Captain Smith sends an away team to check out a completely new, uncharted planet
ID: 3550544 • Letter: C
Question
Captain Smith sends an away team to check out a completely new, uncharted planet,
and they discover the ruins of an ancient pre-warp civilisation. Artifacts and pictures
suggest the creators of the civilisation were four legged beings, that had one long tentacle
coming out of their back, with the tentacle branching at the end with a number of grasping
"fingers". Many of the artifacts contain inscriptions that appear to be mathematical, and
the first to be translated (with limited away-team gear) comes out as the equation
5x^2 - 50x + 125 = 0
with the indicated solutions x = 5 and x = 8. The first solution looked right, but
the second was puzzling. Seven-of-Nine (as she often does) quickly came up with an
explanation: like that of humans, the base of the number system for this civilisation is
equal to the total number of fingers. So how many fingers did these beings have? Please explain.
Explanation / Answer
Let n be the base of the number system (n has to be greater than or equal to 9, since we know they have 8 as a number). Then the quadratic equation becomes:
5x^2-5nx+(1n^2+2n+5) = 0.
Dividing by 5 gives
x^2-nx+ (n^2+2n+5)/5 = 0.
We know that x=5 and x=8 are solutions to the quadratic.
Thus, (x-8 )(x-5) = x^2-nx+(n^2+2n+5)/5
So x^2-5x-8x+8*5 = x^2-(5+8 )x+8*5 = x^2-nx+(n^2+2n+5)/5
So n = 5+8 = 13.