### Engaging math activities for the summer break - Day 3

Counting a group of objects can be done in many different ways. The most fundamental idea is that counting is at all possible in the sense that, regardless of the manner in which it is performed, the result is always the same. For example, place random numbers in a rectangular array and then compute separately the column and row sums. Then adding the column sums gives the same total as adding up the row sums. For little children the array and the numbers inside should be small. Letting all the numbers be 0 or 1 not only makes the activity more accessible to younger children but also adds a twist with a mathematical flavor.

**What is the setup?** Have a grid of N×M squares or several of them for small N and M. (For an interactive version see a separate page.)

**What is the activity?** In each square of the grid place either 0 or 1 and count the column and the row sums. Count the number of distinct results. The task is to fill the grid with 0s and 1s so as to maximize the number of distinct sums.

**What's to observe?** The result of the calculations does not change if two columns or two rows are swapped. This allows to do away with an apparent randomness of number distribution and for a more deliberate and organized placement of the digits.

**A proof by construction**. For N = M+1 or N = M, it is possible to get N+1 distinct sums. For example, in the 7×6 grid below there are 8 distinct sums: 0, 1, 2, 3, 4, 5, 6, 7.

For M smaller relative to N, the number of distinct sum is first equal to N till M is about N/2, but then continues to decrease.

**An extra observation** Replace all 0s with 1s and all 1s with 0s. Call such digit distributions *complementary* - or *complements* of each other. For a square N×N grid, the complement of a an optimal distribution (i.e., the one that affords N+1 distinct sums) is also optimal.

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July 10th, 2011 at 9:54 pmThanks for the article, it might get kids engaded in math more.

November 23rd, 2011 at 12:21 amI certainly hope so

November 23rd, 2011 at 12:10 pm