coins’ movement were introduced. If a coin lands on an occupied position, we can either let it replace the
existing coin or allow the two to share the same unit. In both cases, the argument above still holds – that
said, we still couldn’t jump a coin to reach level 5. Consider the weight of the grid when you try proving the
two scenarios and you will see what I mean.
Figure 8 visualises the grid with assigned
weights according to a different rule, that is
to allow only diagonal jumps. A horizontal
jump decreases the weight by the amount
equal to the coin jumped over, and a jump
downwards causes the weight to decrease by
double the amount. In this model, we are
lucky to be able to reach level 6 at max – the
best solutions consist of pretty much only
upward jumps.
There are more versions of the game you can explore and analyse yourself, such as ones that allow longer
jumps or more complex movements (that you can decide and try yourself). You can even adapt a different
grid format to the game and you will still find Conway’s proof helpful.
But what if we think out of the box? The article I wrote last year for Dulwich College’s Mathematica
Magazine had a bit of multi-dimensional sequences involved and I think I will inevitably go into the n -
dimension cliché again.
In a space with dimensions where all existing coins are placed in the lower ‘half’ below a hyperplane with a dimension of − 1 , we
allow coins to do the same movements – jumps are made in one of
the directions parallel to the axes. In this case, the highest level
(‘above’ the hyperplane) that can be reached is given by a simple
3 − 2 using John Conway’s weighing argument. Unfortunately,
the proof to this theorem is extraordinarily long and difficult, I am
not going to embarrass myself going through any of that – I’ve put
some links at the end to help you discover a bit more about
multidimensional proofs.
To briefly conclude this article, I hope you’ve enjoyed reading and experiencing the complexity of
mathematical game models – they are so closely related to our everyday life, from ‘Connect 4’ to single-
player puzzle games. Mathematics is such a powerful tool, unlike physicists we ‘generalize’ cases – and
John Conway’s proof is a very inspiring start for those of you who wish to begin viewing the world from a
very different perspective.
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