# Shuttle Puzzle

The following example is taken from Martin Gardner's Mathematical Games column in Scientific American, v 201, No 6, Dec 1959.

Draw a number of vertical lines and a number of horizontal lines (shuttles) connecting pairs of the vertical ones. For every vertical line, start at the top and trace the line downwards. Wherever an end of a shuttle is encountered, trace this shuttle horizontally till its other end. From there, turn downwards again and continue in this manner until you reach the bottom of one of the vertical lines. The interesting thing about this procedure is that, starting at the top of two different lines, one always ends at different "bottoms."

As a combinatorial class activity, it goes without saying that all members of the group take part in preparation of the diagram: drawing and labeling lines and putting in shuttles. The activity can be used to distribute a set of tasks between a group of kids.

(In the applet bellow, to draw a shuttle drag the cursor from one line to another. To verify the assignments, select the "Verify" button, and click in turn on each of the vertical lines.)

### This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

 What if applet does not run?

(A variant of the puzzle lets you check your understanding of the transposition mechanism that makes the puzzle tick. Another highlights the representation of the permutations as a product of transposions.)

Explanation