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As a continuation of yesterday’s discussion, I created a simple graphic to illustrate the relative timescales of significant (to us) events in our cosmic history. The two inner squares show the Cretaceous-Tertiary Extinction (65.5 Ma) and Emergence of Homo (2.5 Ma) (Ga = gigaannum = 1 billion years; Ma = megaannum = 1 million years). Any events more recent would be smaller than a pixel.
I wonder how many galactic empires rose and fell in the blue area, before our own place in the galaxy even began to take shape.
This time of year is marked by the advent of the traditional and stereotypical holiday letter, wherein families take the opportunity to share their successes, travels, and extravagances with the world. I’ve never been a fan of these letters, but this year my mom, brother, and I decided to construct an easily customized letter of our own. (View blank letter as a PDF.)
MERRY CHRISTMAS!
(2007)
(2007)
Dear Friends / Family,
It’s that time of year again! What a wonderful year it has been for all of us. The whole family has had a great year, full of memories and adventure. Where should we begin to recount how very blessed we have been? We have never done a Christmas letter before, but fortunately there are letter-writing tools available for people like us.
To start off, ______Deb___________ has been very involved in:
___ the mafia as a money launderer
___ restoring peace to Iraq
_X_ starting a homeless shelter for abandoned and battered cats
_X_ playing extreme Scrabble
___ empathizing with the warming globe
_X_ scratch and sniff lottery, but not for the money
Meanwhile, in ___Pennsylvania_____, ______Jacob_________ has been pursing:
___ world conquest
___ perfecting sleek and discreet flossing technique
_X_ an enriching career in hydrotherapy
_X_ rehabilitation of glue-sniffing ants
___ moderate success in minimal endeavors occasionally
___ study at the Palestinian glass-blowing institute
And last, but certainly not least, _______Luke_________ has enjoyed living in _Scotland & Minnesota_ while succeeding in:
_X_ teaching mute people how to walk
___ further developing fungus taxonomy
_X_ acquiring a wide variety of superpowers including, but not limited to:
___ infrared vision
___ super strength
___ the ability to melt
_X_ an uncanny ability to talk to fish, although they don’t usually talk
back
___ continued development of Opti-Grab™ technology
___ being macho
_X_ selling glue to ants
In short, we could not have possibly asked for a better year. Some people wish for a perfect life; we lived it, and we are so blessed. We could go on, but we are sure your lives are so much fuller than ours—and so busy too. We only wish that we could have shared each special moment with every one of you.
Sincerely / Love / Affectionately / Yours,
_______Deb__________ _______Jacob________ _______Luke_________
For some reason, my friend Seth Schwartzhoff Boyd and I came up with this in middle school.
The perfect response to the question: Are you hungry?
Oh, I’m Chad. Why don’t you Russia on over to visit U.S. sometime. We’ll go in the Palestine, sit on the Afghanistan, and have Turkey dipped in Greece served on our best China, followed by a big Bolivia of Chile and Tunisia on the Sandwich Islands; but make sure you wash your Honduras before you eat or you’ll have Germanies all over them.
My brother sent me a recursive problem yesterday, and I worked out this solution last night.
Assume a population begins with resources at 100 (some unitless quantity). Half of these resources are spent, while the other half are passed to the next generation. The next generation begins with 100, plus 20 to account for progress, plus the half passed down to them. Likewise, the following generation begins with 100, plus 40 to account for progress, plus the half passed down to them. What is the resource value of the nth generation?
We can induct a recursive function by simply looking at the first few terms. Let f(x) == “resource level for generation x”, a = 100, and b = 20:
f(1) = a
f(2) = (a + b) + f(1)/2
f(3) = (a + 2*b) + f(2)/2
f(4) = (a + 3*b) + f(3)/2
f(5) = (a + 4*b) + f(4)/2
Following the pattern, we can write the general case:
f(n) = a + b*(n-1) + f(n-1)/2
This recursive solution can correctly predict the value of the nth generation, but it is O(n) in complexity (that is, it requires n calculations to find the nth generation). Using a computer to solve the equation is a common solution, but we can also solve this recurrence relation to remove the recursion.
Back-substitution for the recurrence relation reveals that f(n) can be written in the form:
f(n) = λ1(n)*a + λ2(n)*b
From the first few values of λ1 (1, 3/2, 7/4, 15/8, 31/16, 63/32) we can induct:
λ1(n) = (2n – 1)/(2n – 1)
This is not as easily done with λ2, though (0, 1, 5/2, 17/4, 49/8, 129/16), since λ2 has the form
λ2(n) = (n – 1) + λ2(n – 1)/2
yet another recurrence relation. This equation can be solved, though, using the method of undetermined coefficients to give:
λ2(n) = 4*(1/2)n + 2*n – 4
Therefore, we can write the final solution to our initial recurrence relation:
f(n) = [(2n - 1)/2n-1]*a + [4*(1/2)n + 2*n - 4]*b
I saw a banner add for a site called “Tickle Your Brain” (at least according to the ad; I didn’t click the link). It featured the puzzle below, for which I cached the image as well, but I didn’t want an annoying animated advertisement in the post. The obvious answer they are going for is even part of the filename(!), but as a point of contention I present justifications for any of the solutions as valid.
Which does not belong in the group?
U E A S O
U, because none of the other characters can be approximated by a function of a single variable.
E, because it is the only character with all of its lines parallel or perpendicular to one another.
A, since none of the others are used in the standard U.S. letter-grade system.
S, because it is the only one with no axis of symmetry.
O, since it is the only character with just two edges.
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