Square Numbers Surprises
By Ankit Kapoor
Add two consecutive square numbers and then subtract 1:
After trying out the formula with 5 different consecutive sets of square numbers, we were able to figure out that all of the numbers were even. Here is what we got:
1 + 4 - 1 = 4
4 + 9 - 1 = 12
9 + 16 - 1 = 24
16 + 25 - 1 = 40
25 + 36 - 1 = 60
To start off with, we saw that every time you multiply an odd number by an odd number (which in this case happens with every odd square number) the product is always odd. This pattern also occurs when you do the same with even numbers, with the exception of the product being even. After this, you can see that you have an odd number and an even number; when you add them together, the sum is always odd, but since we are subtracting 1 it leads us down a step to an even number. As a result to the order of the square numbers being odd, even, odd, even this sequence will always occur no matter what. Here is a formula to represent this, with O being odd and E being even:
n² = O
(n + 1)² = E
O + E = O
O - 1 = E
2) Square any odd number and subtract 1:
As explained in the earlier question, when you multiply an odd number by an odd number the product is always odd (again, this works when you square odd numbers because you are still multiplying an odd by an odd), so when you subtract 1 from the product, it will always be even. The formula is O x O - 1 = E
3) Multiply two consecutive odd numbers and then add 1:
Again, as previously explained in questions 1 and 2, if you multiply any odd numbers (consecutive or not), the product has to be odd. And since it is a known fact that if you add 1 to any odd number, the sum will be even, the answer will always be even. The formula is O x O + 1 = E.
4) Multiply two consecutive even numbers and then add 1:
Again, as aforementioned, when you multiply two consecutive even numbers, the product will always be even. But, since you have to add 1, the answer ends up being odd, since E x E + 1 = O.