Spiral of Theodorus
If you continued the spiral, what would be the value of the last hypotenuse before the triangles start to overlap?
If the value is , express it as .
The answer is 4123.
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1 solution
The sum of the angles of the concurrent vertices is:
φ ( n ) = k = 1 ∑ n arctan k 1 = 2 n + c 2 ( n ) , where lim n → ∞ c 2 ( n ) ≈ − 2 . 1 5 7 7 8 3
The following is a plot of how the correction term c 2 ( n ) varies with n for the first 20 values of n :
2 1 6 + c 2 ( 1 6 ) ≈ 6 . 1 2 8 7 3 1 ≤ 2 π ⟹ n m a x = 1 6 .
Therefore, h = 1 6 + 1 and ⌊ 1 0 0 0 h ⌋ = 4 1 2 3 .