How to get the adjacent blocks in Floyd triangle?

Question

I need to write a function to find the adjacent blocks in Floyd triangle.

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What is the formula to find the adjacent blocks (top, left, right, bottom) of a given value.

For example:

• input 20 → output left: 19, right: 21, top: 15, bottom: 26
• input 28 → output left: 27, right: -1, top: -1, bottom: 35
• input 19 → output left: 18, right: 20, top: 14, bottom: 25

Thank you so much in advance!

Answer 1

The shifts needed to go up or down are uniquely determined by the identifier o the line. If n >= 1 in the given value, you have to find the largest integer k such that:

k(k+1)/2 + 1 <= n <=> k^2 + k + 2(1 - n) <= 0

This is a second-degree polynomial function:

delta = 1 - 8(1 - n) = 8n - 7 > 0
x1 = (-1 + sqrt(8n-7)) / 2 and x2 = (-1 - sqrt(8n-7)) / 2

x2 < 0 < x1 so the 0-based identifier of the line is: k := floor((-1 + sqrt(8n-7)) / 2).

After that: up it's n - k, down it's n + k + 1, left n - 1 and right n + 1. The corner cases (leftmost/rightmost/...) can be spotted with k as well, left to the reader. ;)