Skipping over the busy-work

Many people know this math problem, but I'm going to post it here anyway, because I've always liked the solution.

Add numbers 1 through 100.

Of course, you could just go through and add...

1+2=3

3+3=6

6+4=10

10+5=15*

... but that would be a lot of busy-work, and mathematicians don't approve of busy-work. 

*One small cool thing about the sequence you get from going through and adding everything (1, 3, 6, 10, 15, 21, 28...): these are the triangular numbers. 

So how can you do this in a much faster, easier, and more effective way?

To find the answer, go to answersdiscarded.blogspot.com. As usual, this is where I beg you to try to figure it out by yourselves before checking the answer.

If you've already done that problem, do this one:
Add all numbers divisible by three from 100 to 200.

Leonardo Fibonacci and the Fibonacci Sequence

Leonardo Fibonacci, or Leonardo of Pisa, wrote a book called Liber Abaci, in which he used a sequence of numbers he learned during his earlier travels as an example. He was a mathematician and was born in 1170. He died in 1250.  [more]

Math problem:

Fibonacci's Problem:

"Start with a pair of rabbits, (one male and one female) born on January 1. Assume that all months are of equal length and that :

1. rabbits begin to produce young two months after their own birth;

2. after reaching the age of two months, each pair produces a mixed pair, (one male, one female), and then another mixed pair each month thereafter; and

3. no rabbit dies.

How many pairs of rabbits will there be after one year?"

Solution: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

So 144 pairs of rabbits.

The Fibonacci Sequence.

Why is the Fibonacci sequence beautiful?

(quick summary.. way too quick).
Fibonacci numbers really do appear in nature. For example, some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, too: daisies can have 34, 55, or even 89 petals.
The next time you look at a sunflower, look closely at the arrangement of the seeds. They appear to be spiraling outwards both to the left and the right. There are a Fibonacci number of spirals!

Another amazing thing about the sequence is the fact that, the higher up in the sequence, the more closely the ratio of two consecutive Fibonacci numbers approaches the golden ratio (approximately 1:1.618 or 0.618:1).

Divisibility:
Every 3rd number of the sequence is even and, more generally, every kth number of the sequence is a multiple of Fk. Thus the Fibonacci sequence is an example of a divisibility sequence.

Comment if you don't understand something.

Counterfeit Coins

You have nine identical coins. One of them is counterfeit. The counterfeit one is a tiny bit lighter than the normal ones. 

In two weightings (see picture), how can you figure out which one is counterfeit?

The answer will soon be at answersdiscarded.blogspot.com.