Peaceful Knights Puzzle Game

Place 8 knights on the chess board so that no knight attacks another.  

A knight moves in an ‘L-shape’.

It can move either (a) two steps vertically 

and one step horizontally, or (b) two steps horizontally and one step vertically.

Possible moves of a knight are shown below

Place 8 knights on the chess board so that no knight attacks another.  

A knight moves in an ‘L-shape’. It can move either (a) two steps vertically 

and one step horizontally, or (b) two steps horizontally and one step 

vertically. Possible moves of a knight are shown below

Peaceful Knights

Place 8 knights on the chess board so that no knight attacks another. Knights move in an 'L-shape': two squares in one direction and one square perpendicular.

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♘

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How to Play

Drag and drop knights from the pool onto the chessboard. Place all 8 knights so that none can attack another.

• A knight moves in an L-shape: 2 squares in one direction and 1 square perpendicular
• Knights can jump over other pieces
• Red cells show where knights are under attack
• Green cells show safe placements
• Remove a knight by dragging it off the board

Knight Moves: From its position, a knight can move to any of the 8 squares marked with dots below (if they exist on the board):

· · · · ·
· · ● · ● ·
· ● · · · ●
· · · ♘ · ·
· ● · · · ●
· · ● · ● ·
· · · · ·

Knights placed: 0/8

Congratulations!

You've successfully placed 8 peaceful knights on the chessboard! None of them can attack each other.

Well done! You solved the puzzle!

MATHEMATICS SUBJECT ENRICHMENT ACTIVITY Class: VIII CH5 PART 2 TALES BY DOTS AND LINES

 

MATHEMATICS SUBJECT ENRICHMENT ACTIVITY

🧩 Activity Title: Game of Hex

Class: VIII
Textbook: Ganita Prakash – Part 2
Theme: Puzzle Time / Strategic Games
Page Reference: (As per textbook image)

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📌 Topic

Strategic Reasoning, Graph Theory (Introductory), Logical Thinking

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🎯 Aim

• To develop logical reasoning, strategic thinking, and planning skills

• To understand path formation and connectivity

• To introduce students informally to ideas related to networks and graph connections

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🧠 Mathematical Concepts Involved

• Connectivity

• Paths and networks

• Strategy and game logic

• Logical decision-making

• Non-draw games (deterministic outcomes)

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🧰 Materials Required

• Printed Hex board (or drawn on paper)

• Two coloured pencils/pens (Red and Blue)

• Eraser

• Worksheet / observation sheet

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📋 Description of the Game

Hex is a two-player strategy game played on a rhombus-shaped board made of hexagonal cells.

• One player uses Red

• The other uses Blue

• Each player is assigned two opposite sides of the board

• Players take turns placing one mark in an empty hexagon

• Once placed, a mark cannot be moved or removed

• The goal is to form an unbroken path connecting your two sides

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🧪 Procedure

1. Divide the class into pairs.

2. Give each pair a blank Hex board.

3. Assign:

   • Player 1 → 🔵 Blue

   • Player 2 → 🔴 Red

4. Explain the objective:

   • Blue connects left to right

   • Red connects top to bottom

5. Players take turns placing one mark per move.

6. The game ends when one player completes a continuous path between their sides.

7. Erase and replay with switched colours.

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👀 Observations (Sample)

• The board never ends in a draw

• One player must always win

• Blocking the opponent is as important as forming one’s own path

• Central hexagons are strategically powerful

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📝 Sample Observation Table

Game No.	Player 1 Colour	Player 2 Colour	Winner	Key Strategy Observed
1	Blue	Red	Blue	Central control
2	Red	Blue	Red	Blocking edges
3	Blue	Red	Red	Early path formation

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✅ Solution / Key Insight (Teacher Reference)

• Hex always has a winner

• There is no possibility of a draw

• This is due to the topological structure of the board

• A complete blocking without forming a path is impossible

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🔍 Explanation of Given Solutions (From Book Image)

• In the first diagram, Blue successfully forms a connected chain across its two sides

• In the second diagram, Red completes a vertical chain before Blue can block it

• The empty board allows students to explore multiple strategies

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💭 Reflection Questions (With Model Answers)

1. What strategy helped you win the game?

Creating connections early and blocking the opponent’s path.

2. Why can Hex never end in a draw?

Because the board structure guarantees at least one continuous path.

3. Which positions are most important on the board?

Central hexagons, as they allow connections in multiple directions.

4. How is this game related to mathematics?

It involves networks, paths, logical reasoning, and strategy.

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🚀 Extension / Higher Order Thinking (HOTs)

1. What happens if the board size increases?

2. Can you design a smaller Hex board that still guarantees a win?

3. How is Hex related to graph theory?

4. Can you create your own rules and still avoid a draw?

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🎓 Learning Outcomes

Students will be able to:

• Apply logical reasoning in games

• Recognise mathematical patterns in strategy games

• Understand the idea of connectivity

• Appreciate mathematics beyond calculations

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📚 Cross-Curricular Links

• Computer Science: Game algorithms

• Art: Hexagonal tiling

• Life Skills: Decision-making, patience, planning

MATHEMATICS SUBJECT ENRICHMENT ACTIVITY Class: VIII CH2 PART 2 THE BAUDHĀYANA-PYTHAGORAS THEOREM

Maths Subject Enrichment Activity (Class 8)

Chapter / Theme

BOX LABEL	INSIDE COLOUR
RED	🔵 Blue
BLUE	🟢 Green
GREEN	🔴 Red

Logical Reasoning & Problem Solving
(Ganita Prakash – Part 2, Puzzle-Based Thinking)

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Title of the Activity

🎨 Find the Colours! – One-Box Logic Puzzle

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Topic

Logical reasoning, elimination method, deductive thinking

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Aim

To develop logical reasoning skills by solving a real-life puzzle using minimum information and systematic elimination.

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Learning Outcomes

Students will be able to:

• Apply logical reasoning to solve puzzles

• Understand the concept of incorrect labeling

• Use elimination and deduction strategies

• Explain reasoning clearly in words

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Materials Required

• Three closed boxes

• Coloured balls (Red, Blue, Green)

• Labels marked RED, BLUE, GREEN

• Pen and notebook (for observation & reasoning)

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Given Situation

• There are 3 closed boxes:

  • One contains only red balls

  • One contains only blue balls

  • One contains only green balls

• Each box is labeled RED, BLUE, GREEN

• Important Condition: ❌ No box has the correct label

• You are allowed to open only one box

• Task: Find the correct colour for all three boxes

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Procedure

1. Carefully read the condition that all labels are wrong.

2. Choose any one box to open.

3. Observe the colour of the balls inside.

4. Use logical reasoning to reassign correct labels to:

   • The opened box

   • The remaining two unopened boxes

5. Record the reasoning step-by-step.

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Observation

Suppose we open the box labeled RED.

• Since no box is correctly labeled, the RED-labeled box cannot contain red balls.

• If we observe blue balls inside:

  • The RED label is wrong → confirmed

  • So this box must be BLUE

• Now consider the remaining boxes:

  • Box labeled BLUE cannot be blue

  • Box labeled GREEN cannot be green

• Using elimination:

  • BLUE-labeled box → must be GREEN

  • GREEN-labeled box → must be RED

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Result / Conclusion

✅ By opening just one box, we can correctly identify the contents of all three boxes.

This is possible because:

• All labels are incorrect

• Logical elimination gives a unique solution

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Final Correct Labelling (Example Case)

Original Label	Actual Colour
RED	BLUE
BLUE	GREEN
GREEN	RED

(Answer may vary depending on which box is opened, but the logic remains the same.)

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Mathematical Reasoning Used

• Elimination

• Deductive logic

• Case analysis

• Constraint-based reasoning

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Reflection

• Opening more than one box is unnecessary

• Careful thinking reduces effort

• Logic puzzles strengthen decision-making skills

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Higher Order Thinking Skills (HOTS)

1. Why is it essential that all labels are wrong?

2. What happens if one label is correct?

3. Can you solve this puzzle if there are 4 boxes and 4 colours?

4. How is this puzzle similar to solving equations?

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Real-Life Connection

• Sorting and classification problems

• Error detection in labeling systems

• Decision-making with limited information

SOLUTION – Find the Colours!

Given

• Three boxes contain only one colour each:
  🔴 Red balls 🔵 Blue balls 🟢 Green balls

• Boxes are labelled RED, BLUE, GREEN

• ❌ No box has the correct label

• Only one box can be opened

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🔍 Step-by-Step Logical Solution

Step 1: Open ONE box

Open the box labelled RED.

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Step 2: Observe the colour inside

Suppose the box contains BLUE balls.

Since all labels are wrong, the box labelled RED
❌ cannot contain red balls
❌ cannot match its label

✅ Therefore, this box must actually be the BLUE box.

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Step 3: Relabel the opened box

• RED label → actually contains BLUE balls

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Step 4: Deduce the remaining two boxes

Now only RED and GREEN colours remain.

• The box labelled BLUE ❌ cannot contain blue balls

• The box labelled GREEN ❌ cannot contain green balls

So:

• BLUE label → GREEN balls

• GREEN label → RED balls

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✅ Final Correct Arrangement

Box Label	Actual Colour
RED	🔵 Blue
BLUE	🟢 Green
GREEN	🔴 Red

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🧠 Why This Works

• Every label is wrong → strong constraint

• Opening one box gives enough information

• Remaining boxes are solved using elimination

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✨ Key Mathematical Idea

Logical Deduction & Elimination

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📝 Student Conclusion (Model Answer)

By opening only one box and using the condition that all labels are incorrect, we can logically determine the correct colours of all three boxes.

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🔥 HOTS – Ready Answers

Q1. Why can’t we open more than one box?
👉 It is unnecessary; one box gives complete information.

Q2. What if one label was correct?
👉 The puzzle would have multiple answers or no unique solution.

Q3. What type of thinking is used here?
👉 Logical reasoning and elimination.

TEACHER ANSWER KEY

Find the Colours!

Class: VIII
Book: Ganita Prakash – Part 2
Type: Logical Reasoning / Puzzle-based Enrichment Activity

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🧠 Correct Solution (For Teacher Reference)

Given Conditions

• Three boxes contain only one colour each:

  • 🔴 Red balls

  • 🔵 Blue balls

  • 🟢 Green balls

• Boxes are labelled RED, BLUE, GREEN

• ❌ No box has the correct label

• ✔ Only one box may be opened

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📌 Step-by-Step Logical Reasoning

Step 1: Choose one box to open

Open the box labelled RED.

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Step 2: Observe the colour

Assume the box labelled RED contains BLUE balls.

┌──────────────┐
│  Label: RED  │
│  Inside: 🔵  │
└──────────────┘

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Step 3: Apply the condition

• The label RED is incorrect (given)

• So this box cannot contain red balls
  ✔ Therefore, this box must be the BLUE box

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Step 4: Deduce remaining boxes

Now colours left are RED and GREEN

• Box labelled BLUE ❌ cannot contain blue

• Box labelled GREEN ❌ cannot contain green

So logically:

BLUE label  → 🟢 GREEN balls
GREEN label → 🔴 RED balls

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✅ Final Correct Matching (Answer Table)

Box Label	Actual Colour
RED	🔵 Blue
BLUE	🟢 Green
GREEN	🔴 Red

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🧩 Diagram-Based Summary (Board Explanation)

Initially (All labels wrong):

[ RED ]   [ BLUE ]   [ GREEN ]
  ?          ?          ?

Open RED box → 🔵

Therefore:
[ RED ] → 🔵
Remaining colours: 🔴, 🟢

[ BLUE ] → 🟢
[ GREEN ] → 🔴

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🎯 Mathematical Concepts Assessed

• Logical reasoning

• Elimination method

• Deductive thinking

• Constraint-based problem solving

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📝 Expected Student Conclusion

By opening only one box and using the condition that all labels are incorrect, we can logically determine the correct colours of all three boxes.

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🔥 Higher Order Thinking (Model Answers)

Q1. Why is opening one box sufficient?
✔ Because the condition “no label is correct” strongly restricts all possibilities.

Q2. Can the solution change?
✔ No. The solution is unique.

Q3. What if two labels were correct?
✔ The puzzle would not have a guaranteed solution.

 

MATHEMATICS SUBJECT ENRICHMENT ACTIVITY Class: VIII PART2 Chapter: FRACTIONS IN DISGUISE

 

Maths Subject Enrichment Activity

Title of the Activity

Peaceful Knights – A Logical Placement Challenge

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Class

VIII

Textbook Reference

Ganita Prakash – Part 2
(Activity Page: Peaceful Knights)

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Topic

• Logical Reasoning

• Chessboard Geometry

• Patterns and Spatial Thinking

• Non-attacking Arrangements

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Aim of the Activity

To place 8 knights on a chessboard such that no knight attacks another, and to develop logical thinking by understanding the movement pattern of a knight.

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Learning Outcomes

Students will be able to:

• Understand the L-shaped movement of a knight

• Apply logical reasoning and spatial awareness

• Recognize patterns and symmetry on a chessboard

• Appreciate the role of mathematics in games and puzzles

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Materials Required

• Printed or drawn 8 × 8 chessboard

• Pencil / pen

• Eraser

• (Optional) Chess pieces or counters

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Prerequisite Knowledge

• Knowledge of a chessboard layout

• Understanding of coordinates / grid positions

• Basic idea of movement patterns

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Procedure

1. Draw an 8 × 8 chessboard.

2. Recall the knight’s movement:

   • 2 squares in one direction and 1 square perpendicular (L-shape).

3. Begin placing knights one by one on the board.

4. After placing each knight, check:

   • Does it attack any existing knight?

5. Continue until 8 knights are placed.

6. Ensure no two knights attack each other.

7. Record the final positions using:

   • Square names (e.g., A1, C3)

   • OR a diagram

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Observation

• Knights placed on same-colored squares do not attack each other.

• Knights attack only opposite-colored squares.

• Symmetrical placements help avoid attacks.

• Logical grouping simplifies the solution.

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Result / Solution

A valid arrangement of 8 peaceful knights is achieved where:

• No knight can attack another.

• All knights follow the movement rule.

• The condition of non-attacking placement is satisfied.

✔ Hence, the objective is successfully achieved.

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Verification

Condition	Status
Knight movement followed	✔
No knight attacking another	✔
8 knights placed	✔
Logical placement used	✔

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Reflection / Thinking Questions

1. Why do knights on the same color never attack each other?

2. Is there more than one correct solution?

3. What happens if we try to place 9 knights?

4. How is this problem related to combinatorics?

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Extension / Enrichment

• Try the same activity on:

  • 6 × 6 board

  • 10 × 10 board

• Compare with 8 Queens problem

• Explore how mathematics is used in chess algorithms

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Real-Life Application

• Game design

• Artificial intelligence

• Strategy planning

• Problem-solving techniques

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Conclusion

This activity shows that mathematics is deeply connected to games and logical thinking. By exploring knight movements, students develop reasoning skills and enjoy learning through play.

SOLUTION 1 – Bottom Two Rows (Black Squares)

Positions:

A1, C1, E1, G1,

A3, C3, E3, G3

8  . . . . . . . .

7  . . . . . . . .

6  . . . . . . . .

5  . . . . . . . .

4  . . . . . . . .

3  K . K . K . K .

2  . . . . . . . .

1  K . K . K . K .

   A B C D E F G H

✔ Valid
✔ Symmetric
✔ Easy to explain

SOLUTION 2 – Top Two Rows (White Squares)

Positions:

B8, D8, F8, H8,

B6, D6, F6, H6

8  . K . K . K . K

7  . . . . . . . .

6  . K . K . K . K

5  . . . . . . . .

4  . . . . . . . .

3  . . . . . . . .

2  . . . . . . . .

1  . . . . . . . .

   A B C D E F G H

✔ Colour-based reasoning
✔ Inverted pattern of Solution 1

SOLUTION 3 – Middle Rows

Positions:

A4, C4, E4, G4,

A6, C6, E6, G6

8  . . . . . . . .

7  . . . . . . . .

6  K . K . K . K .

5  . . . . . . . .

4  K . K . K . K .

3  . . . . . . . .

2  . . . . . . . .

1  . . . . . . . .

✔ Demonstrates vertical symmetry
✔ Useful for extension discussion

SOLUTION 4 – Two Columns Pattern

Positions:

B1, B3, B5, B7,

D1, D3, D5, D7

8  . . . . . . . .

7  . K . K . . . .

6  . . . . . . . .

5  . K . K . . . .

4  . . . . . . . .

3  . K . K . . . .

2  . . . . . . . .

1  . K . K . . . .

   A B C D E F G H

✔ Column-based logic
✔ Excellent for reasoning discussion

SOLUTION 5 – Scattered but Same Colour

Positions:

A1, D2, G3, B4,

E5, H6, C7, F8

✔ Non-linear
✔ Tests deeper understanding
✔ No knight attacks another

Rule	Status
8 Knights placed	✔
Knight L-move respected	✔
No attacks	✔
Logical reasoning	✔