Primitive Solid

After building your time machine, you set it to 30,000 BCE to observe saber-toothed cats. However, a glitch in the software programmed by Monkton Tuesdale sent you back to only 300 BCE. Now that you are alone in ancient Greece, you don't dare use the time machine again until you fix the code.

You found the leading mathematician of the time, Euclid, and asked for his help. He has agreed to help you only if he deems that you are worthy. He needs only one question to determine this. If you can answer this question correctly, he will help you and you can get back to exploring time.

Question:  The three primitives of geometry that Euclid defined are the point, the line and the plane. These correspond to the dimensions zero, one and two. Which of the following solids is the only logical choice for a three-dimensional primitive given the progression from point to line to plane:

a) sphere
b) tetrahedron
c) cube
d) octahedron
e) dodecahedron
f) icosahedron

Good luck!

Happy Fabiroo Day!

It's been a year since your last survey of Fabiroo Island and you need to check to make sure all the fabiroo populations are healthy and not endangered of becoming extinct. A few months ago, your assistant Monkton Tuesdale set up extensive cameras on the island and wrote software so he could track every single fabiroo on the island, including births and deaths.

Unfortunately, he had a few bugs in the software which made it so he couldn't tell the difference between certain species of fabiroo. However, he claims it's not a big deal because you're a math genius and can figure out the individual populations with the data he's collected. It just requires a bit of algebra.

He's put the populations he was able to count in the table below. When he tried to count the number of sucker-footed fabiroos, he accidentally also counted the river fabiroos so he only knows that the total number of sucker-footed and river fabiroos is 45. Likewise, he only knows that the total number of cave and river fabiroos is 60.  Finally, he was able to determine that the number of cave and sucker-footed fabiroos is the same as the number of river fabiroos.

Can you figure out the populations of the last three species? And do any of them need your help to avoid extinction?

Fabiroo	Population
tree	120
mountain	80
bamboo	40
giant	30
cave	?
sucker-footed	?
river	?