Why is 2 the only prime number? What makes 2 special?

From the time we learn about prime numbers, we quickly notice that 2 is the only even prime number. All other prime numbers are odd. So why is 2 so special?

The “specialness” of 2 arises less from the number itself, and more from the limitations of our English language. We have the word even, which means divisible by 2 and we have the word odd which means not divisible by 2. Half of whole numbers are even and the other half odd. Or to put that mathematically, 1/2 of whole numbers are even and 1/2 are odd. Generalizing one step further, which may seem unnecessary at the moment, but the reason for which will become apparent as we continue, (2-1)/2 numbers are odd.  

But we lack words for divisible by 3 and not divisible by 3. So let’s create these words now. Let’s define “theven” as divisible by 3 and “thodd” as not divisible by 3. We can see that 1/3 of numbers are theven and (3-1)/3 or 2/3 of numbers are thodd. Now we can note that 3 is the first and only prime number that is theven. Any number greater than 3 that is theven is not prime because it would be divisible by 3. Or, to put it another way, prime numbers greater than 3 must be thodd. Combined with our prior rule about odd numbers, we can say that prime numbers greater than 3 must be both odd and thodd.

Turning now to 4, 4 is even… so it is not prime.

5 is the next number that is both odd and thodd so it is our next prime. Let’s create the words “feven” and “fodd” for divisible by 5 and not divisible by 5. We can see that 1/5 of numbers are feven and (5-1)/5 or 4/5 of numbers are fodd and only fodd numbers above 5 can be prime. Combining all the rules together, numbers above 5 must be odd, thodd and fodd to be prime.

6 is both even and theven … so definitely not prime.

7 is the next number that is odd, thodd and fodd so it is the next prime. We get the new words “seeven” and “seodd”. Numbers above seven must be odd, thodd, fodd, and seodd to be prime.

8 is even… 9 is theven… 10 is both even and feven…  None of these are therefore prime.

11 is the first number that is odd, thodd, fodd, and seodd so it become our next prime and we get the words “eleeven” and “eleodd”…

We can continue along this path forever. Perhaps our new terms theven and thodd might have some utility in the English language. I could say that my two friends and I only buy pizzas that are cut into a theven number of slices like 6 or 12 so we each fairly get an equal number of slices. If we buy a pizza cut into a thodd number of slices like 8, we are not able to distribute the slices fairly. But, it seems that seeven, seodd, eleeven and eleodd and further of these new word creations would have diminishing utility in every day life.

Now let’s return to the questions of why is 2 the only even prime number and what makes 2 special? Well, 2 is special because of the limitations of our current vocabulary. By now adding these new words to our vocabulary, we can see that “2 is the only even prime number”, “3 is the only theven prime number”, “5 is the only feven prime number”, “7 is the only seeven prime number”, "11 is the only eleeven prime number", and so on. 

Now, all the prime numbers are special in their own terms…

Pet Years to Human Years

 

Pet	Pet:Human Year
Dogs	7 years
Cats	4 years
Gold Fish	8 years
Rabbits	8 years
Gunea Pigs	10 years
Iguanas	6 years

Meatball Poker - Draw Odds

Playing Meatball Poker obviously gives very different starting hands over regular Texas Hold 'em.   Since every player is dealt 3 cards and needs to discard one faceup meatball, clearly players are going to be starting with more pairs and flush draws than in a 2 card deal.   With better hands, many players will be less likely to fold pre-flop as well.  So what are the odds of drawing the various hands?

Hand	Meatball (3 cards)	Hold 'em (2 cards)
Any Pair	17.18%	5.88%
Pair J or better	5.29%	1.81%
Specific Pair(i.e. AK)	0.453%	1.32%
Any Suited	60.24%	23.53%
Suited J or better	5.29%	1.81%

There are also some interesting draws that occur only in Meatball Such as a three of a kind draw or three suited cards.  In these cases, you'll need to turn over one of your cards as your meatball diminishing your odds of making hands post flop.

Another situation that arises in Meatball are draws such as a pair and an ace.  In these cases, you may choose to flip your ace as the meatball signaling your hand strength to the other players.   The odds for these situations is shown here:

Hand	Meatball (3 cards)
3 of a kind	0.24%
Specific 3 of a kind (i.e. AAA)	0.02%
AAK(or KKA, QQA, etc...)	0.11%

Meatball Poker

Meatball Poker is a poker variant based off Texas Hold 'em, designed for smaller groups of players with increased frequency of exciting hands.  It uses all the same rules, but with one big difference.  Each player is dealt 3 cards and must decide to keep 2 and to discard one card (the "meatball") face up for the other players to see.  After that the game proceeds exactly like hold 'em with blinds, betting, the flop, and so on...

Why Meatball?  

If you've played a lot of Texas Hold 'em, you'll know that you can play for many hands, over many hours, with little action.  And too often, when you finally do get a good hand and lucky flop, the other players either muck their cards or you end up on the receiving end of a cooler.   Long time players will tell you, it's all part of the game.  But who has time for that?  More over, if you only have 3 or 4 players, it's often not even worth it to start up a game of hold 'em.

Meatball was created to make the game fun again, when you don't want to wait all night or for a large group of friends to show up.  It's great for casual games down in the basement with a group of friends. And it's easy to bring new players up to speed if they've played any poker before.

The Rules

If you know how to play Texas Hold 'em, Meatball will be easy to learn.  Specifically, the game begins much the same way, with a dealer at the button position, the player to the left obligated to bet the small blind, and the next player to the left obligated to bet the big blind (typically twice the small blind).  For a three player game, those would be the only positions.  The positions rotate clockwise each hand.

The dealer will shuffle, present the deck for the cut, and then deal 3 cards face down to each player.  The big and small blinds are paid.  At this point, each player must choose one of their cards, (known as the meatball), to be discarded.  This card is moved, still face down, toward the center of the table (typically across the betting line), but still in a position clearly in front of each player.  When all players have chosen which of their three cards they will discard, the dealer will announce the flip at which point all players flip their discarded card to now be face up. This could, for example, be announced as '"flip" or "balls up". Each player retains two hidden (face down) hole cards.  

The flipped cards (the meatballs) are dead cards, and can not be used in forming a hand. But they represent an important part of the game and must remain face up in front the player who discarded them for the remainder of the game.  They give important information about not only what cards may be remaining in play, but also about the potential strength of the hand of their owners.  Even players who wish to fold pre-flop must choose a meatball to be displayed face up.  Players new to meatball may mistaken choose to fold or muck all 3 cards.  The dealer should return the cards to the player as no cards may be folded until after the flip is complete and turn play begins.  

The new dynamics of having face up meatballs will be discussed in some detail, but at this point, play continues using traditional hold 'em rules.  Betting begins to the left of the big blind, then the flop, another round of betting, the turn card, betting, the river card, and a final round of betting.  A burn card should be discarded each time before dealing the flop, the turn and the river.  The community cards shared by all players consist of those dealt in the flop, turn, and river.  Flipped meatlball cards, are not community cards and can not be used in forming hands. Players may check, raise, call, or fold.  Any player who folds, must leave their flipped 'meatball' card in place until the conclusion of the hand.  The winner is the player with the best hand per normal poker hand values using the best combination from their two remaining hole cards and the five community cards from the flop, turn, and river.

The Dynamics of Meatball

The presence of the flipped meatballs change the dynamics of the game in three important ways.  First, each player gains a little insight into the other players potential hands. For example, if a player chose to discard an 8 of spades as their meatball, you might deduct that it is most likely that player's remaining cards are 9's or higher.  However, the player may have chosen to hold on to a pair, like a pair of 7's or may have a suited  Ace-6 of hearts or perhaps a suited 2-3 hoping for a flush or straight draw.  It can be much more informative if the player discards a higher card, for example a king.  Is that player holding onto pair of aces (the nuts)? or perhaps a suited A-K?  Being dealt 3 high cards (like A-A-K) is a mixed blessing in meatball since one of those cards will need to be publicly discarded.  This is certainly a rare occurrence, but adds to the interesting dynamic of the game.

The second way the game dynamics are changed, is that the frequency of better hands is significantly increased around the table.  Players of Texas hold 'em will need to adjust their thinking of the likelyhood that another player will be holding an A or a K, for example.  [A detailed analysis of the probability of drawing various hands will be presented in a future post].  For this reason, you'll see big hands more frequently through the night or with small groups of players.

The third important dynamic is in calculating the probability of drawing any of your out cards.  'Outs' are cards that you need to complete your hand.  For example if you are holding two suited heart cards in your hand, and two more hearts come up in the flop you might be hoping for a fifth heart at the turn to make a flush.  In hold 'em, you have 9 'outs' (13 hearts total - 4 known heart cards).  You also know that there are 47 unknown cards at this point (52 - 2 hole cards - 3 flop cards), so the odds of drawing a heart at the turn are 9 / 47 or about 19%.  Many hold 'em player use the 'rule of 2' where you multiply the number of 'out' cards by 2% to calculate your odds and decide if a call or a raise makes sense in terms of value.  The rule of 2 is conservative, assuming there are about 50 unknown cards remaining (1/50 = 2%), although it gets a little better at the flop (2.13%), turn (2.17%), and river (2.22%).

In meatball, however, the meatball cards are added to the number of known cards.  So, using the prior example, if you have two hearts and two more show up at the flop, but you are playing against 5 players, there are now only 42 unknown cards (52 - 2 hole cards - 3 flop cards, - 5 meatballs).  So the odds of getting an out cards are 2.38% (1/42).  However, if 2 of the discarded are also heart cards, then your total number of outs goes down from 9 to 7, so your odds of drawing that flush on the turn are now 7/42 only 16.7%.  

The point here is that you have a lot more information to take into account when making your betting decisions.  Maybe you held a pair of 9's as your hole cards, but on the flip you see the other two 9's discarded as meatballs.  Your odds of drawing that 3 of a kind set just went to zero and a fold is probably your best move pre-flop. On the other hand, maybe you're sitting on a pair of J's and there are no J's among the meatballs, your odds of flopping a set are now pretty good since there are only 45 unknown cards left.  You'll need to temper that enthusiasm knowing that pre-flop the other players would have been likely to hold onto a J and discard a lower card if given the chance.  

That's it.  Give it a try.  Be prepared for more big hands.  More coolers.  More excitement and a faster tempo.  Balls up...

Plain Text List of National Parks

I put this plain text list of National Parks together for easy copy and pasting into other documents.  Enjoy.

(Without state name)

Acadia National Park

Arches National Park

Badlands National Park

Big Bend National Park

Biscayne National Park

Black Canyon of the Gunnison National Park

Bryce Canyon National Park

Canyonlands National Park

Capitol Reef National Park

Carlsbad Caverns National Park

Channel Islands National Park

Congaree National Park

Crater Lake National Park

Cuyahoga Valley National Park

Death Valley National Park

Denali National Park and Preserve

Dry Tortugas National Park

Everglades National Park

Gates of the Arctic National Park

Gateway Arch National Park

Glacier National Park

Glacier Bay National Park

Grand Canyon National Park

Grand Teton National Park

Great Basin National Park

Great Sand Dunes National Park andPreserve

Great Smoky Mountains National Park

Guadalupe Mountains National Park

Haleakala National Park

Hawaii Volcanoes National Park

Hot Springs National Park

Indiana Dunes National Park

Isle Royale National Park

Joshua Tree National Park

Katmai National Park and Preserve

Kenai Fjords National Park

Kings Canyon National Park

Kobuk Valley National Park

Lake Clark National Park

Lassen Volcanic National Park

Mammoth Cave National Park

Mesa Verde National Park

Mount Rainier National Park

National Park of American Samoa

North Cascades National Park

Olympic National Park

Petrified Forest National Park

Pinnacles National Park

Redwood National Park

Rocky Mountain National Park

Saguaro National Park

Sequoia National Park

Shenandoah National Park

Theodore Roosevelt National Park

Virgin Islands National Park

Voyageurs National Park

White Sands National Park

Wind Cave National Park

Wrangell-St. Elias National Park and Preserve

Yellowstone National Park,

Yosemite National Park

Zion National Park

(With State Names)

Acadia National Park (Maine)

Arches National Park (Utah)

Badlands National Park (South Dakota)

Big Bend National Park (Texas)

Biscayne National Park (Florida)

Black Canyon of the Gunnison National Park (Colorado)

Bryce Canyon National Park (Utah)

Canyonlands National Park (Utah)

Capitol Reef National Park (Utah)

Carlsbad Caverns National Park (New Mexico)

Channel Islands National Park (California)

Congaree National Park (South Carolina)

Crater Lake National Park (Oregon)

Cuyahoga Valley National Park (Ohio)

Death Valley National Park (California)

Denali National Park and Preserve (Alaska)

Dry Tortugas National Park (Florida)

Everglades National Park (Florida)

Gates of the Arctic National Park (Alaska)

Gateway Arch National Park (Missouri)

Glacier National Park (Montana)

Glacier Bay National Park (Alaska)

Grand Canyon National Park (Arizona)

Grand Teton National Park (Wyoming)

Great Basin National Park (Nevada)

Great Sand Dunes National Park and Preserve (Colorado)

Great Smoky Mountains National Park (Tennessee, North Carolina)

Guadalupe Mountains National Park (Texas)

Haleakala National Park (Hawaii)

Hawaii Volcanoes National Park  (Hawaii)

Hot Springs National Park (Arkansas)

Indiana Dunes National Park (Indiana)

Isle Royale National Park (Michigan)

Joshua Tree National Park (California)

Katmai National Park and Preserve (Alaska)

Kenai Fjords National Park (Alaska)

Kings Canyon National Park (California)

Kobuk Valley National Park (Alaska)

Lake Clark National Park (Alaska)

Lassen Volcanic National Park (California)

Mammoth Cave National Park (Kentucky)

Mesa Verde National Park (Colorado)

Mount Rainier National Park (Washington)

National Park of American Samoa (American Samoa)

North Cascades National Park (Washington)

Olympic National Park (Washington)

Petrified Forest National Park (Arizona)

Pinnacles National Park (California)

Redwood National Park (California)

Rocky Mountain National Park (Colorado)

Saguaro National Park (Arizona)

Sequoia National Park (California)

Shenandoah National Park (Virginia)

Theodore Roosevelt National Park (North Dakota)

Virgin Islands National Park (US Virgin Islands)

Voyageurs National Park (Minnesota)

White Sands National Park (New Mexico)

Wind Cave National Park (South Dakota)

Wrangell-St. Elias National Park and Preserve (Alaska)

Yellowstone National Park,  (Wyoming, Montana, Idaho)

Yosemite National Park (California)

Zion National Park (Utah)

You'll Blow Your Eyes Up, Kid 1821

You'll Blow Your Eyes Up, Kid

Santa has been around for a long time and there isn't much he hasn't seen before.  Way back in 1821, according to Children's Friend III, Santeclaus already had his Christmas Eve gig down; a reindeer-pulled sleigh, parking on roof tops and going down chimneys, sneaking presents into stockings...  He also had a carefully selected list of approved presents. Wooden guns were a popular choice, but he was certain to avoid toys which might cause self-inflicted harm to children.

Where e'er I found good girls or boys,
That hated quarrels, strife and noise,
I left an apple, or a tart,
Or wooden gun, or painted cart;
 

To some I gave a pretty doll,
To some a peg-top, or a ball;
No crackers, cannons, squibs, or rockets,
To blow their eyes up, or their pockets.

Perhaps 120 years later, he had relaxed those safety standards just a little bit when in 1940, despite his initial reluctance in the department store ("You'll shoot your eye out, kid"), he gave Ralphie Parker a Red Ryder carbine action 200-shot range model air rifle:

You've got to give Santa some credit for looking out for the parents too.  Ol' Santeclause avoided any annoying gifts that produced loud noises, going educational whenever he could:

No drums to stun their Mother's ear,
Nor swords to make their sisters fear;
But pretty books to store their mind
With knowledge of each various kind.

  

The naughty and nice list was also well established in 1821.  The list of infringements which got you on the naughty list would have knocked young Ralphie out of contention for getting his Red Ryder thanks to his potty mouth:

 But where I found the children naughty,
In manners rude, in temper haughty,
Thanklefs to parents, lairs, swearers,
Boxers, or cheats, or base tale-bearers,

 

  

Ohhh fudge.  And it wasn't lumps of coal Santa was giving out back then either:

I left a long, black, birchen rod,
Such as the dread command of God
Directs a parent's hand to use
When virtue's path his sons refuse.

Well, there's always next Christmas.

1. (1821) "The Children's friend. Number III. : A New-Year's present, to the little ones from five to twelve." click here to read the full story.
2. (1983) A Christmas Story 

1821 - Santeclaus - Childrens' Friend

The Children's friend. Number III. : A New-Year's present, to the little ones from five to twelve. 

 (1821) 

 

 

 

 

Old Santeclause with much delight
His reindeer drives this frosty night,
O'er chimney tops,and tracks of snow,
to bring his yearly gifts to you.

The steady friend of virtuous youth,
The friend of duty, and of truth,
Each Christmas eve he joys to come
Where love and peace have made their home.

Through many houses he has been,
And various beds and stockings seen;
Some, white as snow, and neatly mended,
Others, that seem'd for pigs intended.

Where e'er I found good girls or boys,
That hated quarrels, strife and noise,
I left an apple, or a tart,
Or wooden gun, or painted cart ;

To some I gave a pretty doll,
To some a peg-top, or a ball;
No crackers, cannons, squibs, or rockets,
To blow their eyes up, or their pockets.

No drums to stun their Mother's ear,
Nor swords to make their sisters fear;
But pretty books to store their mind
With knowledge of each various kind.

But where I found the children naughty,
In manners rude, in temper haughty,
Thanklefs to parents, lairs, swearers,
Boxers, or cheats, or base tale-bearers,

I left a long, black, birchen rod,
Such as the dread command of God
Directs a parent's hand to use
When virtue's path his sons refuse.

Source: Doolittle, I. (1821). The Children's friend. Number III. : A New-Year's present, to the little ones from five to twelve. Part III. : Containing eight coloured engravings. Price 25 cents: plain, 18 3/4 cents. Published by William B. Gilley, no. 92 Broadway. Link: Yale, Beinecke Library