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Thread: fun question

  1. #1
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    fun question

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  2. #2
    Senior Member
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    All hexagons can be made by cutting away 3 equilateral triangles from a larger equilateral triangle.

    All possible n is of the form:
    $\displaystyle n = k^2 - k_1^2 - k_2^2 - k_3^2$
    where $\displaystyle k_i + k_j + 1 \leq k$ for $\displaystyle i\neq j$

    e.g. $\displaystyle 6 = 3^2 - 1^2 - 1^2 - 1^2$
    $\displaystyle 10 = 4^2 - 2^2 - 1^2 - 1^2$
    $\displaystyle 13 = 4^2 - 1^2 - 1^2 - 1^2$

    The next few would be
    $\displaystyle 5^2 - 2^2 - 2^2 - 2^2 = 13$
    $\displaystyle 5^2 - 3^2 - 1^2 - 1^2 = 14$
    $\displaystyle 5^2 - 2^2 - 2^2 - 1^2 = 16$
    $\displaystyle 5^2 - 2^2 - 1^2 - 1^2 = 19$
    $\displaystyle 5^2 - 1^2 - 1^2 - 1^2 = 22$
    ...
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