This makes the total weight of the board:
= + 2 × = 3 + 2 × [( 6 + 7 + 8 + ⋯ ) + ( 7 + 8 + 9 + ⋯ ) + ( 8 + 9 + 10 + ⋯ ) + ⋯ ]
Using the same formula……
= 3 + 2( 4 + 5 + 6 + ⋯ )
= 3 + 2 2
This equation is not so helpful… But if we use the amazing property of = √5−1 2
again , we can come up
with this:
= 2 ( + 2)
= (1 − )( + 2)
= 2 − − 2
We can continue by using the property once more:
= 2 − − (1 − )
= 2 − − 1 +
𝒘 = 𝟏
Turns out that the total weight of the infinite grid at the very beginning, is 1 . Remember that this is also the
exact value of the final grid with only the coin that has reached level 5. This then forms a contradiction,
since we’ve previously discussed how the total weight of the board has its special properties – it can’t
increase. That said, no matter how many coins you have, you will never be able to practically reach level 5 –
whereas level 4 seemed like nothing but a small challenge.
This concludes the proof of the fact that one cannot jump a coin five levels above the initial line.
There are many variations of this Conway’s Soldiers game – lots can be analysed using the same concept.
John Conway had encouraged such a innovative style of interpreting games, I feel the need of just talking a
bit more about several cases that can be created on top of Conway’s Soldiers.
The first one would probably explain an issue that might have occured when the grid full of coins and the
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