!!!SPOILER about the cave "Math Marathon"!!!
As I promised in the game, here are the reasons for the correct answers in the cave "Math Marathon".


Level 1

1. 12+34=56 NO
Obviously false, the correct result is 46.

2. 2+3*4=20 NO
It would be correct if you could calculate from left to right, but multiplication has preferrance over addition, so the correct result is 14 instead.

3. 3+1.5=3*1.5 YES
The result is 4.5 on both sides, so the equation is indeed correct.



Level 2

1. 5*0=5 NO
5*0 means 0+0+0+0+0, so the correct result is 0.

2. 3²+4²=5² YES
3²=3*3=9, 4²=4*4=16, 5²=5*5=25, so 9+16=25 is correct. This is a so-called Pythagoras triple and it means that a triangle with side lengths of 3, 4 and 5 metres is a right triangle.

3. -3*{-4}=-12 NO
Two negative numbers multiplied result in a positive number, so the correct result is +12.

!!!SPOILER about the cave "Math Marathon"!!!

Level 3

1. 413^0=1 YES
Any number to the power of 0 is equal to 1.

2. pi=3.14 NO
The circular number pi is about 3.14, but not exactly. It is an irrational number, which means it is infinitely long and not periodic. It starts as follows - 3.1415926535...

3. square root of 4 divided by 0.5 = 4 YES
The square root of 4 is 2. If you divide 2 by 0.5, the question is how often the 0.5 fits in the 2. As 4*0.5=2, the result is indeed 4.



Level 4

1. i²=-1 YES
That's how the imaginary unit i is defined.

2. 40-32/2=4! YES
This one is a trap! If you would calculate from left to right, you would get the result of 4, but as division has preferrance over subtraction, the correct result is 24. But! On the right side we don't have "4", but "4!". The exclamation mark is the mathematical symbol for "factorial". 4! means 4*3*2*1, which is 24. So the equation is indeed correct!

3. 16^2.25=512 YES
16^2.25=16^2*16^0.25=16*16*{4th root of 16} = 16*16*2=512, so the equation is correct.



Level 5

1. e^{i*pi}+1=0 YES
Many mathematicians think this is the most beautiful mathematical equation, as it combines such important numbers like 0, 1, the circular number pi, Euler's number e and the imaginary unit i.

2. Integral 2x dx = x²+c YES
This is not easy to explain. For people who had integral calculus at school, this is one of the easier integrals, for those who hadn't, it's impossible to explain here!

3. Gamma {0.5} = pi
Hard to explain in this comment either. The correct value of the Gamma function at 0.5 is not pi, but the square root of pi.

Awesome! :-D