12/26/04
SUNDAY
DECEMBER 26, 2004
Some of the volunteers the next morning, after a nice meal of eggs, bacon and sausage, left to go back to site. I had traveled the farthest to be here and planned on staying another day or two, volunteers staying here or not.
Cheeta convinced Braam to drive to Bansang to pick up the DVD player at the PC Regional House. This entailed driving over an hour, crossing a ferry, backtracking 15 minutes, picking up the DVD and doing everything again in reverse. Jessica went along while I watched TV with Kelly and the remaining volunteers.
ESPN was a big hit with most of them, while I was just waiting for something better to come on. When there was football on, or any other sport, I was more intrigued when the ball was in the air then when a team had control over it. When it was in the air only the natural forces of gravity took control. Thoughts of at first freshman level physics took over of Newton’s Laws and F=ma, with the ball being controlled at every second by a force; then thoughts of senior level classes where you learn another interpretation of Newton’s Laws saying the ball actually knows ahead of time which path it’s going to take. When the ball leaves your hand, there is something called the Hamiltonian Principle which states that of all possible paths the ball could go, the actual path it does take does so such that that a certain quantity is a minimum. It still bugs me to this day of things knowing in advance which path to take, that nature chooses the most economical path (in some sense) for every object to follow. Nonetheless there’s the ball flying to the receiver; exactly as Newton or Hamilton would predict.
I gave up watching football and went on to my dartboard problem which had bugged me since the first day arriving at Kharafi. Why was the numbers in the dartboard in the order they are in? If you look at a dartboard starting from the 12 o’clock position and go clockwise the numbers go in the pattern of [20, 1, 18, 4, 13, 6, 10, 15, 2, 17, 3, 19, 7, 16, 8, 11, 14, 9, 12, 5], why are they in the pattern that they are in? I thought of the question: What if you start with the 12 o’clock position and count a certain number of steps, going around repeatedly as needed, and that is where you place the 1. Go around that number of times again this time skipping the 1 each time you pass it, and after that many steps place a 2, etc. Is it possible to construct the sequence of numbers on a dartboard in that fashion?
For example, if you start at the 12 o’clock position you can go just one spot over to get to the correct position of ‘1’. However, you can also go around the whole board again, which would make 21 steps total, or around a third time for 41. Each one of the numbers [1, 21, 41, 61, 81, 101, 121, …] lands you on the correct position for the position of 1. After landing at 1, you now have to go seven more positions to get to the two. That, or 19 more since you’re now skipping the 1 when you cross it, so one less space to count. This leads the possible steps to get to the 2 from the 1 of [7, 26, 33, 40,…] It doesn’t look like it but the number 121 is on both lists. So, if you start at the 12 o’clock position and count 121 steps around you will get to the correct position of the ‘1’. Continue 121 more steps, ignoring the 1 that you placed now, and you get the correct position for the ‘2’.
Is it possible to continue with the 3? Yes. However, that’s where it stops. You can’t find a number that will correctly place the ‘1’, ‘2’, ‘3’, and the ‘4’, let alone all the rest of the numbers.
What if you worked backwards? You can get a little farther. Take an ordinary dartboard and take out all the numbers greater than 13, such that there are now empty spaces where the numbers were. Start at position 13 and count 243 empty spaces going around as many times as needed. That will get you the correct position for the 14. Continue on counting 243 empty spaces and you’ll get 15. That number, 243, is the smallest number that will place all the spaces from 13 to 20 in their correct positions. Any multiple of 840 added to 243 would also work, but 243 was the smallest.
If you try to find a number that will place all the numbers from 12-20 on the board you’ll find it’s impossible, but 13-20 is possible. The one course I had in college on Abstract Algebra and Number Theory came in handy! I never did like the Abstract Algebra part of that course, with topics ranging from Groups, Fields, and Rings. The Number Theory I kind of enjoyed, with prime number theorems and interesting proofs about numbers in general.
For example, given an odd number not ending in five you can find a string of ones such that that odd number can divide the other number. Pick one at random, say 7; 111111 can be divided by 7. Choose 37 for example; 111 can be divided by 37. Pretty interesting, but no application though.
A new volunteer, Greta, has her masters in mathematics in the one area I didn’t particularly like; Abstract Algebra. She’s been very helpful when I need to remember a theorem, or someone to bounce ideas off of.
After working on the Dart Board problem a little bit, and with the volunteers finally being bored of ESPN they switched to another channel. The show now being displayed on TV? A special on cannibalism. Any other day I wouldn’t mind if the channel was changed, but today was different. I had just read the book ALIVE, in which they had to eat the dead to stay alive, and was hoping it would be mentioned on the show.
Just as the show was getting under way Cheeta, Braam, and Jessica arrived with the DVD player and started plugging it in and testing it to make sure it worked. This was interfering with me trying to watch a show on cannibalism. At one point they were testing the DVD and switched the TV setting with the first sentence I heard was
“… another, more recent occurance of cannabilism occurred in Andes Mountains…”
I yelled “STOP!” and sat closer to the TV to hear the five-minute story about the true Andes survivors. I was then happy, and allowed them to again fiddle with the electronics. A somewhat perfect coincidence that the story I read the day before Christmas appeared on a documentary the day after Christmas.
The next six hours we sat, engrossed in three movies back-to-back:
ELF: After inadvertently wreaking havoc on the elf community due to his ungainly size, a man raised as an elf at the North Pole is sent to the U.S. in search of his true identity.
Dodgeball: A group of misfits enter a Las Vegas dodgeball tournament in order to save their cherished local gym from the onslaught of a corporate health fitness chain.
Miracle: Miracle tells the true story of Herb Brooks, the player-turned-coach who led the 1980 U.S. Olympic hockey team to victory over the seemingly invincible Russian squad.
I went to bed after the third movie.
DECEMBER 26, 2004
Some of the volunteers the next morning, after a nice meal of eggs, bacon and sausage, left to go back to site. I had traveled the farthest to be here and planned on staying another day or two, volunteers staying here or not.
Cheeta convinced Braam to drive to Bansang to pick up the DVD player at the PC Regional House. This entailed driving over an hour, crossing a ferry, backtracking 15 minutes, picking up the DVD and doing everything again in reverse. Jessica went along while I watched TV with Kelly and the remaining volunteers.
ESPN was a big hit with most of them, while I was just waiting for something better to come on. When there was football on, or any other sport, I was more intrigued when the ball was in the air then when a team had control over it. When it was in the air only the natural forces of gravity took control. Thoughts of at first freshman level physics took over of Newton’s Laws and F=ma, with the ball being controlled at every second by a force; then thoughts of senior level classes where you learn another interpretation of Newton’s Laws saying the ball actually knows ahead of time which path it’s going to take. When the ball leaves your hand, there is something called the Hamiltonian Principle which states that of all possible paths the ball could go, the actual path it does take does so such that that a certain quantity is a minimum. It still bugs me to this day of things knowing in advance which path to take, that nature chooses the most economical path (in some sense) for every object to follow. Nonetheless there’s the ball flying to the receiver; exactly as Newton or Hamilton would predict.
I gave up watching football and went on to my dartboard problem which had bugged me since the first day arriving at Kharafi. Why was the numbers in the dartboard in the order they are in? If you look at a dartboard starting from the 12 o’clock position and go clockwise the numbers go in the pattern of [20, 1, 18, 4, 13, 6, 10, 15, 2, 17, 3, 19, 7, 16, 8, 11, 14, 9, 12, 5], why are they in the pattern that they are in? I thought of the question: What if you start with the 12 o’clock position and count a certain number of steps, going around repeatedly as needed, and that is where you place the 1. Go around that number of times again this time skipping the 1 each time you pass it, and after that many steps place a 2, etc. Is it possible to construct the sequence of numbers on a dartboard in that fashion?
For example, if you start at the 12 o’clock position you can go just one spot over to get to the correct position of ‘1’. However, you can also go around the whole board again, which would make 21 steps total, or around a third time for 41. Each one of the numbers [1, 21, 41, 61, 81, 101, 121, …] lands you on the correct position for the position of 1. After landing at 1, you now have to go seven more positions to get to the two. That, or 19 more since you’re now skipping the 1 when you cross it, so one less space to count. This leads the possible steps to get to the 2 from the 1 of [7, 26, 33, 40,…] It doesn’t look like it but the number 121 is on both lists. So, if you start at the 12 o’clock position and count 121 steps around you will get to the correct position of the ‘1’. Continue 121 more steps, ignoring the 1 that you placed now, and you get the correct position for the ‘2’.
Is it possible to continue with the 3? Yes. However, that’s where it stops. You can’t find a number that will correctly place the ‘1’, ‘2’, ‘3’, and the ‘4’, let alone all the rest of the numbers.
What if you worked backwards? You can get a little farther. Take an ordinary dartboard and take out all the numbers greater than 13, such that there are now empty spaces where the numbers were. Start at position 13 and count 243 empty spaces going around as many times as needed. That will get you the correct position for the 14. Continue on counting 243 empty spaces and you’ll get 15. That number, 243, is the smallest number that will place all the spaces from 13 to 20 in their correct positions. Any multiple of 840 added to 243 would also work, but 243 was the smallest.
If you try to find a number that will place all the numbers from 12-20 on the board you’ll find it’s impossible, but 13-20 is possible. The one course I had in college on Abstract Algebra and Number Theory came in handy! I never did like the Abstract Algebra part of that course, with topics ranging from Groups, Fields, and Rings. The Number Theory I kind of enjoyed, with prime number theorems and interesting proofs about numbers in general.
For example, given an odd number not ending in five you can find a string of ones such that that odd number can divide the other number. Pick one at random, say 7; 111111 can be divided by 7. Choose 37 for example; 111 can be divided by 37. Pretty interesting, but no application though.
A new volunteer, Greta, has her masters in mathematics in the one area I didn’t particularly like; Abstract Algebra. She’s been very helpful when I need to remember a theorem, or someone to bounce ideas off of.
After working on the Dart Board problem a little bit, and with the volunteers finally being bored of ESPN they switched to another channel. The show now being displayed on TV? A special on cannibalism. Any other day I wouldn’t mind if the channel was changed, but today was different. I had just read the book ALIVE, in which they had to eat the dead to stay alive, and was hoping it would be mentioned on the show.
Just as the show was getting under way Cheeta, Braam, and Jessica arrived with the DVD player and started plugging it in and testing it to make sure it worked. This was interfering with me trying to watch a show on cannibalism. At one point they were testing the DVD and switched the TV setting with the first sentence I heard was
“… another, more recent occurance of cannabilism occurred in Andes Mountains…”
I yelled “STOP!” and sat closer to the TV to hear the five-minute story about the true Andes survivors. I was then happy, and allowed them to again fiddle with the electronics. A somewhat perfect coincidence that the story I read the day before Christmas appeared on a documentary the day after Christmas.
The next six hours we sat, engrossed in three movies back-to-back:
ELF: After inadvertently wreaking havoc on the elf community due to his ungainly size, a man raised as an elf at the North Pole is sent to the U.S. in search of his true identity.
Dodgeball: A group of misfits enter a Las Vegas dodgeball tournament in order to save their cherished local gym from the onslaught of a corporate health fitness chain.
Miracle: Miracle tells the true story of Herb Brooks, the player-turned-coach who led the 1980 U.S. Olympic hockey team to victory over the seemingly invincible Russian squad.
I went to bed after the third movie.
posted by Mike | 5:26 PM
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