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We had a variety of solutions sent in with
different explanations. Here's the first that came to us and it's
from Y6B from Newton Primary School:
If you start with an even number, the square will always be even.
When you subtract any number from an even number, the answer is
always even. It turns out even every time because if you start with
an odd number, the square is odd, and if you subtract an odd number
from an odd number, the answer is always even.
VISUALISATION: The visualisation of the dots helped us because you
could see the dots that had been subtracted. You could see that
there were the same number of dots on each side. There will always
be the same number of dots on either side of the line because a
square is symmetrical and so they have to have the same number
either side of the line... odd+odd=even and even+even=even.
Abhishek got into algebra and sent in
this neat solution.
The answer will always be even.
For example, let's say the number is $x$
so, $x^2 - x = x(x-1)$
which is the multiplication of two consecutive numbers, one of
which will always be even.
And multiplication of an even and odd number is always EVEN.
Hafizur from Stepney Green Maths and Computing
College London, also sent in a good explanation.
It will always be even because:
If an even number is multiplied by itself, another even, then you
wil always end up with an even number. If you then take away from
it another even number, itself, then you will be left with an even
$16-4=12$ (even take away even is always even)
If an odd number is multiplied by itself, another odd number, then
you willl always end up with an odd number. If you then take away
from it another odd number, itself, then you will be left with an
$49-7=42$ (odd take away odd is always even)