»  Solutions to puzzles in my National Review Online Diary

  December 2008


As usual at year end, your task in the December Math Corner was to find something interesting to say about the number 2009.



Several readers resorted to the Online Encyclopedia of Integer Sequences. This never occurs to me: which is odd, as I own the original (1995) book, which I am pretty sure predated the online version.

Here is the most comprehensive of those reader responses. The "Annnnnn" preceding each entry is the number of the relevant sequence in the encyclopedia.

• Miscellany:

    A009720:   2009 is the 7th derivative of tan(tanh(x)*cos(x)) at x=0

    A046735:   2009 does not divide any Tribonacci number

    A051336:   There are 2009 arithmetic progressions in {1,2,3,...,35}

    A067593:   There are 2009 partitions of 39 into Lucas numbers

• Possible Puzzles:

    A025405, A025409 etc:
Trivial partitioning of 2009 into 4 cubes: 13 + 23 + 103 + 103.  Knowing the Ramanujan-Hardy taxicab story, do it in two distinctly different ways.

    A056745:   Show 2009 divides 62009 + 52009 + 42009 + 32009 + 22009 + 12009 [over 3K digits]

• Mysterious(?) Coincidences:

    A006768:   There are 2009 5-dimensional polyominoes with 8 cells
    A057865:   There are 2009 simple Hamilton-connected graphs on 8 nodes [And Mathworld says, "every 8-connected claw-free graph is Hamilton-connected (Hu et al. 2005)."]
    A008764:   There are 2009 nonisomorphic symmetric 3 × 3 matrices over N0 with row and column sums equal to 49 [Note that 49 divides 2009; is this the largest such case?]

• Boundary Connections:

    A039768:   gcd(phi(2009),2008) = tau(2008)
    A125680:   31 Dec 2009 is the next Blue Moon

Other readers noted that:

    "The 2009th prime (17471) is a palindrome."

    "2009 may not be prime, but if we double it and add 1, triple it and add two, or quadruple it and add three, in each case we get a prime."

    "The Euler phi-value of 2009 = 1680 which is a highly composite number [i.e. has lots of factors — actually 48] … I have also attached a graph (in eps format) of the modular hyperbola xy = 1 (mod 2009) with x,y between 0 and 2008." [This reader frustrated me, as I have no software that can read .eps files.]

And really stretching the definition of "interesting" to breaking point:

    "2009 is the only year of this millennium in which the number of digits preceding the last digit, raised to the power of the sum of all but the last digit, yield the last digit. (3(2+0+0) = 9.) The most recent year to satisfy this criterion is 1003."