The Father of Algebra
Math brain teasers require computations to solve.
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?
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
EnderofGames
Feb 21, 2007
 I did x/6+x/12+x/7+4+5=x/2 and still got 84. 
EnderofGames
Feb 21, 2007
 Good teaser, though! 
notsosmart111
Feb 21, 2007
 Um... I'm afraid of my self now. I actually got that right! 
vlerma
Feb 21, 2007
 Sorry, been too long since high school for me, even tho then Algebra was a favorite. Nice job, I think. 
markXmyXwords
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
Feb 24, 2007
 Is that true? Is that really the only thing about his life? 
shopaholic
Mar 04, 2007
 wow im NOT good @ math lolz 
yongrenjie
Mar 25, 2007
 did this before in a book 
Fernandez
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
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! 
doehead
Mar 05, 2009
 I have a doctorate in math science, and there are actually five different equations to find the correct answer. 
xdbtcp
Mar 05, 2009
 What are the 5 equations? I'd love to know!! 
javaguru
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?

cjpacheco
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 crossover) and enjoyed this teaser a lot! Got it with only one initial misstep (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
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 posthighschoolers back into an algebra mindset... Wait, is that a good thing? Good puzzle. 
UptheHill
Mar 05, 2009
 
auntiesis
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
Mar 05, 2009
 i really liked this 
tonjawithaj
Mar 05, 2009
 I loved that! 
kauphi1976
Mar 05, 2009
 spent a few years doin these yonder years ago 
oudeis
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. 
Paladin
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
(D4)  Db = S = .5D
17/28 D  9 = .5D
17/28 D  1/2 D = 9
(1714)/28 D = 9
3/28 D = 9
D = 28 * 3
D = 84
Loved this puzzle... thanks for sharing! 
scholarium
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. 
buu441
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. 
HABS2933
Mar 05, 2012
 I got it, but it took a while, bit daunting at quarter of seven in the morning, but lots of fun. 
here2
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 algebraicallyfun, fun, fun!!! (Still love algebra!!) 
gghali
Mar 05, 2012
 Great teaser. Very fun! 
eighsse
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? 
Babe
Mar 05, 2015
 I do not do math teasers so no point in further comment. 
jaycr
Mar 05, 2015
 I found the answer quickly by clicking the "answer button". 
gaylewolf
Mar 05, 2015
 Babe and JayCR, I love your responses. Short and sweet. And, you, Wordmama, are just fun and good at multi tasking! Nice sunny day here even though the temp is in the frigid zone. So off to a great start!
PS Always a bright response, Habs! 
gaylewolf
Mar 05, 2015
 Hi Auntie Sis, Hope you're well and keeping warm. This has been a long, white winter! 
mepapa
Mar 05, 2015
 Um...I was riding my bike home from school when a bully came riding up and he like pushed me over..... my books all spilled out of my bookbag and the wind like, blew all my notes away and then a largish ... NO I mean gigantic, dog, took off with my algebra book. I was chasing it but my cellphone fell out of my pocket and ,like , the battery went flying? And a car ran over the phone so like, I couldn't call anyone for help with my home work, and so , like I had to get on my ps3 and do minecraft for like, a few hours you know so I could feel better,and so I couldn't like study.
( turns a bright shade of red and puts head on desk.) 
mepapa
Mar 05, 2015
 True story.... 
SfkFred
Jul 15, 2016
 In real World, every story sound complicated but only math always makes it appear simple. 
jamesw71
Sep 14, 2016
 Sorry having problems proving the math here. If you say the answer is 84....then the following should be true.
1/6 of Dio's life in youth = 14 years of age
1/12 of Dio's life he grew a beard +7 years = 21
1/7 of Dio's life later he got married + 12 years = 33
Dio had a son +5 years later = 38 when son was born
According to the "facts" Dio's son was 1/2 of Dio's age when his son died. That makes his son 42 years of age.
So if Dio was 38 when he had his son and his son lived for 42 years..that would have made Dio 80 years old.
Again according to the facts Dio died 4 years after his son which made Dio 76 when his son died. But that doesn't add up, because Dio had his son at age 38, which would only make the son 38 at his son's death which conflicts with the "fact" that his son was 1/2 of his Dio's age, which would have been 42 not 38???
I've plugged different numbers and nothing adds up here...this problem is unsolvable because we can't verify which "facts" are true and which are not.
Please tell me if I errored, I'm not the best at algebra but I can work the problem out line by line...and it doesn't make sense. 
jamesw71
Sep 14, 2016
 Sorry having problems proving the math here. If you say the answer is 84....then the following should be true.
1/6 of Dio's life in youth = 14 years of age
1/12 of Dio's life he grew a beard +7 years = 21
1/7 of Dio's life later he got married + 12 years = 33
Dio had a son +5 years later = 38 when son was born
According to the "facts" Dio's son was 1/2 of Dio's age when his son died. That makes his son 42 years of age.
So if Dio was 38 when he had his son and his son lived for 42 years..that would have made Dio 80 years old.
Again according to the facts Dio died 4 years after his son which made Dio 80 not 76 like I said above.
So I guess that adds up. 
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