Logic Puzzles | How to Solve a Logic Puzzle

How to Solve a Logic Puzzle

If you're new to grid-based logic puzzles, this tutorial will teach you the basics. Start with the "Introduction" first, then move on to the tutorials discussing specific clues or solving methods. Each tutorial contains a number of different slides - you can advance to the next slide by clicking "Next slide" at the bottom of each page, or by using the circled numerical links below each slide. Choose your specific tutorial from the list below to get started.

Introduction

logic game problem solving

A grid-based logic puzzle can seem daunting if you've never solved one before, but don't get discouraged - once you learn a few basic rules you'll be on your way to completing your first grid in no time. Each logic puzzle is comprised of a list of clues and a grid like the one you see here on your left. The grid will display a certain number of categories (in this case, 4) and a certain number of items per category (in this case, 5). Every item is matched to one, and only one, other item in each category, and no two items in a category will ever be matched to the same item in another category. Your goal is to figure out each item's matches, using just the clues given and pure logical deduction.

Next slide »

logic game problem solving

Every puzzle has a set number of categories. In this example, there are four - Prices (green), Names (yellow), Colors (pink) and Zodiac Symbols (blue). Notice how the last two categories (pink and blue) are repeated on both the top and left sides. All logic puzzle grids will follow this same pattern. Why? The point of the logic grid is to determine whether any given item is or is not matched with any other given item. This configuration of categories allows every single item on the grid to intersect with every other item on the grid once, and only once.

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logic game problem solving

Every item on the grid is labeled on either the left-side or the top-side, or both, depending on the category it is in. In this example there are five items for every category - i.e. Bonita, Daryl, Laura, Mario and Sheila are the five items in the "Names" category. Blue, Green, Orange, Pink and Violet are the five items in the "Colors" category, and so on.

logic game problem solving

The larger areas where each category intersects are called "subgrids". Each subgrid will always be a square that is outlined in a slightly heavier black line. In this example, the yellow subgrid at the top-left is the Prices-Names subgrid, because it is where the Prices category and the Names category intersects. There are six subgrids total in this sample puzzle.

logic game problem solving

Every item on the grid has a column (yellow) and/or a row (green) representing it. Each column and row travels across the full-width or full-height of the grid at that point (heights and widths will vary depending on the category). A sub-section of a column or row that is housed entirely within a single subgrid is referred to as either a sub-column (pink) or a sub-row (blue). The smallest squares on the grid, where individual items intersect, are called boxes (purple).

logic game problem solving

The purpose of the grid is to represent (via boxes) the relationships between every possible combination of every item. Your goal is to fill each of those boxes with either a TRUE marker (green circle) or a FALSE marker (red X), based on your reading of the given clues.

logic game problem solving

There are two hard rules to always remember in logic puzzles: 1. Every item in the puzzle is matched to one, and only one, other item in each category. 2. No two items in the same category will ever be matched to the same item in another category. Following those two simple rules, check out the four sample subgrids shown to the left. Each of these samples is invalid because it breaks one or both of those rules.

How do you finish a logic puzzle? As you progress through each clue, your task is to translate the relationships described there into TRUE or FALSE markers on the grid. (We'll discuss how to do that starting in the next tutorial). As you proceed through the puzzle, more and more of the grid will be filled in, until all the top subgrids (in this case, there are three) are completely filled in with TRUE markers. At that point, you have successfully revealed the relationships between each and every item on the grid, and puzzle is solved.

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Logic Games

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Escape from Castle Claymount

Escape from Castle Claymount

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Abandoned

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Tube Master

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Cut & Dunk

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Math Duck

Ice Cream Man

IN

Bob the Robber

Link It Up!

Link It Up!

One Trick Mage

One Trick Mage

Life in the Static

Life in the Static

Dreams and Reality

Dreams and Reality

Escape!

Transmorpher

Transmorpher 3

Transmorpher 3

Truck Loader 4

Truck Loader 4

Truck Loader 5

Truck Loader 5

Gravity Arrows

Gravity Arrows

Drawing games.

Car Drawing Game

Car Drawing Game

Sugar, Sugar

Sugar, Sugar

Air Traffic Control

Air Traffic Control

Brain For Monster Truck

Brain For Monster Truck

Scribble

Draw The Hill

Protect Emojis

Protect Emojis

Draw Parking

Draw Parking

Pumpkin Planner

Pumpkin Planner

Physics puzzle games.

Bump Bump Bump

Bump Bump Bump

Cannon Basketball 4

Cannon Basketball 4

White Ball

Cut the Rope

Bounce

Catch the Treasure

Doctor Acorn 3

Doctor Acorn 3

Zombie Launcher

Zombie Launcher

Helping games.

Ragdoll Rise Up

Ragdoll Rise Up

Code Panda

Else Walker

Chicken Hike

Chicken Hike

Unstable Engineer

Unstable Engineer

If Potato Meets Flag

If Potato Meets Flag

Doctor Acorn 2

Doctor Acorn 2

Sheep, Go Home

Sheep, Go Home

Lost Joe

Moley the Purple Mole

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Trace

Abandoned 2: The Forest

Maze

Maze Challenge

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Coolmath Games: The Game

Coolmath Games: The Game

There Is No Game

There Is No Game

Viking Ship Escape

Viking Ship Escape

Must Escape Dinosaur Land

Must Escape Dinosaur Land

Think outside the box games.

Red

Open 50 Doors

4 Pictures 1 Word

4 Pictures 1 Word

Just Hit The Button

Just Hit The Button

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Blocked Up

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Finders Critters

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Foxy Blocks

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Domino Dash

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Mr. Tulip Head

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Cut the Rope 2

Cut the Rope 2

Cut the Rope: Time Travel

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Cut the Rope: Magic

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Cut the Rope: Experiments

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Rope Bowling

Rope Bowling

Nom Nom Yum

Nom Nom Yum

More about logic games.

Logic games are the games that you really have to put your thinking cap on for. They involve taking in knowledge and using it to make informed decisions on what to do next. A great example is Sudoku, where players are forced to use the process of elimination to determine what number to put in each box. While logic games may look boring on the surface, once players start to understand them they often realize just how much fun they are. 

One of the best parts of our Logic Games Playlist is the variety that is offered here. There are platforming games, drawing, building, physics, the list goes on. Logic games have such a wide variety, so we recommend exploring them to your heart’s content.  

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Online Logic Puzzles and Printable Logic Problems

LogicLike gives you 2500+ logic puzzles online varying from easy to hard! Questions on logical thinking, interactive tasks on intelligence and attentiveness.

Educational platform LogicLike.com helps children to build critical thinking, math & logic skills. Adults improve mental abilities and develop ingenuity.

All types of puzzles to Build Reasoning Skills

Solve logic problems and achieve great results.

Many years ago, Cyprus and China "experimented" with puzzles and noticed that children think faster, and old people live longer and have a clear mind. Since puzzles require us to use various strategies to solve them, we boost our skills.

When it comes to 5-10 year children, solving good riddles for kids is a great start for developing creativity and ingenuity. For growing minds, puzzles are essential for developing problem-solving skills and short-term memory, neural connections and the two hemispheres of the brain working at once.

You may think that our brains are continually straining to solve that puzzles. However, research shows that by focusing on solving one problem, the mind immerses in a kind of meditation. In other words, you have a rest! A successful puzzle-solving process produces dopamine, which improves your mood!

5 Easy Logic Puzzles for Kids to Start

Regular training in solving logical reasoning puzzles helps children develop critical thinking. Have a look at some logic puzzles for kids that the LogicLike team creates for children and their parents.

Logic problems, tasks on patterns searching, and building sequences are especially recommended for preschoolers and primary school students.

Logic problem to solve

teapot, cup and glasses picture to a logic puzzle

The kettle is more expensive than the cup. The cup is cheaper than the glasses. Which dish is the most expensive?

The kettle.

Find a pattern

circles with fractions pattern puzzle

Can you guess the pattern and decide what comes next?

Charlie and Oleg are brothers. Each of them has two sisters. How many children are there in their family?

What is heavier?

red, green and purple boxes on the mechanical balance logic puzzle

Which box is the heaviest of three?

The red one.

Logic grid puzzle

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Tim, Max, and Bill participated in a race and took the top 3 places. Max passed Bill. Tim came after Bill. Identify the place of each driver using the + sign.

You can print the task or go ahead with training online.

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Observe extra logic grid puzzles for kids and adults .

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3 Logic Puzzles for Middle School & Adults just to Warm-up

Random pick puzzle.

red, yellow, green and purple balls in the box - logic puzzle illustration

A box contains 1 green, 1 purple, 3 yellow, and 4 red balls. What is the smallest number of balls you need to take without looking to make sure you have 3 balls of different colors among them?

Logic puzzle with numbers

In an office 5 employees only play soccer, 6 only play tennis, and 3 play both soccer and tennis. Every employee plays at least one kind of sport. How many employees are there in the office?

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Odd one out

picture with 3 cars and a plane for the Choose the odd one out puzzle

Choose the odd one out.

How to start solving logic puzzles and problems?

As soon as you start, recognize what the task is and what kind of answer you have to find. Carefully read the terms and conditions and determine what information you already have, what you can additionally collect by looking at the picture (if there is one, for example). Identify primary facts and discard the rest to concentrate better.

Once you have determined what you need to do and what area of activity is connected, make a step-by-step plan-solution to the problem. You can think over several solutions, assume the result for each and choose the best one. Once you have decided a solution way - go ahead! You are armed and able to solve the logic puzzle in a moment!

We teach you to solve logic problems!

Answers, solutions, comments. The online format at LogicLike.com.

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Logic Grid Puzzles

We’re here to share our passion for logic puzzles!

Our collection of logic grid puzzles for adults and teens are free to play online using our interactive grids. There is no need to create an account – just start playing. Your answers will be checked automatically when you have completed the grid.

You can browse all of our puzzles here: Logic Grid Puzzles .

If you would like to learn more about solving logic problems, check out our guide Solving Logic Puzzles For Beginners . For the initiated, improve your skills with our tips in Advanced Logic Puzzle Techniques !

We have a free Logic Puzzle Grid Printables section for downloading blank grids you can print and use to work on grids with a pen or pencil. You could also use them to create your own puzzle for fun or to distribute to a class.

Logic puzzles are a great challenge (and often relaxation method) for those who enjoy deductive reasoning puzzles, practicing logical or analytical thinking, or are simply drawn to the type of puzzle that has a single solution with no guesswork!

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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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Netflix's hit sci-fi series '3 Body Problem' is based on a real math problem that is so complex it's impossible to solve

  • The three-body problem is a centuries-old physics question that puzzled Isaac Newton .
  • It describes the orbits of three bodies, like planets or stars, trapped in each other's gravity.
  • The problem is unsolvable and led to the development of chaos theory.

Insider Today

While Netflix's "3 Body Problem" is a science-fiction show, its name comes from a real math problem that's puzzled scientists since the late 1600s.

In physics, the three-body problem refers to the motion of three bodies trapped in each other's gravitational grip — like a three-star system.

It might sound simple enough, but once you dig into the mathematics, the orbital paths of each object get complicated very quickly.

Two-body vs. three- and multi-body systems

A simpler version is a two-body system like binary stars. Two-body systems have periodic orbits, meaning they are mathematically predictable because they follow the same trajectory over and over. So, if you have the stars' initial positions and velocities, you can calculate where they've been or will be in space far into the past and future.

However, "throwing in a third body that's close enough to interact leads to chaos," Shane Ross, an aerospace and ocean engineering professor at Virginia Tech, told Business Insider. In fact, it's nearly impossible to precisely predict the orbital paths of any system with three bodies or more.

While two orbiting planets might look like a ven diagram with ovular paths overlapping, the paths of three bodies interacting often resemble tangled spaghetti. Their trajectories usually aren't as stable as systems with only two bodies.

All that uncertainty makes what's known as the three-body problem largely unsolvable, Ross said. But there are certain exceptions.

The three-body problem is over 300 years old

The three-body problem dates back to Isaac Newton , who published his "Principia" in 1687.

In the book, the mathematician noted that the planets move in elliptical orbits around the sun. Yet the gravitational pull from Jupiter seemed to affect Saturn's orbital path.

Related stories

The three-body problem didn't just affect distant planets. Trying to understand the variations in the moon's movements caused Newton literal headaches, he complained.

But Newton never fully figured out the three-body problem. And it remained a mathematical mystery for nearly 200 years.

In 1889, a Swedish journal awarded mathematician Henri Poincaré a gold medal and 2,500 Swedish crowns, roughly half a year's salary for a professor at the time, for his essay about the three-body problem that outlined the basis for an entirely new mathematical theory called chaos theory .

According to chaos theory, when there is uncertainty about a system's initial conditions, like an object's mass or velocity, that uncertainty ripples out, making the future more and more unpredictable.

Think of it like taking a wrong turn on a trip. If you make a left instead of a right at the end of your journey, you're probably closer to your destination than if you made the mistake at the very beginning.

Can you solve the three-body problem?

Cracking the three-body problem would help scientists chart the movements of meteors and planets, including Earth, into the extremely far future. Even comparatively small movements of our planet could have large impacts on our climate, Ross said.

Though the three-body problem is considered mathematically unsolvable, there are solutions to specific scenarios. In fact, there are a few that mathematicians have found.

For example, three bodies could stably orbit in a figure eight or equally spaced around a ring. Both are possible depending on the initial positions and velocities of the bodies.

One way researchers look for solutions is with " restricted " three-body problems, where two main bodies (like the sun and Earth) interact and a third object with much smaller mass (like the moon) offers less gravitational interference. In this case, the three-body problem looks a lot like a two-body problem since the sun and Earth comprise the majority of mass in the system.

However, if you're looking at a three-star system, like the one in Netflix's show "3 Body Problem," that's a lot more complicated.

Computers can also run simulations far more efficiently than humans, though due to the inherent uncertainties, the results are typically approximate orbits instead of exact.

Finding solutions to three-body problems is also essential to space travel, Ross said. For his work, he inputs data about the Earth, moon, and spacecraft into a computer. "We can build up a whole library of possible trajectories," he said, "and that gives us an idea of the types of motion that are possible."

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What you need to know about The Three-Body Problem before watching 3 Body Problem

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A woman in a VR headset looking up, mouth agape in 3 Body Problem

With Netflix’s 3 Body Problem imminent, you might naturally be intrigued by The Three-Body Problem , the science fiction novel by Cixin Liu.

First of all: I cannot explain why the show is 3 Body Problem and the book is The Three-Body Problem . Frankly, this decision drives me bananas. But it does make distinguishing the two in articles like this one easier, and as you’ll soon see, things are going to get complicated enough as is.

What is The Three-Body Problem about?

The simple version is that it’s a story about humanity’s first contact with an alien species. What makes it special is that it’s a very odd first contact story, centering on a wildly immersive VR video game and how it may be connected to the mysterious deaths of the world’s leading scientists. First contact is the light at the end of the story’s tunnel: read the back of the book and you know it’s coming, but how it happens is something you discover by reading.

Why is it such a big deal?

First serialized in China in 2006, The Three-Body Problem quickly racked up accolades upon its 2014 English debut, becoming the first Asian novel to win the Hugo Award for Best Novel. But unusually for such a hard sci-fi novel, The Three-Body Problem quickly escaped the orbit of speculative fiction circles and received glowing write-ups in mainstream press outlets like The New York Times , NPR , and, famously, a shoutout from President Barack Obama . Bob’s Burgers even did a Three-Body Problem episode . It was a big thing!

It’s also a work that some would call unadaptable: The Three-Body Problem has some weird shit going on in its VR game and eventually sets up a conflict that may or may not span centuries.

Remind me what you mean by ‘hard’ sci-fi.

The Three-Body Problem is a brainy book that is very committed to elaborating on the work of a lot of smart characters solving very opaque mysteries. This does not mean it’s impenetrable to people uninterested in becoming conversant in astrophysics, but it does mean there’s quite a bit of what I call “process porn,” focused primarily (but not exclusively) on existing and/or plausible technology. For the most satisfying version of that, consider films like Arrival or even Spotlight (a great movie for making spreadsheets thrilling). How well The Three-Body Problem ’s take on it works for you will depend on how much you connect with translator (and very good novelist in his own right ) Ken Liu’s interpretation of Cixin Liu’s prose.

Yang Hewen as Bai Mulin talking to Zine Tseng as Young Ye Wenjie in a still from 3 Body Problem

It helps that the book begins vividly in the past, during China’s Cultural Revolution in the 1960s — telling a small, personal tragedy that reverberates throughout the first novel as it reaches into the present day and beyond. With that human core in place, it’s easier to get lost in The Three-Body Problem ’s trippy mysteries, and grasp their shocking consequences.

I’m looking up characters from the book and can’t find many in the show. What’s up with that?

Part of what the Netflix adaptation does is relocate a big chunk of the action from China to London, which means lots of changes that ripple outward. Characters are race- and genderbent or reworked and given different names, which makes finding one-to-one analogues for some characters very difficult. But there are very practical concerns that also needed to be addressed, according to the showrunners, and these changes account for that.

“The characters in the book are all spread out in a way, but they don’t know each other, and they don’t connect with each other. Which works really well in a novel, [where] you get inside someone’s head,” co-creator D.B. Weiss told Polygon at a recent press event. “And in television — it’s hard to think of a television show about people who don’t know each other and never meet. Television is about people who know each other, who have strong feelings about each other, interacting with other people about whom they have strong feelings. So we kind of needed to make that happen.”

How much of The Three-Body Problem will 3 Body Problem season 1 adapt?

Sea Shimooka as Sophon walking across lava in a still from 3 Body Problem

Pardon? But what if there’s another season?

Oh, there are more books. The Three-Body Problem is the first in a trilogy, known as Remembrance of Earth’s Past. Netflix’s 3 Body Problem is actually an adaptation of the entire trilogy, folding in the novel’s two sequels — The Dark Forest and Death’s End — into its narrative. The first season of the show will cover The Three-Body Problem but will also introduce threads from later books. It’s a comprehensive adaptation, not a piecemeal one.

So you mean it’s done? There’s an end?

Yes, I do mean that. Don’t worry; while David Benioff and D.B. Weiss are most notorious for what happened when the former Game of Thrones showrunners ran out of track laid by their source material, that problem is not present here. They’re also not the only ones in charge, thanks to co-creator Alexander Woo, who has previously worked on excellent series like Manhattan and Wonderfalls . 3 Body Problem will live or die by the adaptation choices made, not for lack of material.

3 Body Problem will premiere on Netflix on March 22.

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Packers focus on solving Christian Watson’s hamstring problems

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Packers receiver Christian Watson missed eight games last year due to hamstring injuries. The Packers are trying to keep that from happening again.

Coach Matt LaFleur explained this week that Watson visited a lab in Madison, with the goal of solving the issue.

“They’ve got some special lab there or whatever that all the medical guys can give you the better diagnosis on [Watson],” LaFleur said, via Pete Dougherty of the Green Bay Press-Gazette . “But yeah, we’ve looked at a lot of different things in terms of just his body comp and maybe areas that were stronger than others to try to help get us in front of that .”

When he plays, Watson is extremely effective. The goal is to have him available for every possible game.

“We’re trying to look at everything we can do to mitigate those risks,” LaFleur said, regarding Watson’s injuries. “I know [Watson] is working his tail off right now down in Florida, trying to make sure that he comes in the best shape possible. Certainly, we looked at elements of the things that we can control because, I mean, soft-tissue injuries happen in our sport. It’s like, how can we mitigate those? And that’s what we’ve looked at, you know, from him personally to what we’re asking of our players from a load perspective, whether it’s in training or practice.”

Watson had 28 catches for 422 yards and five touchdowns in 2023, his second NFL season. No Packers receiver gained more than 793 receiving yards in 2023; the leader with that number was rookie Jayden Reed.

For more audio journalism and storytelling, download New York Times Audio , a new iOS app available for news subscribers.

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