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Old 2021-08-06, 05:44   #1
MattcAnderson
 
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"Matthew Anderson"
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Smile Exercise for 'Daryll's' age

Hi again,

Many of you will know how to solve this sort of problem. Determine how old Daryll was when he died.

His boyhood lasted one sixth of his life, he married after one seventh more, his beard grew after one twelfth more, and his son was born five years later, the son lived to half his father’s age, and the father died four years after the son.

I found this famous exercise in a (free to me) online book called An Introduction to Diophantine Equations by Andresscu, Andrica and Cucurezeanu.

My answer Daryll lived 84 years, and his son lived 42 years

Regards,
Matt
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Old 2021-08-06, 06:49   #2
SmartMersenne
 
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You don't have to multiply any large numbers:

x/6 + x/7 + x/12 + 5 + x/2 + 4 = x
(14+12+7+42)*x/84 + 9 = x
75*x/84 + 9 = x
9 = 9*x/84
x = 84


In this particular case, you may also use the clues to guess the answer: the answer is most likely divisible by 12 and 7, hence by 84. Unless he lived 168 years or more, the first plausible candidate is 84.
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Old 2021-08-06, 07:24   #3
MattcAnderson
 
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Well done SmartMersenne. I'm glad you worked that out.
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Old 2021-08-06, 11:50   #4
Dr Sardonicus
 
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Looks remarkably like (Spoiler alert!) Diophantus's Riddle...
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Old 2021-08-06, 23:16   #5
SmartMersenne
 
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Not just "like", but "exactly"!
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Old 2021-08-07, 11:02   #6
LaurV
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To get admitted into high school (9th grade) we had to solve the following problem (beside of doing other stupid things), which we still remember: "Good morning one hundred geese". "We are not a hundred, if we would be once more we are, and a half more, and a quarter more, we would be a hundred together with you". Something like that, excuse the translation which sounds crappy in English. We found out how many geese were, of course.

Last fiddled with by LaurV on 2021-08-07 at 11:02
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Old 2021-08-07, 11:55   #7
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Quote:
Originally Posted by LaurV View Post
To get admitted into high school (9th grade) we had to solve the following problem (beside of doing other stupid things), which we still remember: "Good morning one hundred geese". "We are not a hundred, if we would be once more we are, and a half more, and a quarter more, we would be a hundred together with you". Something like that, excuse the translation which sounds crappy in English. We found out how many geese were, of course.
I thought of two different possible meanings of "if we would be once more we are." Both possibilities led to positive integer solutions. It is possible of course that neither possibility I thought of is what was intended

First possibility: "Once more we are" means "twice as many":

2*x + x/2 + x/4 + 1 = 100

11*x/4 = 99

x = 4*9 = 36


Second possibility: "Once more we are" means "one more than":

x + 1 + x/2 + x/4 + 1 = 100

7*x/4 = 98

x = 4*14 = 56
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Old 2021-08-08, 07:00   #8
LaurV
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Quote:
Originally Posted by Dr Sardonicus View Post
"Once more we are" means "twice as many":
Yep, that was correct. My English translation skill sucks. My excuse is that in English it sounds bad either way , in Romanian there was no doubt which possibility it was, Romanian is a lot more flexible/poetic. And that exam was to test math skills not language skills.

Last fiddled with by LaurV on 2021-08-08 at 07:04
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Old 2021-08-08, 14:20   #9
R. Gerbicz
 
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Quote:
Originally Posted by SmartMersenne View Post
In this particular case, you may also use the clues to guess the answer: the answer is most likely divisible by 12 and 7, hence by 84. Unless he lived 168 years or more, the first plausible candidate is 84.
Why, because you see somewhere in the problem the 12 and 7 numbers?
Then solve this:

His boyhood lasted one sixth of his life, he married after one ninth more, his beard grew after one eighteenth more, and his son was born five years later, the son lived to half his father’s age, and the father died six years after the son.

And you get that the answer is not divisible by 18 or 9.
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Old 2021-08-08, 17:44   #10
Viliam Furik
 
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Quote:
Originally Posted by LaurV View Post
To get admitted into high school (9th grade) we had to solve the following problem (beside of doing other stupid things), which we still remember: "Good morning one hundred geese". "We are not a hundred, if we would be once more we are, and a half more, and a quarter more, we would be a hundred together with you". Something like that, excuse the translation which sounds crappy in English. We found out how many geese were, of course.
Funny, my paternal grandfather, who will soon be 86 years old, gave me this exact riddle a few years ago (I think it was around 2012, so roughly 10 years old me). The wording of the problem was exactly the same, a man seeing a lot of geese flying around and talking to them.

He was born in Carpathian Ruthenia/Zakarpattia, now part of west Ukraine - this may be the reason why LaurV also knows this riddle, it's a region neighbouring today Romania.

I don't quite know, how my grandfather knows it, because he was born in 1935, only a few years before WWII, so he didn't have many opportunities to study, even if he had the money for it. As far as I know, he used to be a car mechanic and electrician. I am not sure what education he had, but I know he didn't go to university (nor college, because I think those are considered different things in some countries), neither did my grandmother. My father did.
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Old 2021-08-08, 22:49   #11
retina
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Quote:
Originally Posted by Viliam Furik View Post
I don't quite know, how my grandfather knows it, because he was born in 1935, only a few years before WWII, so he didn't have many opportunities to study, even if he had the money for it. As far as I know, he used to be a car mechanic and electrician. I am not sure what education he had, but I know he didn't go to university (nor college, because I think those are considered different things in some countries), neither did my grandmother. My father did.
Some of my most knowledgable minions never went beyond more than a rudimentary formal schooling.

Smart people (the real meaning of smart, not the fake "smart" devices) can find a way to excel, if they are willing to try.
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