Pharaoh Shuffle

I just learned the trick of Faro Shuffling.

Lets say you have a brand new deck of cards, each suit organized ace to king.  A faro shuffle would be what you would consider a perfect shuffle.  Split the deck into even halfs, then create a new deck using one card at a time from each half deck.  Seems random enough, but if you were to do this 8 times - you would get your original order back.

I just tried this with 26 cards, with hearts and clubs.  I laid them out on the table face up each time to witness the "randomness" of them.  I thought I had made a mistake somewhere, but to my astonishment, they returned to their original order.  I also doubted myself in the middle thinking "Maybe this only works with 52 cards."  But my intuition told me that this would work with any even number of items.  And so I have proven it to myself.

Any even number of items in a certain order, when faro shuffled 8 times, will reproduce that order.

I totally thought there would be a nice java applet somewhere on the net showing this amazing truth, but there is not.  Google searches for faro shuffling produce card tricks and not math.   I would write a program for this, but I'm not too good with arrays.

Perfect Number play

So there is this set of numbers out there that just happen to have their proper divisors add up to itself.  For some reason, everyone wants to call them "Perfect numbers."  I think thats a bit presumptuous, but whatever.  I didn't make it up.  So a perfect number, again, is defined as a number who's divisors add up to itself.  See the wikipedia article and the OEIS list for more info on them.

Well, what if you continued to apply the "perfect number operation" onto a number and then its outcome over and over again?  Random example:


216 has the divisors 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72 and 108.
so
1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 27 + 36 + 54 + 72 + 108 =  384

so then we look at 384 who's divisors are

1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128 and 192

so
1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 + 32 + 48 + 64 + 96 + 128 + 192 = 636

and so on with 636 which turns into 876
876 -> 1196
1196 -> 1156
1156 -> 993
993 -> 335
335 -> 73
73 -> 1

and bingo bango we have come to an end.

Not all numbers come to a small end like this though.  Some of the numbers get HUGE (30 out of the first 1000 actually)  I didn't calculate past 100,000,000 ( 100 million)  - better things to do you see.  Also, some of the numbers get caught into infinite loops.  Once you come across one of the perfect numbers, they are their own outcome and its futile to work on it further.  Also 220's outcome is 284 - and 284's is 220!!  Also 1184's is 1210 and 1210 is 1184!!!  very interesting relations there.

SO I present to you the list from 10 to 1000 of continued perfect number operations.  I need to come up with better names for things....what should I call this?

(seems the service I was using to link my large text files doesn't want me to.  I need to find a different one.  Anyone have any suggestions?)

One of the neat ones:

864 -> 1656 -> 3024 -> 6896 -> 6496 -> 8624 -> 12580 -> 16148 -> 14764 -> 11080 -> 13940 -> 17812 -> 14304 -> 23496 -> 41304 -> 62016 -> 120864 -> 196656 -> 343488 -> 565832 -> 495118 -> 316322 -> 158164 -> 118630 -> 94922 -> 52150 -> 59450 -> 57730 -> 51134 -> 27754 -> 13880 -> 17440 -> 24140 -> 30292 -> 22726 -> 14498 -> 9262 -> 5930 -> 4762 -> 2384 -> 2266 -> 1478 -> 742 -> 554 -> 280 -> 440 -> 640 -> 890 -> 730 -> 602 -> 454 -> 230 -> 202 -> 104 -> 106 -> 56 -> 64 -> 63 -> 41 -> 1

Palindrone

So about 6 months ago I made a list of numbers between 100000 and 999999 that are prime in both lexiconographical order.

here's a sample:

384187, 384203, 384253, 384257, 384301, 384359, 384383, 384473, 384497, 384581, 384589, 384611, 384673, 384737, 384751, 384821, 384841, 384889, 384913, 384961, 385039, 385079, 385087, 385159, 385171, 385223, 385249, 385267, 385291, 385321, 385331, 385393, 385471, 385621, 385709, 385741, 385811, 385837, 385843, 385901, 385991, 386017, 386041, 386119, 386149

The full list can be found here:
http://ompldr.org/vNXZjbw

I don't recall how I isolated this list, but here it is none the less.
Yes I know that Palindrome isn't spelt with an "n"
And no I didn't know what lexiconographical even meant until someone on the net told me.

Usefulness: 0.003
Fun: 2.3

Look and say

So the local minimums and maximums for the polynomial of of the Conway Constant are APPROXIMATELY

88.4609
-36.3293
-2.3595
and
-917730.85932

Whats the polynomial of the Conway Constant?  Well, I'm glad you asked.  Its

THIS bad boy.


As text thats:

X^71-X^69-2*X^68-X^67+2*X^66+2*X^65+X^64-X^63-X^62-X^61-X^60-X^59+2*X^58+5*X^57+3*X^56-2*X^55-10*X^54-3*X^53-2*X^52+6*X^51+6*X^50+X^49+9*X^48-3*X^47-7*X^46-8*X^45-8*X^44+10*X^43+6*X^42+8*X^41-5*X^40-12*X^39+7*X^38-7*X^37+7*X^36+X^35-3*X^34+10*X^33+X^32-6*X^31-2*X^30-10*X^29-3*X^28+2*X^27+9*X^26-3*X^25+14*X^24-8*X^23-7*X^21+9*X^20+3*X^19-4*X^18-10*X^17-7*X^16+12*X^15+7*X^14+2*X^13-12*X^12-4*X^11-2*X^10+5*X^9+X^7-7*X^6+7*X^5-4*X^4+12*X^3-6*X^2+3*X-6

But I really wouldn't recommend putting that into Wolfram Alpha :OD

Usefulness Rating: 0.02
Fun Rating: 5.01

I would like to thank Mathworld and KmPlot

Optimus

Why are prime numbers so notoriously hard to write and find proofs for?

Take the twin prime conjecture:
There are an infinite amount of consecutive primes.
ie 11&13 ; 41&43 ; 197&199

This is an open question in math and hasn't been proved or disproved yet.  There are however A LOT of open questions about prime numbers.

I would like to see some solved and or proven conjectures using primes to see how easy / hard it is to write such proofs.

Emergence

SO!  Here is it.  The first post of my blog about math, science, philosophy and politics.

The name I chose was based on my analogy regarding my explorations into math.  A few months ago I started to explore into a certain direction of math.  In essence I asked "Where does this path lead me?" and to my astonishment I found someone else's well worn path from a different direction with a sign post and notebook.  I was hooked.

I see mathematics as a truly infinite forest with paths criss crossing and diverging everywhere.  I only hope that one day I can place a sign and flag with my name on it somewhere in that forest so that others may cross it in their own exploration.  Or at the very least, chip away at the eternal and infinite mountain known as IGNORANCE and help do more research into the unknown.