The Q*Bert Problem

Ah, new year, new math toys.  This time I finally fulfilled my math teacher imperative for cubes, figure 1.

Figure 1

I had some ideas about what I wanted to do: something simple involving triangular numbers.  But with the cubes, I knew that the temptation to build in 3-D was just too great to avoid.  And maybe I’m just playing too many video games these days, but I started building something and came up with this.

The Q*Bert Problem

How many cubes to change?Q*Bert is a video game that was popular in the 1980s.  To beat each level, the main character had to jump on all of the tops of the cubes so he can change their color.  Look at the picture above — how many cubes must Q*Bert change to beat level 7?

Before they made the final version of the game, one of the “designers” had an idea:

  1. Level 1 should have only one cube.
  2. Each next level should add one more layer of cubes.
  3. There aren’t any hidden cubes in the back.

Problem Statement:

A. How many cubes does Q*Bert have to change to beat level 99?

B.  What is the highest level that requires less than 1000 cubes to change?

The Goods

Pretty Posters

Notes for Future Use

  1. The order of questions 3 and 4 could be reversed — finding the rule is more abstract than finding the solution to the original problem.
  2. After working for a couple of days, some students had an answer but nobody could provide a rule and justify it.  So I used the “Rectangular Numbers Activity” to make the rule and a justification within their reach.  I’ve learned that I like the tension of students not knowing but searching, but at some point there needs to be resolution — an answer!
  3. I entered 4 different grades for this problem, one for each question.  So, I have learned something from the SBG approach.
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3 Responses to The Q*Bert Problem

  1. Paul Hawking says:

    I really like this. You’ve made it personal by bringing in popular culture from the “dark ages,” i.e. when you were their age. You gave them an intriguing riddle to solve. You gave them BIG paper to create posters of their work, which just adds to the fun factor. And you gave a supporting activity to unstick their thinking when they hit a dead end.

    Paul Hawking
    The Challenge of Teaching Math
    Latest post:
    Posts I really like from (Math Be Brave)

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