He can improve his program even further with more granular validity checking and backtracking.
You might want to interest him with solving sudoku. First manually and then with a computer program. Then he will be able to abstract and reapply what he learned to magic squares.
The method basically boils down to:
1. Knowing few things about your solution (for example some numbers in sudoku puzzle or placement of some digit in magic square if for example you look for magic square with number 5 in top left corner).
2. Developing new knowledge from what you already know by applying rules (constraints) you know about the solution (for example if you have just one number missing in a row of magic square, you know what it must be because it has to sum to to magic number with all others already known in that row).
3. When you can't get new knowledge in this way just assume some fact. (I have no idea what must be in the bottom right corner but let's assume it's 7 beacause I don't have 7 yet).
4. Try to build up new knowledge on that (exactly like in step 2.) until you reach a solution (this means the problem might have multiple solutions). Or until you generate piece of knowledge that violates your constraints. This means the assumption was wrong and you discard it and all the knowledge that you generated after. But now you have a new solid piece of knowledge (bottom right corner doesn't contain 7) so you can take a note of it and continue with step 2.
Continue this until you either reach a solution, a contradiction (without active assumption) or have no options to assume anything because all possible assumptions have been ruled out. First result means that there's exactly one solution (if you don't have active assumption). All the latter mean that there's no solution.
If you don't have any knowledge about your solution, just rules it has to obey, like starting with blank magic square, then you just start at step 3.
Funny thing is that it's so general that it's how ALL mathematical proofs work (except those that use induction which have some very specific trick for bundling of infinite amount of forward knowledge derivarion steps into one easily applicable step for convenience).
The whole trick with math is that we encode our knowledge about it in ways that are true for infinte number of possible substitutions and transformations. We write a^2+b^2=c^2 not 3^2+42=5^2-4^2+42 and all the others because there's infinitely many of them. So matching all the math knowledge to your premises to generate new knowlegde about your problem to eventually reach the thesis is rarely an easy thing. But the structure of each math proof is the same as solving sudoku just with clever substitutions that match the rules (which can come from entirity of math) to your problem.
You might want to interest him with solving sudoku. First manually and then with a computer program. Then he will be able to abstract and reapply what he learned to magic squares.
The method basically boils down to:
1. Knowing few things about your solution (for example some numbers in sudoku puzzle or placement of some digit in magic square if for example you look for magic square with number 5 in top left corner).
2. Developing new knowledge from what you already know by applying rules (constraints) you know about the solution (for example if you have just one number missing in a row of magic square, you know what it must be because it has to sum to to magic number with all others already known in that row).
3. When you can't get new knowledge in this way just assume some fact. (I have no idea what must be in the bottom right corner but let's assume it's 7 beacause I don't have 7 yet).
4. Try to build up new knowledge on that (exactly like in step 2.) until you reach a solution (this means the problem might have multiple solutions). Or until you generate piece of knowledge that violates your constraints. This means the assumption was wrong and you discard it and all the knowledge that you generated after. But now you have a new solid piece of knowledge (bottom right corner doesn't contain 7) so you can take a note of it and continue with step 2.
Continue this until you either reach a solution, a contradiction (without active assumption) or have no options to assume anything because all possible assumptions have been ruled out. First result means that there's exactly one solution (if you don't have active assumption). All the latter mean that there's no solution.
If you don't have any knowledge about your solution, just rules it has to obey, like starting with blank magic square, then you just start at step 3.
Funny thing is that it's so general that it's how ALL mathematical proofs work (except those that use induction which have some very specific trick for bundling of infinite amount of forward knowledge derivarion steps into one easily applicable step for convenience).
The whole trick with math is that we encode our knowledge about it in ways that are true for infinte number of possible substitutions and transformations. We write a^2+b^2=c^2 not 3^2+42=5^2-4^2+42 and all the others because there's infinitely many of them. So matching all the math knowledge to your premises to generate new knowlegde about your problem to eventually reach the thesis is rarely an easy thing. But the structure of each math proof is the same as solving sudoku just with clever substitutions that match the rules (which can come from entirity of math) to your problem.