Blogger keeps mucking up my posts...

Ugh. When it comes to math blogging, Blogger totally sucks right now (until I learn some MathML). I always have some formula mutilated. Anywho, I am recovering from a trip to Tigin for trivia night on Tuesdays. I had a blast, joined a team, and had some fun. Met some cool dudes and some cute ladies too... It was totally fun, and will repeat next week. :)

LHIRT POD #3 Solution

Here's the solution to LHIRT POD #3, again written white on white (highlight to read):

According to Catalan, _2NC_N = ∑_{I N}C­_I ². Since _N C ₀ = _N C _N = 1, there’s where our 2 comes from. It turns out that _N C _x where 1 < X < N. Therefore N C _I = 2 + ∑ _{1 N} C _I = 2 + (N * k) for some k. This means that the remainder will always be 2. As far as the converse, consider 343 as an example. ₆₈₆ C ₃₄₃ (mod 343) = 2, and yet 343=7 ³ and thus not prime.

One issue with this: I have yet to find a counterexample that isn’t in the form p^q where p and q are both prime. Interesting?

Cool problem huh? Anyone want to conjecture on that final form of the answer?

LHIRT POD #3

It's time for a little number theory for the POD:

Let _nC_r be the number of ways to choose r items from n objects (or "n choose r"). Assume p is prime. Determine what the remainder of _2pC_p divided by p is, and prove it. For extra credit, is the converse true?

This one may be a little tough if you don't see it off the bat, so I'll leave it up for a while. Besides, I'm not entirely sure about the converse part myself. :)

LHIRT POD #2 solution

Solution for POD #2 below (white on white below):

Let the point on the inner circle be T. Because PQ is perpendicular to OT, OTP is a right triangle. Therefore OP²=OT²+PT². Now, the true area of the doughnut is (π/2)(OP²-OT²)= (π/2)(OT²+PT²-OT²)=(π/2)(PQ/2)².

Amazing, huh? Of course the smart-ass answer is that because a formula must exist for this area based on the length of that tangent alone (or else the problem wouldn't have an answer). Since the radius of the inner circle does not matter, make it radius 0, and the area will stay the same or just the area of the remaining circle. In a non-authoritative setting this form of reasoning is invalid, because you can't be sure if the questioner is being honest in implying the existence of that formula.

Another problem will be posted tomorrow.

Ha ha

It seems that I can't keep up with a daily puzzle, it is a lot of work and my schedule prevents it. I will be instead doing a couple of PODs (Problems of Difficulty :P) a week. This will also allow me to vary my posts more. There is too much in my life going on just to flood it with daily puzzles, no matter how much fun they are.

There's a lot of cleaning in my life recently. Recently I sold my PS2 and all games. I've vastly reduced my diet. My activities are paired down to affordable basics. My theory is that the less I'm invested in, the more invested I am in those activities. Maybe it'll pan out, maybe not, but it's something to try.

LHIRT's POD #2

The problem derives itself from an old puzzle book that solves this problem in a very clever way. I am looking for the real solution, which isn't more difficult than the original method to solve it.

Take a pair of concentric circles, that share a center O. Take a point on the inner circle, and extend its tangent to its intersection with the outer circle at points P and Q. Given only the length of this chord PQ of the outer circle, calculate the area between the two concentric circles.

I will give both solutions in the morning, but I am looking for the one that does not assume that a formula exists in the first place.

Lack of acrimony...

Acrimony is the word of the day, and what's remarkable is its absence; it's amazing that even after all the stuff I've burroughed through in the last few days, that I have nothing sarcastic or cynical to think. My breakup with Dawn is prolonged and melancholy, but not sharp or critical. We just mostly avoid each other as she remains here until she moves in with a friend. Her boxes are piling up, the lacuna of her stuff more striking than the stuff itself. I just removed her from my Wii. I think that both of us are more than ready to have this over as soon as possible, but that doesn't mean we wanted it to happen in the first place.

The important thing to learn from all of this is that you have to be careful with whom you choose to live. What remarkable relationship you have with someone will be cast in a much different light once you share living quarters with that person, and no matter how great your link is with someone, no matter how much you love them and they share that love, you must accept the fact that some people cannot live with one another, and to do so will create great and miserable sadness that the love itself can not overcome. One should ease into these things slowly, carefully, and mindfully, knowing that some things are meant to be and some aren't, no matter how truthful that knowledge is.

Now that I've cleared my mind about said issue, I will speak of it no longer.