Fun with Number Theory: Guessing Your Age

1. Take the last digit of your age, multiply it by 2, then for every decade you've lived add 1 to it. Color this result Blue in your mind
2. Think of your favorite digit, any digit at all, multiply it by 11.
3. Add the last digit of your age to the previous result.
4. For every decade you've lived subtract 1 from the previous result. This is your Red number.

Alright, now tell me your Blue and Red numbers and I will guess your age!

Blue Number: Red Number:

age:

© 2008 by Thoreaulylazy. All Rights Reserved.


Spoilers:

Are you sure? It could be a lot of fun to figure out the trick yourself...
Yes, I'm sure.

Don't say you haven't been warned.
I won't.

You're very persistent. Are you sure you don't have any psychological problems?
Just show me the spoilers already!!! AARRGH.

I tried to warn you earlier, remember?
Sigh.

What is the airspeed velocity of an unladen swallow?

1. I don't know


2. 10 meters per second


3. African or European?

Spoilers:
Why did you waste your time like this?
For starters, I was sick and bedridden for the day, so it wasn't as if I had many productive alternatives. Secondly, I was inspired through the sheer level of disgust the trick "Your Age by Eating Out" instilled in me. To save you the trouble of reading about that other trick elsewhere, I'll give you the gist of it. It's a puzzle that makes you compute (2*randDigit + 5)*50 + (bdayThisYear?1757:1756) - DOB. Yes, I was rather aghast at having to subtract my date of birth, especially from a number that differed by one depending on whether or not I had my birthday this year. What sort of 'puzzle' is this when it's the spanking facsimile of how age is normally computed? Unfortunately, if you view the google hits when searching for that puzzle, you'll find – almost without exception – glowing reviews of how amazing the puzzle is. I needed to right this travesty of mathematics, and so I devised a puzzle that could baffle nearly anyone who wasn't a math or cs major, and managed to successfully validate this claim on a sample of well-educated engineers.

How did you get my age?!
Well, age = ( 77*(Blue div 2) + 770*(Blue%2) + 133*Red ) % 209

77, 770, 133, 209? .. Where do these numbers come from, and why do they work?
Well, I first picked a number theory topic. A popular one is the Chinese Remainder Theorem, which states that, for an unknown x, if x % p_i is known for each prime p_i in some sequence of primes p, then x can be solved in the domain 0..Π(p)-1. This is powerful because the domain grows factorially: an innocuous sequence of small primes, like 11 and 19 can solve an integer in the range 0..(11*19-1) = 0..208. That is, if I knew what the modulo of someone's age was with respect to 11 and 19, then I can solve their age for anyone aged between 0 and 208. I chose 11 and 19 because 10 ≡ -1 mod 11 and 20 ≡ 1 mod 19, which means I can easily harness people's comfort with base 10 to extract information regarding their age modulo primes 11 and 19. Moreover, such simultaneous congruences are fun because they're very similar to a system of linear equations, except middle schoolers are well-acquainted with solving SLEs whereas simultaneous congruences are tackled usually only in math and computer science baccalaureate curricula. In order to read the text below, you should understand modulos and the totient function.

Blue = 2 * (age % 10) + (age div 10)
Red = 11r + (age % 10) - (age div 10)

age % 19 ≡ age - k₁*19 ≡ age - k₁*20 + k₁ ≡ (age % 20) + (age div 20) ≡ (age % 10) + (age div 20) + 10*((age div 10)%2) ≡ (Blue div 2) + 10*(Blue%2)  mod 19

age % 11 ≡ age - k₂*11 ≡ age - k₂*10 - k₂ ≡ (age % 10) - (age div 10) ≡ Red.  mod 11

We can express age in terms solely of Blue and Red:
age = Σ_∀pi∊p( (age%p_i)*(Π(p)/p_i)^φ(p_i) )
= ( (age%19)*(Π(p)/19)^φ(19) + (age%11)*(Π(p)/11)^φ(11) ) % Π(p_i)
= ( (age%19)*11¹⁸ + (age%11)*19¹⁰ ) % 209
= ( (age%19)*77 + (age%11)*133 ) % 209
= ( ((Blue div 2) + 10*(Blue%2))*77 + Red*133 ) % 209
= ( 77*(Blue div 2) + 770*(Blue%2) + 133*Red ) % 209

astrology in the modern world

Below is my comment left on a Slashdot article regarding astrology. I was replying another poster who wrote something along the lines of "astrology is 100% wrong."

I think you meant to say astrology is 50% wrong, because if it were 100% wrong, it would have perfect anti-correlation (akin to scoring a perfect zero on a T/F test, and is just as difficult as scoring a perfect 100). If astrology is 50% wrong, it therefore is 50% right, and depending on the brain chemistry of the person, happy memories may get weighted more than unhappy memories, and therefore the weighted average of astrology working can be significantly higher than 50% - assuming a person who adheres to astrology derives happiness from when it is correct. In fact, for such a person whose happy memories are weighted more than unhappy memories, any catalyst for increased variance will lead to a happier life, including a coin-toss on whether to drive or walk to work. If astrology is a method to higher variance in the day to day experiences of its adherents, then so be it, it results in a happier life among those humans who benefit from high variance. Conversely, for those whose brain chemistry weights unhappy memories more than happy memories, lowered variance in day to day experience is the best method for maximizing happiness. The world needs both people, those who enjoy variance and are willing to eat a mysterious berry, be it a sweet, tasty berry or a bitter, sour berry, and those who hate variance and will only eat the safe, known berry. The risk-takers help society learn about new, tasty berries, and the risk-averse help society continue the species in case the berries were poisonous after all. Astrology is merely a shrub blooming random berries, half of which are sweet (+1 correlation), half of which are bitter (-1 correlation).

Re: Oscar's

Below is my comment left for a Time.com article about Barrack Hussein Obama and the tale of Gaydolph Titler's unsucessful 1940s presidential bid retold at the Oscar's by Jon Stewart:

As an octogenarian, I remember vividly when I was at the voting poll having to decide whether to vote for Gaydolph Titler or not. I recalled the numerous positions Titler had held and while I agreed on his stance on nearly every issue, when it came down to pulling the lever and casting my vote, I just couldn't in good faith vote for Titler. It was a shallow decision which I have regretted to this day. Now in my ripe old age, I worry far less what others may think of me, and this time around I shall cast my vote for Barrack Hussein Obama with my chin held high, or at least as high as my Osteoporosis permits me.