This week’s activity features a city planner creating an urban neighborhood populated by skyscrapers of different heights. We will be exploring the problem as presented by Julia Robinson Mathematics Festival (JRMF) through an online applet. We will also be giving some tips to get students started with this activity.
The city planner is creating a neighborhood of skyscrapers but is told by the city council to follow one rule:
- Each row and column must contain skyscrapers of each different height. In other words, no row or column can contain two skyscrapers of the same height.
In Example 1, the middle skyscraper of height 2 is in the same row as another skyscraper of height 2.
In Example 2, we have fixed the rule and moved the skyscraper so that each skyscraper of height 2 is in a different row and column.
The goal of the activity is to place all skyscrapers, of height 1, 2, 3, etc., in the grid so that all rows and columns have skyscrapers of different height.
The idea of the activity comes from a combinatorial idea called Latin squares. Starting off with 3×3 grids serves as an introduction by allowing some allocation of skyscrapers and making sure all of them follow the rule.
Once students have found one or more ways to solve how to place the skyscrapers, introduce the next variation of the problem:
- The city council wants the city of skyscrapers to look a certain way. They want to see a given amount of skyscrapers if you stand on specific side of the city (or grid). Taller skyscrapers block the view of smaller ones behind them.
A common first attempt to the problem is to use a trial-and-error approach. Looking at example 3 above, we see that all conditions are fulfilled, but there needs to be some rearranging of skyscrapers to complete the grid since the blue (or height 2) skyscrapers cannot be placed in the second column without having another skyscraper of the same height in the same row.
So that trial-and-error isn’t overly used or relied on, ask or introduce this scenario:
- City council is swamped with work! They are not able to check your work so try to place all skyscrapers without needing to change (or replace) them with other skyscrapers of a different size.
Attempting skyscraper problems can be very reminiscent to Sudoku puzzles and the attempt to completing it. It is best to have students think ahead in placing their skyscrapers (or numbers in Sudoku) so that it will not contradict something later in the puzzle. For example, if you can only see 1 skyscraper in your view, then the tallest tower must be placed there. Also, if you can see all four skyscrapers, then they must be arranged in ascending order (height 1, 2, 3, etc.).
Watch students attempting to solve the problem during Math Monday Live.
Slight changes to the way that skyscrapers are placed allow for more depth to the problem as well as more of a challenge. Furthermore, thinking ahead of where certain skyscrapers must be will help complete the grid.
An extension to skyscrapers, for those that want to try to find some sort of pattern, is asking how many different ways are there in placing skyscrapers on a 2×2 grid? 3×3 grid?…nxn grid? Another extension is having students create their own skyscraper puzzles and sharing it with their classmates so that they can try to solve it.
For more detailed information and resources on Skyscrapers puzzles, the JRMF website provides PowerPoint slides. You can view the slides here.