I recently volunteered for an MRI study at Vanderbilt. I got to lay in the machine for 90 minutes while it made loud rattling and humming noises. It used a supposedly ultra-powerful magnet: 7 Tesla. The superpowers haven’t kicked in yet, but while we’re waiting, click the picture below for the rest of the slices.
Saturday morning, I gave a keynote at the 28th Annual Mathematics Symposium at Western Kentucky University. My talk, “How Big Can You Think?” began with a simple problem whose answer is so large it cannot be written down, even approximately, with elementary notation, such as towers of exponents. I tried to follow by saying that this inexpressability can be resolved with more powerful notation that greatly extends our ability to concisely describe numbers. I finished with an example (the “Harvey Friedman number”) which defies even the arrow notation used for Graham’s Number.
I met many interesting people, including
Dr. Peter Hamburger
Dr. Molly Dunkum
Dr. Dominic Lanphier
Dr. John Armstrong
Dr. Jason Rosenhouse
Dr. David Benko
Dr. Peter Lax
Dr. Peter Dragnev
Dr. Claus Ernst
Dr. Barry Burnson
Finally, here’s a picture Natalie took of me just after I spoke.
A few days ago, when I wrote a post proposing a metric to rank countries in the 2008 Olympics, it wasn’t yet over. Now that we have final results, let’s see how we rank:
China 100%
United States 94%
Russia 60%
Britain 43%
Australia 38%
Germany 36%
South Korea 29%
France 29%
Italy 23%
Japan 21%
Several rankings have been proposed as to how to determine which country is winning the Olympic medal race. Here I’m only going to look at the medals in isolation. Folks have suggested dividing the medals by the countries’ populations, by the GDP or by the number of athletes. You could factor in the sport popularity, fourth place rankings or time zone differences. You could even rank the countries by how well they exceeded expectations on online betting sites. I would like to see that. But it’s too complicated to do here.
Back to the medals alone. That means three numbers per country.
Clearly,
1) Using the total number of medals leaves out some important information. 3 golds is obviously better than 3 bronzes.
2) Using the number of golds leaves out some important information. 3 golds and 3 bronzes is obviously better than 3 golds.
I’m not the first one to suggest weighting the numbers. It’s progress, but it punts on how to do it. I’ve seen 4, 2 and 1. I’ve also seen 5, 3 and 1. It’s pretty easy to make up numbers, but it’s more satisfying to have a rationale. Here I propose the <i>distinguishing power</i> rationale. There’s only one gold medal. There’s also only one silver medal, but somebody else did better, so silver has half the distinguishing power of gold. Similiarly, bronze has 1/3 of gold. The actual weights don’t matter as long as the ratio is right. 6, 3 and 2 are the smallest integers that express the distinguishing power ratios.
And finally the rankings (today), as a percentage of the leader:
China 100%
United States 93%
Russia 49%
Britain 43%
Australia 36%
Germany 31%
South Korea 27%
Japan 24%
France 24%
Italy 20%
Gregorus seems to have already posted about this, but I can post my solution. We came up with the puzzle of finding the 8-digit number (with zeros allowed at the beginning) which has the largest possible number of permutations that are possible dates. Some numbers like 99,999,999 don’t have any. Others like 01,234,567 have a lot (starting with 01/23/4567, and most of those digits can be swapped around).
I wrote my program in the Bash shell language and it has two lines.
Line One:
perl -e 'use Date::Manip; my $d = ParseDate("01/01/0001");while(UnixDate("%Y") < 10000) {print UnixDate($d,"%m%d%Y\n"); $d = DateCalc($d, "+ 1 day");}' >> all-dates.txt
This just makes all the possible dates, one day at a time. Perl’s Date::Manip module is notoriously slow, and it took almost two hours to make all 3,652,059 dates in this millenium (not quite actually: the last year of the millenium has a five digit year). As for the delay, I rationalized that I had optimized my code for “programmer time”.
Line Two:
perl -ne 'print join("", sort (split("", $_)))' all-dates.txt | sort | uniq -c | sort -n
This sorts the digits in each date to make ordered digit clusters, then orders the clusters of digits by how many times they appear.
The results? 00,123,578 will make 2,472 dates! Of course, it is a 20,160-way tie, because any permutation of 00,123,578, such as 57,800,123 will make the same 2,472 dates.
Unfortunately, I can’t make any dates near to us from 00,123,578. The closest I can get on both sides is 07/08/2135 and 03/20/1875. Can anyone do better?
Here’s another problem I’ve been looking at lately:
Find a number 
, such that if 
is any digit from 0 to 9, 
has as its first digit past the decimal point. For instance, 63.18 works for the first power 
and the third power 
, but it doesn’t work for the second 
where the first digit is 7 instead of 2.
Hint: The smallest such number is less than 10.
To prove that the problem is possible to solve quickly, I’ll provide the solution to a more complicated version. Suppose that the first two digits of must match instead of just the first one. This means that 
has “03” just to the right of the decimal point all way to up to 
having “99” just after the decimal point. Then the smallest solution is the following number:
95.01063709417263364386569653065071891390859441647675346849
01552762611728856071543459219055118914594166160224386992990
49592128996983422861693601042135900702684741868297684742429
253949832737577847233354…
The smallest solution’s digits actually never end so I had to cut it off somewhere. I choose the cut off point to be the first place where the remaining digits still worked. In other words, if you truncated the 4 off of the end, the number would be changed too much and would fail to meet all the conditions. Furthermore, I was forced to round up (because anything smaller doesn’t work).
I’ve been forwarded a classic problem, in bold below.
The Monkey Problem
A rope over the top of a fence has the same length on each side, and weighs one-third of a pound per foot. On one end hangs a monkey holding a banana, and on the other end a dead-weight equal to the weight of the monkey. The banana weighs 2 ounces per inch. The length of the rope in feet is the same as the age of the monkey, and the weight of the monkey in ounces is as much as the age of the monkey’s mother. The combined ages of the monkey and its mother are 30 years. One-half the weight of the monkey, plus the weight of the banana is one-fourth the sum of the weights of the rope and the dead-weight.
The monkey’s mother is one-half as old as the monkey will be in when it is three times as old as its mother was when she was one-half as old as the monkey will be when it is as old as its mother will be when she is four times as old as the monkey was when it was twice as old as its mother was when she was one-third as old as the monkey was when it was as old as its mother was when she was three times as old as the monkey was when it was one-fourth as old as it is now.
How long is the banana in inches?
In fact, I’ve seen this problem back when I was kid, though I’ve never tried to solve it, mostly because that last paragraph looked tedious and as though it would require a deeply nested set of multiplied and offset variables. It turns out however, to be simpler than I imagined. Here we go….
A rope over the top of a fence has the same length on each side, and weighs one-third of a pound per foot.
Everything else in the problem uses ounces, so let’s make it consistent. We’ll make up variables as go along. First we need rope weight and rope length.
On one end hangs a monkey holding a banana, and on the other end a dead-weight equal to the weight of the monkey.
Add in variables for Dead Weight and Monkey Weight.
The banana weighs 2 ounces per inch.
Add in variables for banana weight and banana length.
The length of the rope in feet is the same as the age of the monkey
Rope Length and Monkey Age.
… and the weight of the monkey in ounces is as much as the age of the monkey’s mother.
Add in MMA for Monkey Mother Age.
The combined ages of the monkey and its mother are 30 years.
One-half the weight of the monkey, plus the weight of the banana is one-fourth the sum of the weights of the rope and the dead-weight.
The monkey’s mother is one-half as old as the monkey will be in when it is three times as old as its mother was when she was one-half as old as the monkey will be when it is as old as its mother will be when she is four times as old as the monkey was when it was twice as old as its mother was when she was one-third as old as the monkey was when it was as old as its mother was when she was three times as old as the monkey was when it was one-fourth as old as it is now.
How long is the banana in inches?
No doubt these equations are easy to solve by hand, but why settle? We can write them in a form that can be solved by computer. Right now, I prefer the mathematical software Sage . Formatted for Sage, the equations look like this:
var('RW, RL, DW, MW, BW, BL, MA, MMA')
equations = [
RW == RL * 1/3 * 16,
DW == MW,
BW == 2 * BL,
RL == MA,
MW == MMA,
MA + MMA == 30,
1/2 * MW + BW == 1/4 * (RW + DW),
MMA == 1/2 * ( 3 * ( 1/2 * ( 4 * ( 2 * ( 1/3 * ( 3 * ( 1/4 * MA)))))))]
s = solve(equations, RW, RL, DW, MW, BW, BL, MA, MMA, solution_dict = True)
print s
And the output is:
Notice the banana weighs 11.5 ounces and the monkey only weighs 18 oz., so something is little implausible about the problem, but as mathematicians, that’s not our problem…
During my stay in Thailand last week, I took 1,040 pictures and video segments. A representative 112 are posted on flickr. All the pictures in this blog entry below are also on that set.
For the first half of the week, I was in Bangkok at Natalie’s apartment. After her work finished for a break, we flew to Krabi, far to the south, then took a ferry to the island of Phi Phi (pronounced Pee-Pee). We stayed there for two nights and went bushwhacking in the mountainside jungle. There were far more european tourists than americans, (but they all stayed in town).
For the next phase, we took a ferry to the peninsula of Railay. The end of this ferry trip involved climbing on open water from the ferry into a covered canoe with a motor on the back (called a longtail boat). Then after a short canoe trip, wading up the beach with all our stuff, because there was no dock. I think we were safe, but there were other tourists with children and some with small babies that had to be passed from boat to rocking, heavily-laden boat.
In Railay, we saw a wild monkey climbing on the walls of a Hotel. Railay is quite isolated and there are apparently no roads to access it. In order to make our 10AM flight the following morning, we chartered a private longtail boat (with wading on both ends of the trip). We specially requested that the boat leave at 7:30 and paid in advance for this, but I guess that information got lost in the shuffle somewhere. We left around 8:10 with a german family and made it to the flight in time.
The area around Railay and Phi Phi is open ocean punctuated by numerous islands with steep cliffs on several, and sometimes all, sides.
Apparently at least one James Bond film was shot in the area. More details can be found on the Wikipedia page for The Man with the Golden Gun.
Some observations about Bangkok:
There’s a 7-11 on almost every corner
It’s very hot and humid, but everybody wears pants and long sleeves
Street vendors sell fruit, grilled meat and various other dishes. For 10 bhat (about $0.30) you can get a 1/3 of a pineapple, carefully peeled and chopped up in a bag with a shish-kebab stick to eat it with.
Stoplights have a large timer next to the lights which counts down the seconds until the light will change. The timers, which are about 5 feet across, are red, yellow or green to match the lights. You can see one below:
Compared to Beijing, Bangkok residents are friendlier, seem to be slightly more wealthy, are more likely to speak English and have somewhat cleaner air.
Several times in Bangkok, we saw elephants being led on the street.
The Thai language is harsh sounding.
Soy Dogs (soy means street) sleep under cars and in doorways. The one I saw looked healthy and well-groomed. Apparently, Thai people have a strong sense of generosity.
In Tokyo, as the plane was leaving the gate, I saw 3 members of the airport ground crew who had been loading luggage, wave in unison to the plane, then bow together before they turned and went back to the terminal.
Is because it’s a neat place for a sign that is already wired into every plane and they might want to use it for something later? Or is it really stopping people from lighting up? Is there even an off switch for it in the cockpit?
Mr. Fuller asks for the most unworkable Web 2.0 idea.
How about the one where you enter the kinds of pollen you are allergic to and it matches you people with similar allergies so you can carpool to work through parts of town far away from those plants?
Or the historical-data-driven horoscope site, where people with your same birthday who are in time-zones ahead of you write what actually happened to them that day, which is then turned around to be your prediction for the next few hours.
Then there’s the one where you upload geo-locating data about what part of your world your dog’s breed is originally from, so you can find neighbors whose dogs speak the same dog-language (e.g., a peruvian dog speaks the Spanish form of dog).
Finally, there’s the site where you upload your Gmail contact information and for each a friend a movie you know he or she is excited to see, and it does nothing but spam them with mail appearing to be from you that reveals spoilers. The site propagates virally, on the premise that each friend will be angry enough about it to do the same to his or her own friends.
