Operation: Combination
How it works
A combination machine like the one shown above outputs a sum for each row and each column as follows:
- The sum for each (horizontal) row is the sum of the gray numbers above the squares in that row that have an O
- The sum for each (vertical) column is the sum of the gray numbers to the left of the squares in that column that have an O)
In particular, for each row and column, we take a combination of the numbers 1, 2, 3, and 4 based on the Xs and Os in that row or column, and then we add the numbers in the combination together to get the sum for the row or column.
For example, the pattern of Xs and Os in the top row is OXOX (from left to right), which tells us to include 1 and 3 in our combination (and to exclude 2 and 4); since 1 + 3 = 4, the sum for the top row is 4.
In a combination machine puzzle, you are given a set of row sums and column sums, but no Xs and Os. Your goal is to design a combination machine (that is, to figure out where to put Xs and Os) so your combination machine outputs the desired row and column sums.
In this activity, students begin by computing the output of various combination machines, and then explore a variety of different combination machine puzzles. Some of these puzzles are possible to solve, and some are impossible!
Combination Machine Practice handout
Why we like this activity
It’s fun! Students enjoy trying to solve the puzzles.
It helps students to develop logical reasoning.
It helps students to develop numerical reasoning.
It requires students to engage in mathematical habits of mind:
Finding and using strategies to solve the puzzles efficiently (without guessing)
Using logic and understanding and explaining when trying to determine which puzzles are impossible
It has a low floor and a high ceiling: Students can start solving puzzles by trial and error, but as the puzzles get more challenging, more careful strategizing is required!
