CTK Insights

24 Jun

Engaging math activities for the summer break - Day 2

An engaging activity has been described by Martin Gardner in his Mathematical Games column in Scientific American, v 201, No 6, Dec 1959 and later included in one of his collections, New Mathematical Diversions. Rather recently, an upgraded variant has emerged as the Japanese ladders game. Amazingly, neither Gardner has mentioned the Japanese sources in 1959, nor half a century later his article has been referred to in the latest development.

What is the setup? Draw a number of vertical lines

Add (or better yet let children do that) a number of horizontal lines (shuttles, or ladder rungs) connecting pairs of the vertical ones:

What is the activity? 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.

What to observe? The characteristic property of this procedure is that, starting at the top of two different lines, one always ends at different "bottoms."

How to prove that? There is a couple of ways to explain why the procedure - for any set of the shuttles - always leads from different "tops" to different "bottoms". One is to realize that a single shuttle has this property. If "tops" and "bottoms" are labeled anyhow, then a single shuttle simply exchanges labels at the bottom of the two vertical lines that it connects. Since

  1. at the beginning all the labels at the bottom were different, and
  2. a single shuttle does not change that fact,

applying a sequence of shuttles does not change this fact either.

The second explanation appeals to the intuition via a physical metaphor. Think of every vertical line as a rope with a shuttle indicating the cross-point of two ropes. Since the ropes may only cross but not merge, one can find their number either by counting the upper or the lower ends. The result is the same in both cases.

The puzzle provides a simplified model of a set of braiding ropes. The model is simplified in that two ropes may cross in two distinct ways depending on which one goes beneath the other. Shuttles do not differentiate between the two possibilities.

What can be done with it? Oh, there is a plenty of possibilities. One is to demonstrate how useful mathematics may be in real life. The shuttle setup is perfect for an equitable distribution of chores among a group of children. Let the kids draw the lines for a minute or two and then associate the kids with the top labels and the jobs to perform with the bottom labels.

Lat one kid draw the shuttles and select one top and one bottom ends.

Let another kid find one seek 1 or 2 shuttles adding which would make the assignment of the selected top to the selected bottom. It can always be done with a single shuttle (Why?) Doing this with two may be more difficult than with just one. (See a separate page for interactive practice.)

The assignment problem can be turned upside down (or is it inside out?) Label both the tops and the bottoms of the poles.

Ask children to draw the shuttles that induce the specified job distribution. (Check the interactive version if you think that's simple.)

The Japanese ladders variant adds a twist by only allowing shuttles between the neighboring vertical lines.


  1. M. Gardner, New Mathematical Diversions: More Puzzles, Problems, Games, and Other Mathematical Diversions, MAA, 1995

One Response to “Engaging math activities for the summer break - Day 2”

  1. 1
    Engaging summer Math activities at CTK Insights | Wild About Math! Says:

    [...] in 1959, nor half a century later his article has been referred to in the latest development. [ Full article [...]

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