#36: The Numbers = ???

" There are exactly six solutions to the following equation:

x^6 - 108x^5 + 4405x^4 - 87270x^3 + 881464x^2 - 4239552x + 7418880 = 0 {1}

The six solutions (roots) are 4, 8, 15, 16, 23, 42.

The {1} equation can also be rewritten as:

(x - 4) * (x - 8) * (x - 15) * (x - 16) * (x - 23) * (x - 42) = 0 {2}

The 108 and 7418880 numbers in the expanded form {1} are a direct result of simply expanding the {2}. In any equation in the form {2}, the coefficient of x^5 is always the addition of all the solutions (roots) and the loose term (with no x^n) in {1} is always the multiplication of the solutions (roots). For example:

{1}: x^6 - 30x^5 + 354x^4 - 2080x^3 + 6309x^2 - 9090x + 4536 = 0
{2}:(x - 1) * (x - 3) * (x - 4) * (x - 6) * (x - 7) * (x - 9) = 0

The solutions to the above equations are 1, 3, 4, 6, 7 and 9. 30 is 1+3+4+6+7+9 and
4536 is 1*3*4*6*7*9.

Alternatively, perhaps the Numbers are the first six numbers infinitely long sequence. One such sequence is defined by S(n)=(1/40)(2400 - 4896n + 3670n^2 - 1175n^3 + 170n^4 - 9n^5) (derived using the Lagrange Polynomial). According to this formula, then next equation in the sequence is 46.

Or maybe the Numbers are a mixture of several sequences, for example:
2 4 8 16 32 64...
where next number is previous multiplied by 2, and
2 8 15 23 42 52 63...
where you have to add 6, 7, 8, 9... accordingly to the previous number.
Also, a random equation was found between these equations.
First of all 4 x 8 x 15 x 16 x 23 x 42 = 7418880
now look 4+8+1+5+1+6+2+3+4+2=36
7418880 7+4+1+8+8+8+0=36

coincidence? maybe. "



Yep. Uh huh (cricket chirp, cricket chirp). Can you explain that on an abacus?