For my A-Levels I have been studying Pascals triangle, and I found that if you select any of the numbers from each prime number row, and divide this by the row number it is ALWAYS an integer. My maths teacher and I are stumped, any help would be appreciated.

http://www.metacafe.com/watch/yt-H1u8d04sKf8/geometrical_illusion/

http://www.metacafe.com/watch/828182/geometrical_trick/

Are there any explanations to the missing area? Is it just an illusion?

Every measuring instrument/device have percentage errors. And due to these percentage errors we can never consider any measurement to be a perfect whole number.

Eg: If we manage to measure out 1.0g of matter, we still can't say that it can't be 1.01g(which would no longer make it a whole number.).

So the question is, are whole numbers possible?

How many lottery tickets would have to be bought for there to be a >=50% probablility that every single possible combination has been chosen?  Or for smaller numbers how many people would you need in a room for there to be a >=50% probability that every day of the year is one or more of these people's birthdays?  

If these numbers are too large then how many people/tickets would be needed that there is a >=50% probability that >=50% of the lottery tickets/birthdays have been chosen?

Assume that birthdays and lottery tickets are chosen at random and not bunched, and forget about leap years to keep things slightly simpler.

Suppose the initial temperature of a substance is 0 degree celsius. The temperature then increases, to 20 degree celsius.

So here's the question. What is the temperature difference between the two, in Kelvin?

Most people know about 3Ds. And some people have theories on 4Ds. So my question is, how many theories on new dimensions(if there is any) have been created? What are these new dimensions called? And have these theories ever been acknowledged by other people?

Whilst studying Mechanics (M1), you often use "light and inextensible" to describe string. So my friend posed a hypothetical question that if you had some string that was 1million light years long with me on one end and him on the other. If he pulled one end how long would it take for the my end to move. I thought it would be instantaneous but my teacher thought different.

Time slows down the faster you travel. Living atop a tall building makes you age slower because you're travelling faster as the Earth spins. However, being further from the gravity well of the planet makes time move faster. Granted the changes in time are microscopic but which is the greater effect? Would you age fractionally slower or fractionally faster atop Burj Dubai relative to someone at ground level.