Let's look at one interesting isomorphism.

But first, let's play a game. The rules are very simple: we take turns
picking numbers between 1 through 9, without replacement (that is, if
one of us already picked say 2, nobody can pick 2 again). The first
person to use three numbers to add up to 15 wins! If we go through all
9 numbers without any one of us being able to add up to 15, it's a
tie.

Here's the thing: you're probably already very good at playing this
game but don't know it yet. ;-)

Here's a sample game:

Player 1's numbers | Player 2's numbers | Available numbers
--------------------+----------------------+------------------
5 | | 1 2 3 4 6 7 8 9
5 | 9 | 1 2 3 4 6 7 8
5 7 | 9 | 1 2 3 4 6 8
5 7 | 9 3 | 1 2 4 6 8
5 7 6 | 9 3 | 1 2 4 8
5 7 6 | 9 3 2 | 1 4 8
5 7 6 4 | PLAYER 1 WINS (with 6 + 4 + 5 = 15).

Notice that when player 1 grabbed 6, player 2 grabbed 2 in order to
block 6 + 7 + 2 = 15 but then couldn't block Player 1 grabbing 4 and
winning.

Play a few more rounds... see if you can figure out what this game
is. Believe me, this is a game you know very, very well. In fact,
you know this game so well, that you would normally only play it
during times of intense boredom, or you haven't played it at all much
since childhood.

Still haven't figured it out yet? Explanation below.

Right, the isomorphism between the 1-9 game and tic-tac-toe goes
like this. Draw a 3x3 magic square:

8| 1| 6
--+--+--
3| 5| 7
--+--+--
4| 9| 2

Then clearly choosing numbers between 1-9 is equivalent to playing X's
and O's in the tic-tac-toe magic square. It is further easy to see
that all 3x3 magic squares are the same as the one given here except
for rotations and reflections, which does not alter the structure of
the corresponding tic-tac-toe game either. To finally prove the
isomorphism, we must show that all 8 winning combinations of
tic-tac-toe correspond to all possible ways to add 15 with three
different numbers chosen from 1-9. This can be seen by systematically
enumerating all such possible sums:

9 5 1
9 4 2
8 6 1
8 5 2
8 4 3
7 6 2
7 5 3
6 5 4

I enumerated and grouped them in such a way that should be clear that
these are the only eight possible combinations of winning moves in the
1-9 game. Thus, the 1-9 game *is isomorphic with
tic-tac-toe*. Amaze your friends with your impressive finesse at
playing the 1-9 game.

There are a few lessons here. First, isomorphism is a concept that is
much more general than "a bijection or a homomorphism such that" blah blah
blah... Serge Lang liked to categorise with extreme generality the notion of isomorphism through
category theory: an isomorphism is an arrow with an inverse
arrow. That's cute (and it was more cute to watch him ask a bunch of
mathematics undergraduates what an isomorphism is and respond that all of
their function- or set-based answers were incomplete), but it still
fails to capture in my opinion what isomorphism is all about: same
shape, same structure, same basic concept, same
mathematics. Isomorphism is at the root of mathematics; indeed, I'm
tempted to say that isomorphism is the birth of mathematics; the
recognition that adding two apples is the same as adding two shekels
is quite profound.

Next, a good isomorphism allows us to understand an unfamiliar concept
in terms of the familiar. This we have already seen above. The last
point here is that one needn't too advanced or esoteric mathematics in
order to understand and appreciate interesting isomorphisms. :-)