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Ball paradox

By David, 30 November 2015 Activity

Someone drawing a straight line on the ground.

Find a large paved area and draw a line in chalk.

You will need

  • Chalk
  • Measuring tape
  • About 10 different sized, round balls. For example:
    • Soccer ball
    • Basketball
    • Netball
    • Tennis ball
      Several balls of differeng sizes in a row.

      Place the balls along the line.

    • Cricket ball
    • Baseball
    • Ping pong ball
    • Billiard ball
    • Squash ball
    • Marble

What to do

Someone is using a tape measure to find the length of the balls.

Measure the line of balls.

  1. Find a large concrete or paved area.
  2. Draw a long, straight chalk line.
  3. Arrange the balls along the line so they are touching each other.
  4. Mark and label the start and end of the line of balls, and measure the line.
  5. Mix the balls up and arrange them on the line again in a different order. Mark the start and end again, and measure the distance. Is this line the same length as before?
    The balls are in a different order, and the lne is shorter.

    Try a different order.

  6. Try some other orders – maybe group them by colour, or arrange them biggest to smallest. What’s the longest line you can make? What’s the shortest?

What’s happening?

Even though the balls don’t change size, the order they are placed in will affect the length of the line.

Balls of the same size will meet at their midpoints, using their full length. When a big ball is placed next to a small ball, the small ball goes underneath the big ball. From above, it will appear that the two overlap.

To make a long line, you want to minimise the overlap. If you put the balls in order of size, each ball will be adjacent to the most similar sized balls. This will give you the longest line.

To make a short line, try to put small balls next to large ones. For example, basketball, squash ball, tennis ball, ping pong ball, soccer ball, will give a much shorter line than basketball, soccer ball, tennis ball, squash ball, ping pong ball.

More information

The maths of sport equipment
A new tool to help mathematicians pack

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