This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with significant food for thought.
A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.
Weekly problem 48 - 2006
Nearly $80$ solutions were sent in! These had many examples that had even answers. We did suggest that you went a bit further and here we have some ideas.
First from Owen
from the Montessori School of Wooster, Ohio
A square number subtracted by its self (X squared - X) always has to be even. It will always be a even number because an even number times a odd number ($3 x 3 = 3 x 2$ ) is always even
(ex. $4 x 4 - 4 = 4 x 3 = 12, 5 x 5 - 5 = 4 x 5 = 20$)
from Claremont Primary
School in the UK
had the following thoughts;
$2 x 2 = 4$
$4 - 2 = 2$ (even)
Answer is even with an even number starting point
$3 x 3 = 9$
$9 - 3 = 6$ (even)
Answer is even with an odd number starting point
Therefore answer is always even. Why?
The answer will always be even because an odd number squared makes another odd number. An even number squared makes an even number. And an even number take away another even number makes another even number. But quite strangely an odd number take away another odd number also makes an even number.
From Jasmine, Arran, Rebecca and Thomas
at St. Mary's School Shawbury
in England , we had a similar thought;
The answer is even. We know this because, when you square an even number you always get an even number, but when you take an even number away from the number that you now have, you still remain an even number, no matter what number you started with.
When you square an odd number, it should result as another odd number, but when you subtract an odd number, from the number you have now, it will result as an even number.
e.g. Even squared = even - even = even.
e.g. Odd squared = odd - odd = even.
We tried this with $1,2, 3$ digit numbers and they all ended even, we also tried different odd and even numbers and found the same result, all the answers were even.
from Bourton Meadow Academy
It is always even because the multiples of an even number are even
$(2a)(2a+1) = 6a$
If you use an odd number it is simply this formula backwards
$(2a+1)(2a) = 6a$
Finally Victor and Elliott
from Kenakena School
in New Zealand
$X² - X =$ even. If $X$ is odd. $X²$ is odd. so $X² - X = $ even.
If $X$ is even. $X²$ is even. so $X² - X = $even.
Side proof : how even - even = even and odd - odd = even:
If n and e is an arbitrary whole number, then even = $2n$, so even - even $= 2n - 2e = 2(n - e)$. Thereby, even - even = even.
So odd$ = 2n + 1$. So odd - odd = $2n + 1 - (2e + 1) = 2n - 2e + 1 - 1 = 2(n- e)$.
Thereby, odd minus odd = even.
We had a few very late in the month suggestions from Jack, Daniel, Sophiie, Isobel and Ryan at Keidmarch Primary School
Thank you all for your hard work and the emails.