We also used a special case (mulling) which indicated the direction of the
proof.
Remember - one example can disprove a general theorem but one example cannot
prove a theorem.
The fact that opposite diagonal squares are the same colours
are a result of what you were given (although not explicity)
This is why one must write down everything that you were given
both explicitly and inplicitly
BUT remember the nine dots example - you must not assume facts
which you think you are given but you really aren't
It is a strong mathematical proof
although it does net write mathematical symbols.
What we have used is "what we are given" again.
The fact that a triangle is right angled means that
Pythagoras' Theorem holds.
This again is a piece of implicit information and is also
an abstraction of reality.
Once we move from a "real triangle" to the formula
a2+b2=c2.
Then we can prove it.
Instead of starting from the beginning
let us look at the end of the process.
At the end of the process, there are two beakers
with equal volumes of liquid.
One is mainly milk with some water
.
One is water with some milk.
If we conside the beaker that is mainly milk.
Where has the water came from?
The answer is of course the water beaker.
But as the water beaker has the same volume of liquid
the space vacated by the water must be filled with milk.
Thus
there is the same volume of water in the milk as milk in the water.
There was no need for the complicated algebra.
amd the proof is just as correct
The power of mathematic to abstract from the real world is often
seen as one of the aspects which makes it difficult and inaccessible.
It is also one of its sublime achievements
and can often help to solve practical problems.
Let us conside the coconut problem.
We have a pile of N coconuts.
Let us hypothesize that we also have a pile that contains -4 (minus four)
coconuts.
This pile cannot exist in the real world but can exist
in the mathematical cloud desvribed previously.
To keep the same number of coconuts, let us add 4 to the N-pile.
Thus we still have N+4+-4=N coconuts.
Now we can simulate the problem.
Conside the negative pile.
There must be at least 3125 coconuts in the large pile
(Rather neat solution - eh!!)
Let us ansume that when we throw one coconut to the monkey,
Throw one away (-4)-1=-5
Take away 1/5 (ie -1)
(-5)-(-1)=-5+1=-4
It is a miraculous pile, it stays invariant.
Thus we have looking for a number in the large pile
that can be divided by 5, five times.
The smallest such number is 55 ie 3125.
Therefore N+4=3125 => N=3121.