a problem from my childhood...

when i was 9 years old my dad called me from england to say hello (he was there for work). it was rather hot in india at the time and we started talking about temperatures. this is how a part of the conversation went.

me: hey dad.
dad: hey son.
...
...
...
...
dad: its so nice out here, about 18 degrees. how is the weather in india?
me: its about 81 degrees.

and then i handed over the phone to my mom and went into my room. for quite a few days the two numbers, 18 and 81, bothered me quite a bit. over a weekend, while doodling, i found something rather cute about those two numbers but little did i know that it would be the cause of unrest for the next 12 years to come.

it is fairly easy to see the following.

9x2 = 18

and

9^2 = 81

see anything cool? the product of 9 and 2 gives you the reverse order of the digits of the 2nd power of 9. in that innocent observation lies a deadly dragon (i stole the reference from one of mazz's posts, hope he doesnt mind).

for 12 years now i have toiled over the following few problems and havent solved any.

1. are there any pairs besides (9,2) which produce similar results? *adding leading zero's is allowed for when you reverse a certain number*

2. is (9,2) the only pair that has this property? if so, prove that there is no other.

3. if there are others, are there infinitely many of them or just a few? if there are finitely many pairs, then are they of a certain form?

one is not restricted to base 10. feel free to explore other bases as well. however over the past few years ive found out that if there is anything to be found it is PROBABLY going to be found in base 2, 3 or 5 just because of the way the numbers behave mod 2,3 or 5.

the list goes on to 6 other questions besides the 4 asked above.

so if you are ever bored at an airport or cant fall asleep at night, give this simple problem some thought. it has provided me with ample entertainment for 12 years and it doesnt seem it will fizzle out anytime soon.

any thoughts, remarks or suggestions that might lead to the taming of the dragon and putting my tired mind to rest will be met with great rewards mostly in the form of a million thanks and perhaps a shiny penny via paypal. **the recipient pays the paypal feel however**

i didnt say anything about the product only being 2 digits long :: infact placing that restriction reduces the problem to writing a short program in c++ just cycling through all possible products and seeing if any of them work out makes it rather boring =/ the real fun starts when you reach the limits of computation and then all you can use is your head to find patterns in the problem.

an excellent example of this is the world famous (or infamous) fermat's last theorem which is extremely similar to the pythagorean theorem we have all come to know and love since kids.

pythagorean theorem states,

for right triangles, the square of the sum of the length of the two lengths equals the square of the length of the hypotenuse or simply,

a^2 + b^2 = c^2 where a,b are the legs, and c is the hypotenuse.

fermat only extended the problem another exponent and proposed,

a^n + b^n = c^n had no solution for n > 2. in simple terms, if in the pythagorean theorem you replace the power of 2 with any other integer greater than 2, you would not be able to construct a triangle with integer (whole number) lengths such that the relation a^n + b^n = c^n holds.

as close as that might be to a geometry problem, it went unproven for over 400 years and then finally andrew wiles solved it after working on the problem for 27 years or so. he met the problem in 4th grade, about the same time i met the problem i mentioned above. does that mean that it will take me 27 years to solve or will it go down in history as another famous unsolved history and my name will be immortalized (i can only hope ), only time shall tell. given the choice, i would pick knowing the solution any given day.

it IS finite if i do this.

"infact placing that restriction reduces the problem to writing a short program in c++ just cycling through all possible products and seeing if any of them work out..."

i was only referring to placing the restriction of the product being 2 digits long when i said that the program would terminate. obviously the average computer could run the program for a considerable larger set of test values, i would still like to come up with a solution without the help of a computer. i dont feel like it is a real proof (especially since the number of matrices to be inverted are infinite) and i dont want to be ostracized from the community just like the fellows that found the solution to the four color problem.

man am i excited about finding another person with similar interests here on head fi. besides music and headphones of course. i hope you dont mind me asking but do you work in an academic setting now mazz? i know you mentioned something about coming to work to the us for a few days when you came to canjam but i wasnt sure of you mentioned what kind of work. :sigh: another member i regret not getting a chance to talk to =/ perhaps at the la meet next year mr. mazz. who knows, we might be the next dynamic duo like nash and graham or the bernoulli brothers! (well you could do all the smart stuff, ill be content with just presenting it ) hope to meet you sometime in the future sir.

/back to projective geometry...bleh

this is rather freaky. i picked up a book today by clive cussler without reading much about it. the cover didnt look familiar and i love him so it was a no brainer for me to pick it up. i just took it out of my backpack to give it a little read before heading to bed and oddly enough, the title turns out to be, "Dragon." whats with all the talk of dragons today?

chinese dragons are kewl. transylvanian horn tails are the annoying scary ones.

i have quite a few other original problems, though not as interesting as the one in the first post. in case anyone is interested in taking a look at them or is a notebook-pencil mathematician (the best kind) let me know, ill either post them or send you a pm.