Equation Solving for Kids

When I gave my applied math lecture to the nine-year-olds (see Catastrophic Streamlines), of course I talked also about our Telescopes solution, and the need to solve systems of linear equations there.

When you talk to children about linear equations, you should not assume that this is a common phrase in their language (this may be true for grown-ups as well). For introducing system of linear equations, I used the following problem, already formulated in ancient China:

There are a number of rabbits and doves in a cage. In all, there are 35 heads and 94 feet. How many rabbits and doves are there?

The arising system of equations can be solved by guessing, and some children are quite good there; but it is also possible to introduce Gaussian elimination (in this special case) in elementry school.

More than 2 unknowns and equations
When I asked the children for examples with more than 2 unknowns, they were too shy to tell me their thoughts. However, showing them a puzzle page of a newspaper, they realized that Sudoku solving is (at least somehow) similar to solving systems of equation. And this led me to magic squares.

In my definition, a magic n x n square (a "magic square of the order n") is a square table with n rows and n columns. You have to place the numbers 1, 2, 3, ..... n^2 into the cells of table in such a way that the sum of each row, the sum of each column and the sum of each of two main diagonals is the same.

It can be shown fairly easily that there is only one solution for the order 3, the so-called Lo Shu square (China, 600 BC). All other solutions are obtained by reflection and / or rotation.

Lo Shu square. Image Source: Wikimedia commons
Magic squares of order 4
The order 4 is already more interesting.

When the children are told that the magic sum is 34, they can solve this system of linear equations with 8 unknowns (more or less) easily.

If you want to know the number of magic squares of order 4, this can be done by just trying all possible permutations (or intelligent versions of such a brute force algorithm) and check if the resulting square is magic. It turns out that the number of essentially different magic squares is 880.

Next week: Magic squares of order 5 and 6 and algorithmic connections to finance.