Key Concepts:

  • One-stroke drawing
  • Connectivity in graphs
  • Odd and even degree vertices

Hello, young mathematicians! Are you ready for an adventure? Today, we’re going to explore a fascinating puzzle that will take us from point A to point B in the most exciting way possible. But before we embark on this journey, let’s understand some important concepts that will help us find the longest path.

What is a One-Stroke Drawing?

Imagine you have a pencil, and you want to draw a shape without lifting the pencil from the paper or retracing any line. That’s a one-stroke drawing! But not all shapes can be drawn this way. There are some special rules that we need to follow.

The Rules of the Game

To successfully complete a one-stroke drawing, we need to have a shape that meets certain conditions:

  1. The shape must be a connected graph, which means all parts of the shape are joined together, and you can get from any point to any other without lifting your pencil.
  2. The number of odd points (vertices where an odd number of lines meet) must be either 0 or 2.

If there are 0 odd points, you can start and end at any point you like. If there are 2 odd points, you’ll start at one and end at the other.

Our Puzzle: The 4x4 Square

Now, let’s look at our puzzle. We have a 4x4 square made up of 1x2 rectangles, and we need to find the longest path from point A to point B without retracing any line. But there’s a twist! Our current shape has 14 odd points, which is way too many for a one-stroke drawing.

The Strategy: Eliminating Odd Points

To make our shape ready for a one-stroke drawing, we need to reduce the number of odd points. We can do this by removing lines between odd points. Since we have 14 odd points, we need to remove at least 7 lines to pair them up.

But wait, there’s more! Our starting point A and ending point B are currently even points, and we need to make them odd points to be the start and end of our one-stroke drawing. This means we need to remove at least one more line, bringing our total to 8 lines that need to be removed.

Choosing the Lines to Remove

We want to find the longest path, so we should remove the shortest lines possible. If we run out of short lines, we’ll have to start removing longer ones. It’s like choosing the shortest straw - we want to keep the long straws (lines) for our path!

The Solution: Finding the Longest Path

Now, let’s get to the fun part - solving the puzzle! We’ll go step by step to find the longest path from A to B.

  1. Identify the Odd Points: Look at the 4x4 square and find all the points where an odd number of lines meet. These are our odd points.
  2. Pair Up Odd Points: Try to remove lines between odd points to pair them up. Remember, we want to remove the shortest lines possible.
  3. Make A and B Odd: Ensure that A and B become odd points by removing a line that touches either A or B.
  4. Check the Remaining Lines: After removing 8 lines, check to see if you can draw the remaining shape in one stroke, starting at A and ending at B.
  5. Draw the Path: Use your pencil to draw the path without lifting it from the paper. If you’ve removed the lines correctly, you should be able to do this in one go!

Practice Makes Perfect

This puzzle might seem tricky at first, but with practice, you’ll become a pro at finding the longest path. Try drawing different shapes and see if you can apply the one-stroke drawing rules to them. It’s a great way to sharpen your mind and have fun with math!

Conclusion

Congratulations, young explorer! You’ve learned about one-stroke drawings, odd and even points, and how to solve a challenging puzzle. Remember, math is like a treasure hunt - it’s full of surprises and rewards for those who are curious and persistent. Keep practicing, and you’ll discover the joy of solving even the toughest of puzzles!

Now, grab your pencil and embark on your own mathematical adventure. Who knows what amazing paths you’ll find from A to B!