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A Touchy Situation

I am indebted to Robert Benjamin, a scientist from the Los Alamos National Laboratory, for A Touchy Situation. Bob first did this activity with his son when his son was in kindergarten. Therefore, he feels that the activity is appropriate for students at all grade levels. He also notes that the activity works best if students use Cuisenaire® Rods. The flat edges on the Cuisenaire® Rods keep them from rolling and their differing lengths are necessary in Part 1 of the puzzle. If Cuisenaire® Rods are not available, pencils of varying lengths can be substituted.

A Touchy Situation has two parts. In the first part, students are challenged to place rods on a flat surface (no stacking allowed) in such a way that each rod touches every other rod in the arrangement. At first glance, students may think that it is possible to get many rods to touch. However, after working on the problem for awhile they will find that although it is easy to get three rods to touch, getting four rods (the maximum possible) is not so easy.

In Part 2, students must follow the same rules as in Part 1 (each rod must touch every other rod in the arrangement), but this time they are allowed to stack the rods. At this point, you might want to challenge students to predict how many rods they think they can get to touch. In Bob’s experience they often predict ten or more, and are surprised when they find that the maximum number of rods they can get to touch is much less.

Students who solve this puzzle early can be challenged to find (and sketch) multiple solutions for four, five, and six rods. They can also be challenged to think of additional solutions if one of the rods were flexible and could be bent.

ATouchySituationgirlPart 1
Place rods flat on your desk in such a way that each rod touches every other rod in your arrangement. No stacking is allowed. What is the maximum number of rods possible? Sketch your solution.

ATouchySituationboyPart 2
In this second part you are allowed to stack the rods, but the other rule still applies –each rod in your arrangement must touch every other rod. What is the maximum number of rods possible? Sketch your solution.


Click the arrow below to view the solution.

In the first part students were challenged to see how many rods (we suggested using Cuisenaire Rods) they could place flat on a table in such a way that each rod touched every other rod with no stacking allowed. The maximum number possible in this part is four (one solution appears below). In the second part, students were to do the same thing, but this time they were allowed to stack the rods. We know that it is possible to get six rods all touching, but I do not know if this is the maximum number. If your students have gotten more than six, please send in their solution and we’ll be happy to share it with our readers.


One Response to A Touchy Situation

  1. Five rods are possible, but it will be kept on top of those four rods and hence will not be on table and vice versa…

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