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## My new signature

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About my custom signature, £åÕ³. This stands for a problem in number theory that dates back to dynastic Egypt. Contributions to its solution were made by such notables as Fibonacci and Sylvester (the latter being the founder of the American Mathematical Society), but a completely satisfactory solution had to await the arrival of computers and computation-intensive techniques (modesty forbids me saying more). I can't really state the problem here in it's full generality, justifying the "sum of the products" notation åÕ, but its most elementary instance can be given in the form of a riddle:

Ben Cartright died, having lived a full and happy life, and he left his estate to his three sons. Included in the estate were 17 horses. The instructions in the will were that the estate be divided in these proportions: one-half to Adam, the oldest son; one-third to Hoss, the second son; and one-ninth to Little Joe, the youngest.

Well, you see the problem. You can't take half of 17 horses. For that matter, you can't take one-third of 17 horses either, or one-ninth. So they called upon the circuit-riding judge to resolve the matter.

The judge's solution was this: She put her horse into the corral with the others, making 18 in all. Then she took half of the eighteen horses (9) and gave them to Adam; she took a third of the eighteen horses (6) and gave them to Hoss; and one-ninth of the eighteen horses (2) and gave them to Little Joe. So everyone was happy, having received exactly the number of horses he had coming to him.

The judge then got back on her horse and rode away.

The riddle is -- wait a minute, didn't the judge give out her horse, too, as one of the 18?

Mathman

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## The Cartright Riddle

Okay, Mathman, I'll bite. Yes, the judge added her horse to the group of horses to make 18 and thus enable the execution of Ben's will (poor Little Joe, really got rooked). But according to the riddle, the judge, after dividing up the horses, gets on her horse and rides away. I'm guessing (guess being the operative word:lol: ) that this has something to do with the horses as a group/sum vs. individual units/products. Technically, the judge divided the group/sum of horses correctly, so does it really matter what she does with each horse/product individually? Are the units part of the sum are separate? Beats me. Or perhaps the judge has beaucoup political power and takes whatever she wants, screw you Anyway, you said this mathematical theory started in dynastic Egypt. Did dividing slaves have anything to do with inspiring it?

Anxiously awaiting the answer to the riddle. I worry about those Cartright boys without Ben to guide them.
Rgirl

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## Hop Sing's non Indian giving solution

"The judge then got back on her horse and rode away."

You did not say the time lapse between her decision, and her departure.

Don't worry, Hop Sing to the rescue, he brought to the wild wild west an ancient Chinese secret of cloning. (Clonaid people you are 2 centuries behind). It was a success, and she left an exact copy of her horse at the Ponderosa, then rode away.

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## Re: Hop Sing's non Indian giving solution

I decided to delete my "Michelle's the greasiest" :lol: quote in view of all my good resolutions on the Kwan/nonKwan detente thread. I was afraid that someone would interpret it as "nyah, nyah, my skater is better than your skater." Which she is, but never mind that.

About cloning, a funny thing happened to me last night. Someone snuck in while I was asleep and stole everything in my house, and replaced it with an exact duplicate!

I asked my wife, what the heck is going on, am I crazy or what? She said, "Who are you?"

Mathman

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Test

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## calculus

Mathman, I will be leaving one more message at the brain thread about pharmackinetics and pharmacodynamics. I am not good at explaining things, so if you care to give the other posters a 3 sentence review of derivatives.

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## divide

Mathman originally said, " The instructions in the will were that the estate be divided in these proportions: one-half to Adam, the oldest son; one-third to Hoss, the second son; and one-ninth to Little Joe, the youngest."

I have no idea how to solve this riddle with 1/2+1/3+1/9 = 17/18 not a total of 18/18. I am sure you are right in setting these parameters, but if I am allowed to change little Joe's portion from 1/9 to 1/6, here is what I will do:

Ben left 17 horses, and the judge put hers into the pot for a total of 18.

Horses were given in 3 rounds, each time the fate of 6 horses were determined. Adam should receive 3, i.e. 1/2, and Hoss should receive 2, i.e. 1/3, and Joe 1, i.e. 1/6
In the first round :

Adam, Hoss and Joe each received 2 horses (one of the 2 that was given to Joe was the judges), Adam was owed one horse and Joe had an excess of one horse.

In the second round they each received 2 horses, again adam was owed one and Joe had 1 excess.

So after 2 rounds and 12 horses gone, Adam demanded 2 more horses in the final round, after all they owed him one horse in each of the first 2 rounds.

In the third round with the remaining 6 horses left, Adam received 4, so he was paid back the 2 horses that they owed him. Hoss received 2 horses in the third round, and he was happy, becuase he was supposed to receive 1/3 of the 18 = 6. Little Joe received nothing because he received an excess of 1 horse in the first 2 rounds. He then gave the judges horse back to her, and Joe was left with 3, and he was happy because 1/6 of 18 = 3.

The judge was happy to have her own horse back, and loved Hop Sing's cooking so much, she asked him to work for her. (Of course she gave an offer to the Cartwright boys and Hop Sing that they could not refuse.)

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## Re: divide and conquer

emiC, you got it! The fractions 1/2 + 1/3 + 1/9 don't add up to 1. So the testator didn't make provision for his whole estate, only 17/18ths of it. Note that in the judge's solution only 17 horses were actually given out (9 to Adam, 6 to Hoss and 2 to Little Joe) leaving the extra 1/18th of the expanded estate for the judge.

Ah, Math humor. The sad part is that even mathematicians don't think it's funny.

The serious question in number theory is, characterize ALL sets of numbers n1,...,nk for which 1/n1+1/n2+...+1/nk+1/(least commom multiple of (n1,...,nk) = 1 exactly. This is called an Egyptian fraction equation in reference to the ancient Egyption system of fractional arithmetic, which only recognized fractions whose numerator are 1. (BTW, in the original of this riddle, a rich sultan died and left an estate of 17 <em>camels</em>. We were subsequently informed by our editor that the word "camel" is now politically incorrect because it might offend middle easterners.

This riddle works because {2,3,9} is a solution to this equation. We could just as well have said, Ben Cartright died and left his estate of 3,263,441 horses to be divided among his five heirs in the proportions 1/2, 1/3, 1/7, 1/43, and 1/1807. In fact, numbers of this sort turn out to have application to some serious mathematical problems that people actually want to know the answer to.

Now I will attempt to puzzle out <em>your</em> riddle. It's got me stumped so far. If I solve it, we can write a joint paper. (Publish or perish, you know.)

Mathman

PS. There is a second part to the riddle. Hop Sing had a beautiful daughter who played the violin...

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## violin or piano

Hop Sing Kwong? I didn't know Hop Sing's last name is Kwong.

Are you sure Hop Sing's daughter is not a pianist? Look at those pianist fingers.

Donna is very slim, what do they eat at Ponderosa, tofu?

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## Re: divide and conquer

Oh, man, one of those <em>trick</em> mathematical riddles where you actually have to know and use basic math:lol: No wonder I sucked/suck at these. So the "whole" of Ben's estate to be divided by his will wasn't the "whole" to begin with, just 17/18ths of a the whole. I must say I do enjoy these once somebody else has figured them out. So the First Annual Mathman Silly Numbers (like Monty Python's Silly Walks) Award goes to emiC! Way to go! Woo-hoo! (Big party with math geeks ) You got that one in a snap. Very cool:smokin:

BTW, Mathman, as official whiner for the nonKwans/neutronKwans/maybeKwans (had to add the last category after Nats) I love the "Michelle's the greasiest" siggy. Besides, according to the KF/NKF Detente Guidelines, "Nyah, nyah, my skater's the best!" is perfectly allowable. However, "Nyay, nyah, my skater's the best and you and your whole family and every skater you've ever liked suck" falls outside the guidelines.

Even cooler job for you on the paper. If modesty permits what's the title?
Rgirl

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## Re: divide and conquer

Rgirl -- Well, since you asked ... I have reached the point in my career where I take greater pride in the accomplishments of my students than I do in my own putterings. You can check this topic out in the following papers by then-undergraduate students.

Premchand Anne, Egyptian fractions and the Inheritance Problem, College Math J., 1998.

Ana Vasiliu-Theodorescu, Znam’s problem, Mathematics Magazine, 2002.

Mi-Kyung Joo, On the system of congruences Sigma 1 to k Pi j <>k nj congruent to 1 modulo ni, Fibonacci Quarterly, 1995.

Shylynn Lofton, Homology 3-spheres and cosmological modeling, Proc. of the First Annual Michigan Space Grant Consortium, 1997.

Remembering these students ... Hey, you want diversity in higher education? Chand is Indian, Ana is Romanian, Miki is Korean and Shy is African-American.

Mathman

PS. You might wonder what the inheritance riddle, or the mysteries of ancient Egypt, has to do with cosmology. (I know you were going to ask that, so I am saving you a post.) Well, it turns out that if you have a solution to the inheritance equation, for instance 1/2 + 1/3 + 1/7 +1/42 = 1, then you can construct a model of the universe which "twists" 2, 3 and 7 times (like a multi-dimensional Moebius band) as you traverse the universe in various directions. The fact that these fractions add up exactly to 1 translates physically into the condition that if you make such a journey around the whole universe and arrive back where you started, you will find that all of the laws of physics are the same as when you started -- you don't arrive back home turned inside out, for instance, or with time going backward.

Extra credit: What is the next number in the sequence 2, 3, 7, 43, 1807, ... ?

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## Re: divide and conquer

The extra credit answer is 3,263,443. This was calculated by multiplying all of the below and adding 1.

2 + 1 = 3
(2 x 3) + 1 = 7
(2x3x7) + 1 = 43
(2x3x7x43) + 1 = 1807
(2x3x7x43x1807) + 1 = 3,263,443 (assuming i calculated correctly)

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## Re: divide and conquer

:D OK. Note that the numbers in this sequence, 2, 3, 7, 43, 1807, satisfy the Egyptian fraction equation

1/2 + 1/3+ 1/7 + 1/43 + 1/1807 + 1/(2*3*7*43*1807) = 1.

Find two more sets of five numbers {n1, n2, n3, n4, n5} that satisfy the equation

1/n1 + 1/n2 + 1/n3 + 1/n4 +1/n5 + 1/(n1*n2*n3*n4*n5) = 1.

Hint: One of them starts with 2, 3, 7, 47, ?

The other starts with 2, 3, 11, ?, ?

MM

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15. 0

## Re: My new signature

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Interesting!!!

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