accepted, rejected, approved

Brain teaser. I was fiddling with the grid pegs (unit legos, essentially) down in 'Brozy's mom's preschool the other day. I built a vertical line of three red pegs, then added an horizontal line of three green pegs on top of it. Adding a diagonal line of three yellow pegs to get the following arrangement, I frowned.

Y	R
G	R, G, Y	G
	R	Y

"I'd like counter-clockwise to correspond to up," I thought, and switched the green and yellow pegs on the center post. "Huh," I realized, "*any* switch on the center post would change the correspondence from incorrect to correct or vice-versa, assuming that we wrap around the ends of both sequences. There are only two ways to traverse a three-item series when starting point isn't considered. I wonder how many ways there are to traverse an n-item series."

Can you answer my question? Basically, I'm asking how many distinct Hamiltonian cycles there are for n arbitrary points. (In layman's terms, how many ways can you beatI've found both recursive and non-recursive perspectives on the problem. I'll give a few hints in the rest of the entry, so stop reading if you're still working on it.

If you haven't heard, Ambrosia and I are engaged. We'll probably tie the knot in October or November, though there are no firm plans yet. You should read Ambrosia's account of my proposal(s).

Hint 1: The number of distinct cycles where you ignore starting point is equal to the number of distinct cycles with the same starting point.

We ran a 5k last weekend for the 'Brozyville County Fair. I got a medal, because there were only three men in my age group. The announcer did a double-take when reading my time as he announced the medal; I was the slowest male runner, at over 31 minutes. The second-slowest runner came in at around 28 minutes. He was over sixty years old. And I'm really, really sore. Perhaps I will train for the next one.

The rest of the fair was a cornucopia of equine delights as usual, though. (I'm fairly sure that the recent 'Brozyville stake Pioneer Pageant involved more horses than did, for example, the actual Mormon pioneer movement.)

Hint 2: You may find the problem easier to visualize if you think of it as arranging points on a circle or items in a list.

House-hunting by car is the most frustrating thing ever. We drove around the neighborhoods between Freedom and State for about an hour the other night looking for signs. (I'm not sure how anonymous my blog should be, so I will neither confirm nor deny that I live in Provo.) We found forty or so advertising singles' housing and ten married or ambiguous, which we called. One of them called back. It was a five-bedroom for a couple thousand a month. The next day, I was delighted to notice a sign just across the street from 'Brozy advertising a married apartment. I called the owner and set up an appointment for last Tuesday.

"He sounds really old," I told Ambrosia. "He probably will be a nice manager." We sat in the yard for a while until he showed up and asked if 'Brozy was my friend. "Yup," I confirmed. The apartment was reasonably nice: spacious living room, bedroom, and kitchen pantry, at least. We thought about it a little more, and I called him back the next day to apply. I met him at the apartment and filled out my name, social security number, driver's license number, date of birth, license plate number. I then called 'Brozy for her social security number and driver's license number, I filled out my current landlord's contact details, my employer's details, and my parent's phone numbers.

"I'm not sure who else to put on the references section, sir," I said. "What are you looking for other than landlord and employer?" He suggested I put my bishop's information down and asked if I'd lived in Helaman Halls freshman year with any of his friends who are bishops there. I called my roommate for my bishop's cell phone number, wrote it down, and added my manager's information, crossing out the office phone number I'd written in the address field for work and putting it next to my manager. He took the application. "We do keep BYU standards here," he noted, "so you can't be living with anyone who's not a man."

Hint 3 (non-recursive): The starting point doesn't matter. How many ways are there to arrange the rest of the points along the path?

I called my landlord and told her that I'd like to move out soon. She reminded me that I am legally obligated to give her a month's notice, but relaxed when I told her I'd be out in time for a new tenant fall semester.

With housing worries out of the way, I relaxed for the weekend. Monday evening, I called the landlord back. "Hi, this is bawb," I said. "Just checking up on my application."

"Oh, mumblerblemum."

"I'm sorry, what?"

"I rejected your application."

{pause}

"And why would that be, sir?"

"Well, there was just too much inconsistent information."

"For example?"

"You didn't provide a phone number for your employer."

"You didn't provide a field to write it in."

"I looked your bishop's name up in the phone book, and the number didn't match."

Welcome to the twenty-first friggin' century, old man. Is this your first time here? Let me show you these magic boxes called cell phones. "My roommate is the ward clerk, so I called him and wrote down the number he gave me. I figured it would be more convenient for you and my bishop if you called his cell phone."

"Well, I've rejected the application."

"I see. Thank you, sir."

Douchebag. At least 'Brozy now believes that I am capable of anger.

Hint 3 (recursive): The base case will be H(2) = 1. Given H(n), how many solutions will there be for H(n+1)? Consider how many solutions are possible each time you add an additional point to a given cycle.

Last night, we went to Craigslist.org. Utah. Click. Provo/Orem. Click. Housing. Click. First listing. Click.

This is a recently remodeled and updated apartment with one very large bedroom, and nice large bathroom. Living room and kitchen are covered with hardwood laminate flooring. New fridge, oven range and microwave, dishwasher, partly furnished with love seat, chair and ottoman, kitchen table and chairs, and coffee and end tables.(see pics) Rent is $600 and includes all utilities including cable and high speed internet. Laundry room is shared with upstairs but there are 2 sets of washer and dryers. Apartment has large oversized windows that let plenty of light in and faux wood levelor blinds, wood beam ceiling in living room, plenty of storage with 3 large closets, fenced back yard, off street parking, and is located in a quiet culdisac.

We called, we came, we saw. We'll call back and commit tonight.

Best of all? They live upstairs with a wonderful golden Labrador. And they say 'Brozy can have cats.

I've never been more grateful to have worked with a crochety old fart. And I'm never looking for houses in person again. The end.

Hint 4: Just think !

ESP game. Collaborative image recognition and labelling.

Slate: growing meat. (See Wikipedia: Quorn.)

DHTML Lemmings

XKCD is fantastic. My triumvirate of yellow webcomics--XKCD, Qwantz, and Toothpaste for Dinner--is complete.

Hitachi perpendicular memory--Schoolhouse Rock style

Wikibooks: Angels and Demons divergences from reality. So satisfying to see them all bullet-pointed.

I think if I weren't Mormon, I'd sincerely consider Singularitarianism.

Holy flying mobulas, Batman

XKCD: vote on the best possible date

Tame silver fox videos

LA Times special: Man vs. Earth's oceans

Donald Rumsfeld fighting technique and BBC quotes

Was the Hamiltonian cycle problem too easy? Try this URL-based brain-teaser, courtesy of Lunkwill.