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	?

Wapiti or Bears

You are trying to get evidence that there are more bears than wapiti in the Jasper National Forest. Your research assistant, the renowned scientist Monkton Tuesdale, insists there are more wapiti. But you think there are more bears, not because you have any scientific reason, but because you like them more. Monkton calls you a fool.

To prove you're right, you set up a fenced area in the forest with a pressure pad at the gate that can measure if a bear or a wapiti enters the area. In the center of the area, you place a large pile of apples.

After leaving the site for 24 hours, you return to find that 63 apples are gone. You know from previous research that bears eat three apples per visit while wapitis only eat one, so you write down in your notebook:  3·b + 1·w = 63. You check the pressure plate computer and see that 29 animals have entered the area so you write down: b + w = 29.

How many bears and how many wapitis entered the fenced area?  And who is the fool, you or Monkton?

Power-Ball Lottery

Explorers who know math know that playing the lottery is dumb.  The chances of winning are almost always lower than the amount you win. But today, January 12, 2016, the winning amount is $1.5 billion! Even after taxes, the odds are in your favor because there's a 1 in 292 million chance of winning. However, if there are two or more winners and you have to split the pot, the odds then become dumb again.

So this math-wise explorer is not playing the lottery and did not buy a ticket.  But if he did, he would have picked these five white-ball numbers: 12, 13, 36, 43 and 57 and for the red power ball: 5.

Let's see if he regrets not buying a ticket worth over a billion dollars!

UPDATE: Upon further reflection, if those numbers DO win, I'm going to feel like an idiot and I hate that feeling! So I bought a single ticket with those numbers. And just to make it more fun, I'll share the money with anyone who can tell me the exact number of possible tickets in the Powerball. The rules are: all five white balls (1 to 69) have to match, but they can be in any order, and the red ball (1 to 26) has to match.  Good luck!

Exact number of possible tickets:

Solve this problem to get one million dollars!