[The Fingers]

It should be

Spoiler

0.  It only takes 1 person on the planet with no fingers on their left hand to make the multiplication equal 0.

 

[Area of a crescent]

The radius of the yellow+green circle is sqrt(2)/2. (r^2 = 2/4 = 1/2)

So the the area of the yellow triangle is 1/4 (b*h*1/2)
The area of the yellow + green part is 1/4 of the area of the circle (1/4 * pi * 1/2 = pi/8)

The area of the blue + green areas is half of the area of the circle (1/2 * pi * (1/2)^2 or pi/8)

So the area of the blue part is (blue+green) - (Yellow + green) + yellow

Or

pi/8 - pi/8 + 1/4

So 1/4

[50% increase]

is 285714

[unanswered]

I forgot the equation for seven, with that, we know that

s = 63/(5*sqrt(6))

a = 150*1080.0*sqrt(6.0)/(7*sqrt(5))

[unanswered]

So to sum up:

for "unanswered, the only individual letters we don't know are s and a, but we know the value for s times a.

u = 1

n = 1/9

a (part of sa)

n = 1/9

s: sa is 30 * 9720.0 / sqrt(5)

w = 1/5.0

e = 9

r = 36*sqrt(6)

e = 9

d = 30*sqrt(30)

[unanswered]

I miss the part where tw=2 (two=2, but I don't think we ever proved o=1)

Never mind, I figured that part out.

twenty/two = 10 = enty/o

10* (en/o) = 10

en/o=1 en=o

one=1 and o = ne, each would have to be the square root of 1.

[unanswered]

I didn't have time to figure it out all the way, but the answers are not integer, (Also u has to be one)

sixty/six = ty = 10 -> ty=10

forty/four = 10 = ty/u

u-> 1

twenty/ten = 2 = twy, 2 = two -> o=y

I had several other letter combinations that equal each other and a few that resolved to actual numbers.

Hope that helps some.

[not so easy Math Problem]

283.5Mi

252 miles of it is downhill from A to B and 31.5 miles are flat.

It takes 1/2 hour to go the flat part (31.5 mi/63mi/hour)

For the downhill portion , it takes 3.5 hours (252/72)

Going back (uphill) it takes 4.5 hours (252/56)

[Guess the number, binary style]

The basic goal is to cut the space in half with each guess.

So the first question is is the card 7 or higher (21 cards are 6 or lower, 24 are 7 or higher)

If the card is 7 or higher, the the next 2 questions are is it 9 and is it 8 (if both answers are "no", then it's a 7)

If the card is 6 or lower, Ask if the card is 5 or 6. This cuts the space in half(ish) again. Out of the 21 cards that are 6 or lower, 11 are 5 or 6.

If the card is 5 or 6, use your last question to see if it's 5 (and if not, you know it's 6).

If the card is 4 or lower (and we only have one more guess), ask if it's 4 (since we have the most of them). If it is, we are good.

If not, just guess 3 (we have 3 3's and 3 aces and dueses, so you have a 50/50 change).

In all there is a 100% chance to guess all cards 4 or higher (which is 39/45 of the time). In the other 6/45 times we get it right 50% of the time. The overall average for this is 42/45 times we get it right.