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? 

The unilluminable room...illuminated (literally!)

You enter a room. It is dark. You cannot see, so you light a lamp that emits light through a tiny hole. You notice the walls are covered with mirrors.Suddenly, you realize there is still a part of the room that is dark. Wherever you go,this is true.Where are you? Your'e in... Tokarsky's unilluminable room!

Tokarsky unilluminable room was discovered by George Tokarsky in 1995, as a response to Victor Klee's problem. There are actually two rooms, a 26-sided one and a 24-sided one, both of which work.
My idea for making it illuminable is the following:
Take Tokarsky's room and turn it into a bubble wand. Take this, dip it in some soap, and run.
Tell your friend to take a picture of the bubble just before it comes out.
Place the match at the top of this dome, and Tokarsky's room will be illuminated!

Click here to see Tokarsky's room.

A Mathematical Thought

I have just had an interesting mathematical thought, which I do not know the answer to.
Here it is:

I was watching a documentary called " The Story of Maths" hosted by Marcus du Sautoy, a documentary about the history of mathematics, and Marcus said that Euler proved that the infinite series 1/1+1/4+1/9... equals (pi^2)/6.
( My mother thought this very odd.) Anyway, my thought was this:
Supposed that you calculated all the results of all of these kinds of series ( thus, you would be calculating 1/1+1/8+1/27... and 1/1+1/16+1/81... and so forth.) My question is, is there a pattern between all these results? And if so, can you use this pattern to calculate the result for the next infinite series? ( that is, if you have the pattern and also have the result for the series before it, can you calculate the result of the series after it?)
Hope you enjoy this problem!

Books I've Been Reading

The Savage Stone Age (Horrible Histories) by Terry Deary

This book gives a very funny overview of the Stone Age. I learned that Neanderthals used to suck out the brains of other Neanderthals! I also learned that Stone Age people used to bury their dead (after a while) in burrows. They used to capture mammoths by making them run into tar pits. They also used to kill bison by making them run over cliffs. (Poor bison!) The book has some very funny cartoons and it should delight audiences of any age.



Number Freak: From 1 to 200: The Hidden Language of Numbers Revealed by Derrick Niederman


Number Freak is about numbers and all their interesting nooks and crannies. It informed me that Franklin D. Roosevelt was afraid of the number 13 and that in a 9x9 Sudoku grid; no puzzle with fewer than 16 givens (number clues) can produce a unique solution. I was also amazed by the de Bruijn cycle (it's made up of 16 ones and zeroes). I am still reading this book.



The Wind in the Willows by Kenneth Grahame and the audiobook narrated by Jim Weiss (both unabridged)
I read the entire Wind in the Willows in two days while listening to the audiobook. It's about a toad's best friends who try to cure his addiction for motor cars. They failed but he got into an adventure that eventually cured him of it. The book was extremely good and I'm sure it will charm audiences everywhere.

What I Learned Today

Life of Fred: Trigonometry by Stanley F. Schmidt
Chapter 1: It helped me remember what 'sine' is. Sine is the side opposite the hypotenuse dividing the hypotenuse. I did some of the Your Turn to Play puzzles in Chapter 1.

Birds, Beasts and Other Relatives by Gerald Durrell
I finished reading this book today. The book is very entertaining and is a biography of Gerald Durrell, the famous naturalist.