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More ways to get Braingle...

The Father of Algebra

Math brain teasers require computations to solve.

 

Puzzle ID:#35386
Fun:*** (2.96)
Difficulty:** (1.98)
Category:Math
Submitted By:buddyboy*
Corrected By:cnmne

 

 

 



Diophantus was a Greek mathematician who lived in the third century. He was one of the first mathematicians to use algebraic symbols.

Most of what is known about Diophantus's life comes from an algebraic riddle from around the early sixth century. The riddle states:

Diophantus's youth lasted one sixth of his life. He grew a beard after one twelfth more. After one seventh more of his life, he married. 5 years later, he and his wife had a son. The son lived exactly one half as long as his father, and Diophantus died four years after his son.

How many years did Diophantus live?

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Answer

The riddle, the "facts" of which may or may not be true, results in the following equation:

x/6 + x/12 + x/7 + 5 + x/2 + 4 = x

where x is Diophantus's age at the time of his death.

Therefore, Diophantus lived exactly 84 years.

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Comments

EnderofGamesAca*
Feb 21, 2007

I did x/6+x/12+x/7+4+5=x/2 and still got 84.
EnderofGamesAca*
Feb 21, 2007

Good teaser, though!
notsosmart111Aie*
Feb 21, 2007

Um... I'm afraid of my self now. I actually got that right!
vlermaAus*
Feb 21, 2007

Sorry, been too long since high school for me, even tho then Algebra was a favorite. Nice job, I think.
markXmyXwordsAus*
Feb 22, 2007

i barly graduated from high school so i coulnt figure it out maybe you can teach me howlololololo
4demo*
Feb 23, 2007

We had to do this exact problem for homework back in Algebra I, so it was easy (but not back then). No kidding.
solarsistim321*us*
Feb 24, 2007

Is that true? Is that really the only thing about his life?
shopaholicAus*
Mar 04, 2007

wow im NOT good @ math lolz
yongrenjieAmy*
Mar 25, 2007

did this before in a book--
Fernandez*us*
Apr 03, 2007

cool one! my most favorite even though i solved it sompletly different!
buddyboy*
Jun 12, 2007

i actually got the idea from a 7th grade algebra book, i'm in 7nth grade.
vg674*us*
Mar 24, 2008

My first step was to find the least common multiple. From then on, it was simply checking to see if the numbers were correct.

Excellent teaser.
melomaniac
Mar 05, 2009

a simpler way/ (without the equations). All values have to be integers, so a number that can be divided by 12 and 7, 84! ta da!
doeheadAus*
Mar 05, 2009

I have a doctorate in math science, and there are actually five different equations to find the correct answer.
xdbtcpAus*
Mar 05, 2009

What are the 5 equations? I'd love to know!!
javaguru*us*
Mar 05, 2009

I didn't have pencil and paper to set up and solve an equation, so I just solved it as a fraction by punching this into my calculator to get the answer:

144 * 7 = M+ * 9 / ( MR - 72 * 9 - 14 * 18 ) =

This was equivalent to the following equation with some shortcuts computed in my head to cut down on the keystrokes:

(4 + 5) * 6 * 12 * 7 * 2 / (6 * 12 * 7 * 2 - 6 * 12 * 7 - 6 * 12 * 2 - 6 * 7 * 2 - 12 * 7 * 2)

6 * 12 * 7 * 2 = 1008 is the common multiple of the fractions (not the least common multiple, but simpler to use) and thus the denominator.

6 * 12 * 7 + 6 * 12 * 2 + 6 * 7 * 2 + 12 * 7 * 2
= 504 + 144 + 84 + 168 = 900

is the numerator. 900/1008 is the fraction of his life not accounted for by the 4 + 5 = 9 other years, so subtract from 1 and multiply the reciprocal by 9:

9 * 1008/(1008 - 900) = 9 * 1008 / 108 = 84

The 72 * 9 - 14 * 18 is just shortening the calculation by rearranging

a * b * c + a * b * d = a * b * (c + d)

and

a * c * d + b * c * d = c * d * (a + b)

to give (6 * 12) * (7 + 2) + (7 * 2) * (6 + 12) = 72 * 9 + 14 * 18

So doehead, was this one of your five equations?

cj-pachecoAus*
Mar 05, 2009

i dont know how but i got 83
wordmama*
Mar 05, 2009

I love algebra (despite my name, I know there's not often a cross-over) and enjoyed this teaser a lot! Got it with only one initial mis-step (used the common multiple of 6& 12 before noticing the 7, and quickly switched his age from 72 to 84 and then followed the wording to prove that it fit & lo and behold! Now I'm embarrassed it's out there that I said I like algebra but didn't solve it algebraically
math12*us*
Mar 05, 2009

well actually the LCM of 6 and 12 would be just 12, not 72.

Anyway, as u can c from my name, i LUV math!!!
Provlear*
Mar 05, 2009

Gets all of us post-highschoolers back into an algebra mindset... Wait, is that a good thing? Good puzzle.
UptheHillA*
Mar 05, 2009

auntiesisAus*
Mar 05, 2009

I have tutored in math for so many years that I had better get this one right, and I did. Quite easy. Very enjoyable. Thanks.
EmilyAnne*us*
Mar 05, 2009

i really liked this
tonjawithaj*us*
Mar 05, 2009

I loved that!
kauphi1976Aus*
Mar 05, 2009

spent a few years doin these yonder years ago
oudeisse
Mar 06, 2009

EnderOfGames, of course you got the same thing. Your equation just subtracts x/2 from both sides.
wordmama*
Mar 12, 2009

Math 12, I did not say LCM, I just said 'common multiple.
PaladinAus*
Mar 13, 2009

I like to write the equations as I read through them, so I did it as such:

D = Dionysus's age
S = his son's age

1/6D + 1/12D + 1/7D + 5 = Dionysus's age at his son's birth = Db

S = .5D

so

(D-4) - Db = S = .5D
17/28 D - 9 = .5D
17/28 D - 1/2 D = 9
(17-14)/28 D = 9
3/28 D = 9
D = 28 * 3
D = 84

Loved this puzzle... thanks for sharing!
scholariumA
May 16, 2009

"The son lived exactly one half as long as his father, and Diophantus died four years after his son."

it seems to be confusing.. particularly when he stated; and Diophantus died 4yrs after his son..

honestly speaking, i got confused with it. i thought, Diophantus died 4yrs after his wife gave birth to their son..

x/6 + x/12 + x/7 + 5 + 4 = x

ignoring x/2, as if its just some kind of a "negligable fact"..
that his son's age doesn't need to be involved in the solution thinking that Diophantus died after his son's birth..

with that, i got 14.82 so i was shocked! "Diophantus died with that age?!"

so I read again the problem..

tried to solve again, this time with his son's age.. which is x/2.

with that, i got the right answer which is 84..

yes, its a good problem. but next time, make sure that it is "comprehensively clear" to all the readers. because a better understanding with the problem is a must to get the right answer.
buu441Ain*
Mar 05, 2012

me love maths. awesome teaser

and as for diophantus, diophantine equations are awesome two variables and only one equation fun to solve.
HABS2933Aca*
Mar 05, 2012

I got it, but it took a while, bit daunting at quarter of seven in the morning, but lots of fun.
here2A*
Mar 05, 2012

thanks. I got it!
wordmama*
Mar 05, 2012

Had not remembered doing this 3 yrs. ago! This time I did do it algebraically--fun, fun, fun!!! (Still love algebra!!)
gghali*
Mar 05, 2012

Great teaser. Very fun!
eighsseAus*
Aug 28, 2013

Good one!

I see a lot of people saying it is easy because you just have to find the least common multiple and use trial and error for 28, 56, 84. Which does work, of course, BUT, it doesn't say in the teaser that it's going to be an integer. It just asks how many years. It could have been 76.9 or 45 1/3 or something. Just sayin'!
tomlaidlaw
Jan 10, 2014

Really, now. What was Diophantus famous for?



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