= −1∓√1 2 −4×1×(−1) 2×1 > 0 ( ′ ℎ ′ )
√5 − 1 2
∴ =
Looks familiar? It is the Golden Ratio . In a seemingly unrelated 2d chessbaord game, it still appears – I hope
that everyone who’s seen this for the first time would be amazed by the beauty of mathematics. Do keep in
mind that this value is smaller than 1 (approximately 0.618) and it will impact our algebraic calculations.
The board has to start with a weight of at least 1 on this massive grid which we can vaguely describe as
infinite at the moment.
With the value of in our hands, we can now zoom out and work on the total weight of the grid. You may
find revisiting figure 5 helpful.
Due to the symmetrical pattern of the coins below the red line (which is the total weight before any jump has
been made) we can basically calculate the sum of the middle column and every neighbouring column
starting from the middle one times two.
The weight of the middle column can be calculated:
= 5 + 6 + 7 + 8 …… ( ′ ′ ) Notice that this is a geometric series with 5 as its starting term and 1 as the ratio between the successive
terms. How is this helpful? Have a look at how a geometric series can be simplified according to the
formula:
∞
+ + 2 + 3 + 4 + ⋯ = ∑ = 1 −
ℎ | | < 1
=0
Therefore:
5 1 −
=
Remember the key characteristic of the golden ratio:
2 = 1 −
Hence:
5 2
= 3
=
The contributors of the two columns on either side of the middle column produce very similar series, but
they start with a slightly higher weight due to the further distances from the destination coin. The columns
further out also increase in the same patter – I will put figure 7 to help you understand it.
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