Now things begin to get more complicated. To reach level 3, you will probably need a minute to figure out a
solution, which is very likely going to look like figure 3 . At this point, you would probably feel that you still
have enough coins to into level 4.
I’ve seen people that have managed to do level 4 in only a matter of minutes but this is indeed quite difficult
in terms of the complexity when finding the minimum number of coins needed to reach level 4, which is 20.
Figure 4 demonstrates the layout of coins so that you can reach level 4 – have a look and try it yourself and
you will begin to appreciate just how dynamic and creative this game can get. Remember, a 6-year-old kid
can technically be quicker than you when working out a solution for the first time due to the very basic
spatial abilities required for this game, but you would certainly beat him in generalizing the games using
mathematics (which is what I am going to talk about next).
It is reasonable for you to think: ‘What if we try level 5?’ I admire your courage, but you’d probably spend a
lot of time trying to figure out a possible solution. In fact, it will become clear as if you are going to need
infinite amount of time with infinite many coins at your disposal. But the fact that we are not living in the
universe of Interstellar* means that we will have to confirm that reaching level 5 in Conway’s Soldiers is
impossible.
Prof. John Conway was able to turn this puzzle into a mathematical model and prove that it was impossible
to make it to level 5. Being able to derive so much information from a simple chessboard and some coins is
truly a magnificent accomplishment, just like his contribution to Conway’s Game of Life , which is a far
more complex and dynamic system that is still being explored to this day.
The first step is to give each coin that is sitting on the grid a value. This value can be literally anything, we
will leave this for later. The sum of the board is given by adding up all the values of these coins – the total
value of the current configuration on the board (we can also call it the weight of the grid). Figure 5 shows
how John Conway was able to do this, from a grid which level 5 has been achieved. Suppose a coin has reached level 5, the value 1 ( 0 ) is assigned to the coin. The goal is to then use contradiction to show that
this can never happen.
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