Viviana Baeza , 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 and linked to your identity:

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

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

Information

English, French, German, Italian, Japanese, Korean, Portuguese, Russian, Simplified Chinese, Spanish

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Some in‑app purchases, including subscriptions, may be shareable with your family group when family sharing is enabled., more by this developer.

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