Trivia and General Interest

Fun with numbers

One of my greater difficulties in reading Gödel, Escher, Bach is that on nearly every page I find myself so fascinated with a concept which the author has pointed out explicitly, or I am struck by some tangential thought which possesses me completely,  that I can't help but get sidetracked pondering or experimenting with the ideas, so I don't make much progress in the book.  I think Professor Hofstadter would approve, however.

Several nights ago this happened to me; the author had made some passing observation about the number 987,654,321 and it struck me as such a playful number (in a childlike sense) that I decided to play with it a bit.  I found some things which are very interesting, and I wish I had a super-genius mathematician friend who could explain some of this quirkiness to me.

I will post the operations below, and add any others which I may find in the future:

987654321 / 123456789
Answer: 8.0000000729
This was the first operation that I undertook.  The answer literally made my jaw drop.  Eight...with a bunch of zeroes right of the decimal, then a few random numbers.  Now, I don't know if the quotient truly resolves at the tenth decimal place, or if the calculator merely stops displaying it there.  When I multiply it back out (8.0000000729 * 123456789) the product is exactly 987654321, so I presume that the quotient truly resolves in the tenth decimal place.
So why eight?  And why not exactly eight? I would expect either a nice, round, whole number or else some kind of gobbledygook, but this is so, very, very, very near to 8, that 729 hundred-millionths would be negligible in any applied mathematics.

987654321 + 123456789
Answer: 1111111110
I just like the symmetry of this one.  If you want to make it consist of pure 1's, just move the decimal one place to the left on each of the addends (98765432.1 + 12345678.9 = 111111111).

(987654321 + 123456789) / 2
Answer: 555555555
This is, by definition, half the value of the previous number.  The reason I find this so fascinating is because I remember when I first realized, as a child (before I knew how fun numbers can be) that by pairing the highest and lowest numbers between 1 and 9 working towards the middle number of 5, (1+9, 2+8, 3+7, ...) you end up with all 10's for sums.  Dividing these by 2 gives you a mean of 5 for each pair, and you still have that 5 left over when you paired 4 and 6.

987654321 - 123456789
Answer: 864197532
This one is not too profound, at least not that I've discovered, but I found it surprising that the difference consists of each of the integers 1-9 in an (apparently) random configuration.

Pascal's Triangle

Credit for this is due to the website "Futility Closet"
(https://www.futilitycloset.com/2017/02/07/noted-13/) which I was directed to by a fellow redditor.  Here's Pascal's Triangle, in which each number is the sum of the two numbers diagonally above it:

1

1  1

1  2  1

1  3  3  1

1  4  6  4  1

1  5  10 10  5 1

1  6 15 20 15  6  1

1  7 21 35 35 21  7  1

1  8 28 56 70 56 28  8  1

 

 

Now if we lean the triangle way to the left, like so:

1

  1  1

      1  2  1

          1  3  3  1

              1  4  6  4   1

                  1  5 10 10  5   1

                      1  6  15 20 15  6   1

                          1   7  21 35 35 21  7   1

                               1    8 28 56 70 56 28  8  1

Then the sum of each column from left to right (or right to left, had we leaned the triangle right instead of left) gives us the Fibonacci sequence, at least to the point where we arbitrarily stopped (9th row).  Continuing Pascal's triangle indefinitely should produce the Fibonacci sequence indefinitely.  Visit the link above if you're interested in a further manipulation of Pascal's Triangle.





 

The Wikipedia Game

As I mentioned in the post The Emergence of Patterns it is possible to navigate from many (most? all?) random Wikipedia entries backward to the entry for Philosophy by following the first link on the page which is neither italicized nor in parentheses, then repeating this process indefinitely.  This is a fun exercise (for a nerd like me) but there is a more mentally challenging Wikipedia game I'd like to share with you.

At the time of this writing, the game is referred to as Wikiracing.  It involves assigning two random articles to the competitors, one as the start point and the other as the "finish line".  Two common forms of the game are to arrive at the second article by only clicking links in either (a) the shortest elapsed time, or (b) the fewest "steps" (that is, clicks on links).

Even if you have nobody to play with, it can be a fun exercise.  If the start/end points are extremely obscure or specific, it requires quite a bit of strategic thinking.  Enjoy!