Will They Meet?
Moves
0
Distance
—
Click on the grid to place Romeo
Juliet’s fraction:
Juliet’s direction:
Distance:
Getting Started
1
Place Romeo at
(0, 0) and Juliet at (8, 0). Move Romeo right by 4 squares, then right by 2, then right by 1. What happens to the distance between them?2
Can you make Romeo and Juliet meet exactly? What starting positions and moves work?
3
If Romeo always moves in the same direction by the same distance, does the gap between them always shrink?
Repeated Moves
4
Place them 8 apart on the same row. Move Romeo right 8 repeatedly. Record the distance after each move. What pattern do you see?
Hint: look at the distance as a fraction of the original distance.
5
Now try moving Romeo right 8, then left 8, then right 8… alternating. What happens to the distance? Do they still converge?
6
What if Romeo moves in a square pattern: right 4, up 4, left 4, down 4? Where does Juliet end up after one full loop?
Changing the Fraction
7
Switch Juliet’s fraction to ⅓. Does she converge faster or slower than with ½?
8
Try ¾. What’s different about the convergence? Is there a fraction where they meet exactly?
9
If Juliet moved the same distance as Romeo (fraction = 1), what would happen? What about fraction = 0?
Opposite Direction
10
Switch to Opposite direction. Place Romeo and Juliet on the same spot. Move Romeo right 6. Where is Juliet now?
11
With opposite direction, does the distance always increase? Can you make them converge?
12
With opposite direction and fraction ½, move Romeo right 8 repeatedly. What happens to Juliet’s position? Does she settle somewhere?
Coordinates and Midpoints
13
Turn on coordinates. After each move, look at where Juliet is. Can you see a connection to midpoints?
Hint: look at Juliet’s position relative to her old position and Romeo’s new position.
14
With fraction ½ and same direction, if Romeo is at
(10, 0) and Juliet is at (2, 0), and Romeo moves right 6 to (16, 0), where will Juliet be? Can you predict before clicking?15
Can you write a formula for Juliet’s new position given her old position, Romeo’s new position, and the fraction?
Deeper Thinking
16
If Romeo keeps making the same move forever, will they ever meet exactly? Or do they get infinitely close but never touch? What does this remind you of?
17
The distance after n identical moves follows a pattern. Can you express it as a formula involving the fraction and n?
Hint: think about geometric sequences.
18
If we allowed infinitely many moves, where would Juliet end up? This connects to the sum of an infinite geometric series.
How to Play
The Story
Romeo and Juliet are on a grid. Romeo makes a move — a horizontal or vertical jump of any whole number of squares. Juliet responds by moving in the same direction, but only half as far (because she is only little). Can you find a way to make them meet?
How to Use
1. Click on the grid to place Romeo (red), then Juliet (blue).
2. Set the number of squares and click an arrow to move Romeo.
3. Watch Juliet follow automatically. The move log tracks every move.
4. Change Juliet’s fraction (½, ⅓, ¼, ¾) to see how it affects convergence.
5. Switch Juliet to Opposite direction for a completely different investigation!
2. Set the number of squares and click an arrow to move Romeo.
3. Watch Juliet follow automatically. The move log tracks every move.
4. Change Juliet’s fraction (½, ⅓, ¼, ¾) to see how it affects convergence.
5. Switch Juliet to Opposite direction for a completely different investigation!
Keyboard Shortcuts
Use arrow keys to move Romeo in any direction (using the current distance setting).