Not a submission but I think this blog is very cool :)
Today's number is 4862
(this number was submitted by @real-fragments)
Very clever submission, I must say. I almost didn't catch the hidden meaning.
If you take the ASCII value of each character in your submission (including spaces, which I've represented as underscores), you get:
N = 78
o = 111
t = 116
_ = 32
a = 97
_ = 32
s = 115
u = 117
b = 98
m = 109
i = 105
s = 115
s = 115
i = 105
o = 111
n = 110
_ = 32
b = 98
u = 117
t = 116
_ = 32
I = 73
_ = 32
t = 116
h = 104
i = 105
n = 110
k = 107
_ = 32
t = 116
h = 104
i = 105
s = 115
_ = 32
b = 98
l = 108
o = 111
g = 103
_ = 32
i = 105
s = 115
_ = 32
v = 118
e = 101
r = 114
y = 121
_ = 32
c = 99
o = 111
o = 111
l = 108
_ = 32
: = 58
) = 41
Then you can add them to get:
78 + 111 + 116 + 32 + 97 + 32 + 115 + 117 + 98 + 109 + 105 + 115 + 115 + 105 + 111 + 110 + 32 + 98 + 117 + 116 + 32 + 73 + 32 + 116 + 104 + 105 + 110 + 107 + 32 + 116 + 104 + 105 + 115 + 32 + 98 + 108 + 111 + 103 + 32 + 105 + 115 + 32 + 118 + 101 + 114 + 121 + 32 + 99 + 111 + 111 + 108 + 32 + 58 + 41 = 4862
This number, 4862, is quite interesting. It is a Catalan number, which is part of a sequence of numbers that occur in various counting problems.
The n-th Catalan number can be expressed directly in terms of the central binomial coefficients as
C_n = 1 / (1 + n) (2n choose n) = (2n)! / ((n + 1)!n!)
If we choose n = 9, we get 4862.
There is a whole host of counting problems that have Catalan numbers as a solution. For example:
C_n is the number of Dyck words of length 2n. A Dyck word is a string consisting of n X's and n Y's such that no initial segment of the string has more Y's than X's. For example, the following are the Dyck words up to length 6: XY, XXYY, XYXY, XXXYYY, XYXXYY, XYXYXY, XXYYXY, XXYXYY
C_n is the number of monotonic lattice paths along the edges of a grid with n x n square cells which do not pass above the diagonal. Note that we define monotonic paths as ones which start in the lower left corner, finish in the upper right corner, and only consist of moves pointing rightwards or upwards. Below is the number of cases for n = 4, but note that if n = 9, we get 4862 solutions.
A convex polygon with n + 2 sides can be cut into triangles by connecting vertices with non-crossing line segments. The number of triangles formed is n and the number of different ways this can be achieved is C_n. Below is the case for n = 4 again.
There's a lot more uses for these numbers that would take up several Tumblr posts, but the above should serve as a small taste of what the Catalan numbers (and C_9 in particular) can offer.
Overall, these numbers are incredibly interesting. Thanks for the submission!
