Magic square

The following program finds magic squares.

array3<array3<int>> magic_square;
int target;

magic_square.each(function^ (row) {
  invariant row.sum == target;
  invariant row.all?(*positive?);
});

magic_square.transpose.each(function^ (column) {
  invariant column.sum == target;
});

left_diagonal = magic_square.map(function (row, index) {
  return row[index];
});

right_diagonal = magic_square.map(function (row, index) {
  return row.reverse[index];
});

invariant left_diagonal.sum == target;
invariant right_diagonal.sum == target;

expose magic_square, target;

The example on the right lets you cycle through the different solutions for a ‘target’ value. You can lock cells to a specific number by clicking on them.

How does it work?

We start by declaring a 3x3 array to hold the numbers in our square. We then loop through each row and column and specify that their sum must equal the ‘target’. We build arrays for the diagonals and specify that their sum must also equal the ‘target’.

This program makes use of transpose, which lets us iterate over the numbers in the square by columns instead of rows. It passes positive? as a function pointer to all? to specify that all of the numbers in the square must be positive.

CLI example

Here is an example of running this program with the command-line interface:

sentient --run magic-square.json --assign '{ target: 15 }' --number 3

# standard output:
{"target":15,"magic_square":[[5,9,1],[1,5,9],[9,1,5]]}
{"target":15,"magic_square":[[4,9,2],[3,5,7],[8,1,6]]}
{"target":15,"magic_square":[[7,4,4],[2,5,8],[6,6,3]]}

Here is an example that finds a solution with a ‘5’ in the middle:

sentient --run magic-square.json --assign '{ magic_square: { 1: { 1: 5 } } }'

# standard output:
{"target":15,"magic_square":[[3,9,3],[5,5,5],[7,1,7]]}