Even numbers are like 0, 2, 4, .. .., and odd numbers are like 1,3,5,7,9. We talk about odd and even numbers in the domain of integers.

All us of think the following is correct:

Even + Even = Even

Odd + Odd = Even

Odd + Even = Odd

With the sum of three numbers, it holds that:

Odd + Even + Even = Odd

Odd + Odd + Odd = Odd

With a little bit more thinking, for any sum of integers, if the count of odd numbers is odd, then the sum is odd; and when the count of odd numbers is even, then the sum is even. The count of even numbers does not matter.

All these seem trivial, but it could be very useful too!

Just for convenience, let us introduce “/” for odd numbers, and ⋈ for even numbers.

Therefore, the sum of three numbers which is odd can be:

/ + / + / = /

or / + ⋈ + ⋈ = /

Consider the magic square problems: fill in the numbers from 1 to 9 into the square (of 3-by-3 cells):

so that the sum of each row, of each column and of each of the two diagonals, are equal.

Find the total of 1 + 2 + 3 + .. + 9: it must be odd (because there are five odd numbers: 1,3,5,7,9) ! So the sum of each row must be odd, and so is the sum of each column, and so is each of the diagonals.

Now try to guess at: of all the nine cells, which one is an odd number, and which one is an even number?

We figured out there are only two possibilities so as to satisfy the “odd-sum” condition of all rows, columns, and both diagonals:

// Pattern 1

/ | ⋈ | / |

⋈ | / | ⋈ |

/ | ⋈ | / |

// Pattern 2

⋈ | / | ⋈ |

/ | / | / |

⋈ | / | ⋈ |

One can go one step further to colour the cells to get a nice pattern!

This is an example to demonstrate how much we can progress using the roperty of odd/even number ONLY. Although the solution is still not at hand, we have significantly reduce the space for searching!