Leibniz Wrong About Dice!

I was reading a book about Leibniz called “Selections” and then “I, Claudius” by Robert Graves. Emperor Claudius wrote a small pamphlet about throwing dice. Leibniz had a small section about dice odds—that he was using to illustrate some other matter.

 

Two dice references in two different books got me thinking.

 

Leibniz got something wrong! He wrote about someone throwing two six-sided dice and rolling an “11”, and he wrote that there was only one way to roll an “11”.

 

Technically, there are two ways to roll and “11”: you can either roll 5 & 6…but you can also roll 6 & 5. Each die has a different outcome.

 

TWO SIX-SIDED DICE WAYS TO THROW EACH NUMBER

2 = 1+1

3 = 1+2 or 2+1

4 = 1 +3 or 3+1 or 2+2

5 = 1+4 or 4+1 or 2+3 or 3+2

6 = 1+5 or 5+1 or 2+4 or 4+2 or 3+3

7 = 1+6 or 6+1 or 2+5 or 5+2 or 3+4 or 4+3

8 = 2+6 or 6+2 or 3+5 or 5+3 or 4+4

9 = 3+6 or 6+3 or 4+5 or 5+4

10 = 4+6 or 6+4 or 5+5

11 = 5+6 or 6+5

12 = 6+6

 

As you can see, there are more ways (easier) to roll a “7” than any other number. Lucky Seven! 

When Leibniz made his error, he stated there was only one way to roll an “11”. But as you see, there are two ways, which makes “11” easier to roll and “2” or “12”—and the same odds as rolling a “3”. That’s a pretty significant difference-that makes the chances or rolling an “11” 5.56% instead of only 2.78% which is double the odds.

Scribbled notes on an old library catalog card I was using as a bookmark:

Claudius was never as dumb as he pretended to be.

In the beginning...there were card catalogs.

In the beginning...there were card catalogs. 

Every book in the library had 3 cards typed or handwritten for it so they could be searched (by flipping through the cards) by either author, title or subject. A metal rod was run through the holes in the cards to keep them from being rearranged maliciously.

We finally wised up and got computers to do the job for us, because they're so easy to use-unlike pieces of paper. (Yes, the above really is a photo of my work computer screen, lol).

The Melvindale Public Library is down to clearing out its LAST small bunch of old cards, which we are using as scrap paper for patrons. I used 50 cards to create this modified (accidentally) polyhedral/rhomboidal lampshade:

The elementary Geometry: if you superimpose the rhombus (diamond) just right over the cards you only need 30 slotted cards and no scotch tape. That didn't quite (whoops) happen so 5 "plus-sign" shaped centers were needed to fill it out a bit better. The design works better with the proportions of poker playing cards, as done here: http://boardgamegeek.com/geeklist/50207/making-polyhedra-from-magic-cards-a-guide-with-t . 

A MUCH easier intro to paper polyhedra (way cooler shapes that the average youngster could assemble) are available in printable templates that usually require only a single sheet of paper here: http://www.korthalsaltes.com/visual_index.php .


If you'd like one of these cards as a souvenir, they're in the scratch paper box next to the OPAC: Online Public Access (cardless computerized) Catalog. Once they're gone, they'll be gone for good. 

As of 10/1/2015 we've still got a small bunch left in the scrap paper box.

YES and/or No!

YES and/or No!

"There are two sorts of eternity, from the present backwards to eternity, and from the present forward." -Abraham Cowley

Yes!

An infinite amount of nothing adds up to something. I find it creepy that the above two numbers, 1 and 0.999999... are considered the same by mathematicians, but they are.

There are many more proofs of the above that are more complex, but you don't need them because 1 = 0.999...

Creeped out? Keep reading.

No!

Even more odd is Zeno's Paradoxes, specifically the one about a race between Achilles and the Tortoise. The wording is difficult so I'll put it a more digestible way:

If you walk halfway across a room, and then half of that distance, and then half of that distance... on and on forever (according to Zeno) two things will happen:

1) You will never reach the end of the room, and
2) You will never stop moving!


Zeno was Wrong because of the same thing that (sort of) explains 1 and 0.999... being equal: transfinite infinities. If the distances become infinitely small, so does the time to traverse them.


The infinities eventually add up to 1 (or getting across the room). 

Now don't you feel better about 0.999.... equaling 1? Me neither, but at least it takes the edge off so I can sleep.

My next post on probability/chance briefly goes into infinities of nothing adding up to the possibility of something big: quantum tunneling.

PAPER FOLDING MATH PEOPLE

PAPER FOLDING MATH PEOPLE

Score of a lifetime: library book sale had a pile of unwanted math books. So I buy them and when I get them home I see most belonged to noted mathematics author Leo F. Boron!

He authored books on math and translated many more. The best part besides his notes, signatures and ex-libris bookplate: a paper polygon tucked inside of one of the books. It's a regular pyramid made from green construction paper.

I also snagged a book once owned by noted University of Michigan professor George Kish. Is the third edition of Burlington's "Mathematical Tables and Formulas". The papers of Professor George Kish are now in the archives of the Bentley Historical Library in Ann Arbor Michigan...well, at least the ones that aren't on my bookshelves.