Pentagon Number Puzzle Game ⭐

A classic mathematical logic puzzle that challenges your problem-solving skills! Place numbers 1 through 10 in the circles (5 inner pentagon vertices + 5 outer extending circles) so that each of the five lines (3 circles per line) sums to exactly 19.

Game Rules

1. Use all numbers 1-10 exactly once
2. Place them in the 10 circles (5 pentagon vertices + 5 outer circles)
3. Each of the 5 lines must sum to 19
4. Each line contains 3 circles: outer circle → vertex → vertex
5. No repeats, no skipping numbers

How to Play

1. Select a number from the grid on the right
2. Click a circle in the star to place it
3. Watch the line status update in real-time
4. Green lines = correct sum of 19
5. Red lines = incorrect sum
6. Click filled circles to remove numbers and try again

Strategy Tips

• Start with the edges or vertices that appear in multiple lines
• Work backwards: if a line is almost complete, calculate what number you need
• The middle points (edge midpoints) are often key positions
• Use the hint button if you're stuck - it shows one valid solution
• Remember: there may be multiple valid solutions!

Technical Features

• Pure Logic Puzzle - No random elements, solvable through deduction
• localStorage Persistence - Resume your game anytime
• Responsive Design - Plays great on desktop and mobile
• Smooth Animations - Satisfying visual feedback
• Dark Mode Support - Easy on the eyes

The Math Behind It

This puzzle is based on magic configurations - arrangements where multiple sets of numbers share elements but all have the same sum. It's related to:

• Magic squares
• Latin squares
• Combinatorial optimization
• Graph theory (star graphs)

The challenge comes from the constraint that each number can only be used once, but multiple lines share the same positions!

Challenge Yourself

• Beginner: Use hints and take your time
• Intermediate: Solve without hints
• Expert: Find multiple different solutions
• Master: Solve it mentally without placing any numbers first!

Solution Guide 🎯

The Answer

Here's one valid solution (there may be multiple solutions):

Pentagon Vertices (Inner Circle):

• Top: 6
• Right-Top: 8
• Right-Bottom: 10
• Left-Bottom: 7
• Left-Top: 9

Outer Circles (Extended Points):

• Top-outer: 4
• Right-Top-outer: 5
• Right-Bottom-outer: 1
• Left-Bottom-outer: 2
• Left-Top-outer: 3

Verification

Let's check each line sums to 19:

1. Line 1: 4 + 9 + 6 = 19 ✓
2. Line 2: 5 + 6 + 8 = 19 ✓
3. Line 3: 1 + 8 + 10 = 19 ✓
4. Line 4: 2 + 10 + 7 = 19 ✓
5. Line 5: 3 + 7 + 9 = 19 ✓

How to Solve It

Mathematical Approach

1. Understand the constraint:

   • Each pentagon vertex appears in exactly 2 lines
   • Each outer circle appears in exactly 1 line
   • Total sum across all lines = 5 × 19 = 95

2. Calculate the totals:

   • Sum of 1-10 = 55
   • Let V = sum of pentagon vertices (5 numbers)
   • Let O = sum of outer circles (5 numbers)
   • V + O = 55
   • Since vertices are used twice: 2V + O = 95
   • Solving: V = 40, O = 15

3. Strategy:

   • Pentagon vertices should use larger numbers (sum to 40)
   • Outer circles should use smaller numbers (sum to 15)
   • Example split: 10 for vertices, 5 for outer

4. Trial and adjustment:

   • Place numbers strategically
   • Check line sums as you go
   • Adjust if needed

Logical Approach

1. Start with corners:

   • Pentagon vertices are the most constrained (used twice)
   • Place larger numbers (6-10) here

2. Balance the lines:

   • Each line needs to sum to 19
   • If a vertex has a large number (e.g., 10), pair it with smaller numbers

3. Use symmetry:

   • The puzzle has rotational symmetry
   • Solutions can often be rotated to create new valid solutions

Tips for Finding Other Solutions

• Swap outer circles: Outer circles 1-5 can often be rearranged
• Rotate the solution: The pentagon has 5-fold symmetry
• Try different vertex combinations: Any set of 5 numbers summing to 40 could work for vertices
• Computer search: There are approximately 60-100 distinct solutions (not counting rotations)

Did You Know?

This type of puzzle has been popular for over 100 years! Victorian-era mathematicians loved creating "magic star" puzzles with different target sums and configurations.

The five-pointed star with 10 positions is one of the most popular variants because:

• It has a unique visual appeal
• The number of lines (5) matches natural counting
• The target sum (19) is achievable with numbers 1-10
• It's challenging but not impossible

Try creating your own variant by changing the target sum or the number of points in the star!