Magic squares have been developed independently by many cultures around the world. There’s a beauty in their balance: no matter how you sum their rows, columns, or diagonals, you always get the same result. The classic 3 by 3 magic square has every row, column, and diagonal sum to the “magic” number 15.
Here are two variations on magic squares that I’ve grown to like quite a bit. I hope you do too!
Puzzle 1. Flower Petal Puzzles
Put the numbers 1 - 9 into the nine circles of the flower below so that each straight line of three “petals” sums to the same number. Every number must be used exactly once.
Bonus question/hint: there are three possibilities for what the “magic” sum could be in this case. Can you find all three?
Puzzle 2. Magic Triangles
Place the numbers 1 - 9 in the nine circles of the triangle so that the four circles on each side of the triangle sum to 20.
Bonus Challenge: it is possible to find a solution to this puzzle such that, when you square each of the numbers in the circles, the triangle is still magic, with the sums on each side adding to 126. Can you find this “bimagic” triangle?
Solutions
Puzzle 1.
The key to solving flower petal puzzles lies with the central petal: once you’ve chosen what number goes there, you have no choice but to pair up the other numbers on opposite petals so they all add to the same thing. That means pairing the largest and smallest remaining numbers, then the second largest and second smallest remaining numbers, and so on.
One solution is to put the 1 in the centre. This will lead to the smallest magic sum: 12. A solution looks like this.
We could also put the 9 in the centre to get the largest magic sum: 18.
And there’s one more solution. Put 5 in the centre to get a magic sum of 15.
Interestingly, these are all the solutions. Here is one way to see why 1, 5, and 9 are the only options for the centre of the flower.
First of all, notice that the sum of all the numbers 1 - 9 is 45. One number will go in the centre, and the rest will be divided into four pairs, with each pair having an identical sum. That means whatever goes in the centre must leave us with eight numbers that sum to a multiple of 4. If 1 goes in the centre, the other eight numbers sum to 45 - 1 = 44, which is, happily, a multiple of 4. That means we can divide them into pairs that sum to 11, which is exactly what we did in the first flower petal solution.
If a 5 goes in the centre, the other eight numbers sum to 45 - 5 = 40, which is still a multiple of 4. And if 9 goes in the centre, the other eight numbers sum to 45 - 9 = 36, which is also a multiple of 4. Notice that 1, 5, and 9 are each four apart from each other. That’s no coincidence. It’s necessary for the remaining sum to be a multiple of 4. In particular, any of the other numbers 2, 3, 4, 6, 7, 8 cannot go in the central petal, which is precisely what you find if you attack the problem using trial and error.
Research: can you analyse the behaviour of flower petal puzzles with 7 petals? 11? More?
Puzzle 2.
I don’t have a slick way to explain how to get this solution. The bimagic triangle was discovered by David Collison, and I don’t know if his methods of finding bimagic triangles and squares were ever fully explained. Possibly it just involved an unusual persistence! In any case, here is the solution.
Add up the four circles on each side and you get 20.
But here’s the real magic: square them and add them up and you get the same number too!
As you can check, each side sums to 126.
There’s plenty more magic if you keep playing with magic squares, triangles, and flower petals. Happy puzzling!
