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Fencing Numbers

Fencing Numbers

This week’s Puzzle Corner activity is a simplified version of a game called Fences. The original version has a 10 x 10 dot grid with the digits 0, 1, 2, and 3 spread repeatedly throughout. Each digit represents the number of line segments that will surround that square in a valid solution. For example, a square that contains a 3 would have line segments on three sides, and a square that contains a 2 would have line segments on two sides, and so on as shown below.

The challenge is to create a “fence” around the digits by connecting the points horizontally and vertically, such that each digit is bordered by the correct number of line segments.

To make the game appropriate for students, the grids in our version are five by five. Additionally, several of them have more than one solution, which always increases the chances that students will experience success. A sample grid is shown below with three different solutions, each of which is corrected based on the numbers given.

ex   ex1   ex2   ex3

There are three student sheets provided for this activity. The first challenges students to solve each of the four puzzles in the manner described above. The second student sheet provides a place for students to record multiple solutions for the same problem. Encourage students to search for and record as many different solutions as they can for each problem (one of the problems has only one solution). The third student sheet is an option extension for those students who finish quickly or are looking for an extra challenge. It provides six blank grids in which students can create their own problems. These problems may be created either by first drawing a fence and then putting in a few numbers, or by first putting in a few numbers and then trying to find a fence to fit those numbers. Once a workable problem is created, students should write the numbers in pen and trade it with a classmate to solve.

Be sure students use pencils when working on this activity so that they can erase their incorrect attempts and try again. Another option is to laminate the first student sheet and have students use dry erase markers to search for solutions.

There are many possible extensions to this problem in addition to the one given on the third student sheet. For older students, expand the grid to six by six or seven by seven to create more difficult problems. Challenge students to create a problem with more than four possible solutions or only possible one solution. Determine the fewest number of digits that must be placed in the grid to result in a problem that has only one solution.


Click the arrow below to view the solutions.

One possible solution for each problem is shown here. Problems one, two and four each have multiple solutions. Problem three has only the one solution shown.

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