Hami, China

The city of Hami in China is also known as the city of Kumul in the province of Xinjiang. Hami is famous for its melons, but not much else. The Han people were in constant strife with the Xiongnu people over this region, as it was very fertile. When the Mongols invaded a small kingdom was established here called Qara Del. Benedict Goës visited on his travels through the area when he traveled to Cathay to see whether or not it was China.  Apparently, in the past the Chinese procured diamonds, gold and agate from this region.

Researched from http://en.wikipedia.org/wiki/Hami_City for Popsicle Stick Project #1
 

Lamarck vs Darwin on Woodpeckers

What would Lamarck have said about a woodpecker's cushioned head? And how would Darwin have replied? A hypothetical conversation:

Lamarck: Bonjour Charles!

Darwin: Cheerio Jean-Baptiste!

Lamarck: Ze woodpecker is a hard-headed old fellow no?

Darwin: Yes, thanks to natural selection.

Lamarck: Aah mon ami (shaking head dubiously). I say it is due to using his head over and over again to make holes in ze trees! He uses it and uses it and develops a cushioning layer from all this use. That cushion protects him. His little baby woodpecker inherits it from him no?

Darwin: Jean-Baptiste, you have made a mistake. The woodpecker inherits the cushioned head as follows:

Imagine a large population of woodpeckers. Some have well-cushioned heads, some have less-cushioned heads.

Woodpeckers with well-cushioned heads can peck holes in trees more effectively than woodpeckers with less-cushioned heads. This enabled the woodpeckers with well-cushioned heads to survive better during times of low food supply than their less-cushioned counterparts.

The woodpeckers with well-cushioned heads have a trait that helps them survive longer. This makes them more attractive to female woodpeckers. And as a result, they reproduce more.

Another anagram puzzle

What does IS PASSPORT RENEWAL scramble to that has 3 words, and the last word is SWANS?

Anagrams

Quick puzzle: How are these words, sentences, and groups of words related?

I DREW

IS PARADE

AA CRIME

(By the way, aa is a kind of lava.)

Answer: They're all anagrams! Anagrams are when you take the letters of any word or sentence and rearrange them to make a diffrent word or sentence.

Once you've solved the above, try this:
Unscramble the words below. (Hint: Sherlock Holmes)

(Clue: It has four words)

YEMEN

ELATE

DRY

MAR

NOT

SAW

2 little mathematical riddles

1. Imagine you have 100029 marbles.Your friend has 11(x) marbles, you don't know what x is.
You compare marbles and discover that your friend has "n" less marbles than you. What is "n" if it is less than 12?
2.10 brothers have had an extremely good apple harvest. The 10 brothers have divided up the crop evenly when one brother suddenly has to do an errand. The remaining 9 divide up the crop evenly again when another brother walks off again. This repeats until one brother is left with all the apples. Every time a brother walked off the apples were divided evenly, and every time the brother that walked off did not eat an apple. What is the least amount of apples where this is possible?

Mathematics...finally! (no.1: Cayley multiplication table)

(updated 2/5/13)
Several people (read:Dad and Mom) have been asking me if it's Math and Other Adventures, where's the math? I finally gave in and decided to start a mathematics column.
Thought #1: In mod n, the left diagonal of its Cayley multiplication table is a palindrome.
Mod (modular) is a system of arithmetic that is repeated. In other words, it goes round and round.
Ex: (mod 6)
012345
012345
012345
012345
012345
...
A number in mod is restricted between 0 and n. If it got larger, it would get reset back to the beginning.
Ex:
012345
012345
012345
...
Bold number 5 = 17

Now, here is a more succinct definition of modulus:

k(mod n)(that means k in mod n) = remainder of k/n

Here's why:
Imagine two sticks, a and b'.

Imagine b' is bigger than a.
Now say we cut off part of b' so there's a part b equal to a:

By definition, c is the remainder of b'/a.

But wait! If b is equal to a, b is equal to 0:(in mod a)
01234...a-1(because then total number of numbers is a)
0
(Bold 0 = b)
But that means that if you add c to 0, you get c! Therefore, c is what b' is in mod a. (This also works when b is a multiple of a, as multiples of a are 0 as well.)

Now, what is a Cayley table? A Cayley table looks like the below. This is a Cayley multiplication table,which is basically a multiplication table for modulus.

Now take a look at the left diagonal of this table:(next picture). As you can see, it's a palindrome (1410141).

Here's why:
The left diagonal is made out of squares(1,4,9,16,25,...) with the last square being (n-1)^2. Now, n-1is the same as -1 in mod n. n-2 is also the same as -2 , and so on. Around some point k is when x gets close to -x.
Now, we know that (-x)^2=x^2.
That means than (n-1)^2=(-1)^2=1^2,(n-2)^2=(-2)^2=2^2, and so on. This means that the values in the left diagonal's left side is mirrored by the values on the left diagonal's right side. But this means the left diagonal is a palindrome. So we have finally proven that the left diagonal is a palindrome.
Thank you for reading!
Dash

Note: Similar logic can be applied to find that the left diagonal and pandiagonals are palindromes.

A Little Mathematical Riddle

I sure seem to be in the habit of taking sabbaticals! :)

Imagine you are Chief of Police Montgomery, and you want to arrest the Rogerson brothers.
You want to know how many brothers there are, in order to fit them all in your prison cell. You can't see how many brothers because they have taken sanctuary.
You do, however, know 6 facts:

1. The Rogerson brothers' father, Peter, was a TzuTzuist. The TzuTzuists believe that you should only have a single digit number of kids.
2. Peter was extremely rich. His fortune was worth an eight-figure odd number.
3. The last digit of the fortune is 9.
4. The digit sum of the fortune is 17.
5. Peter divided the fortune evenly among his sons.
6. The fortune each brother received is a whole number.

How many Rogerson brothers are there?