 ## Friday, May 21, 2010

### How long did he live?

Here is an ancient Greek puzzle about Diophantus, who is believed to be the father of algebra:
This tomb hold Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life.

How long did Diophantus live?
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Jen said...

Is it 65 years and 4 months? I have the following formula, where X is his lifetime:

X-4 = 2(X/6+X/12+X/7+5)

So take the fractions of his life that are his childhood, teens, and bachelorhood (X over 6, 12 and 7 respectively) and add 5 years. Double this to get the age he was when his son died, then add 4 years and get his total lifetime.

Simplified and solved:

X-4 = 2(33X/84 + 5)
X-4 = 66X/84 + 10
X-14 = 66X/84
-14 = 66X/84 -X
-14 = (66X-84X)/84
-14 = -18X/84
-14*84 = -18X
-1176 = -18X
X = 65 and 1/3

Maria said...

Katrina - you are correct but unfortunately I have to remove your solution as only one puzzle answer per person per week is allowed. You just masterfully solved the Working Holiday Puzzle.

Jen - you are close but a bit off at the end. Want to revisit your solution?

Lynnet said...

D=Diaphantus
x=D's total life
boy=1/6X
teen=1/12X
married=1/7X
son born=+5
son dies when son's age is half of D's final age
D dies= +4
D's age at son's birth= 1/6X+1/12X+1/7X+5=11/28X+5

son's age at death=y
D's age at son's death= 11/28X+5+y
D's age at own death=11/28X+9+y=X

Son's age at death= 1/2 of D's total age=y=X/2
D's age at death= 11/28X+9+X/2=

11/28 + 1/2= 11/28 + 14/28=25/28
25/28X + 9=X

9=3/28
3=1/28
X= 28x3

28
x3
__

84

X=84
Diaphantus died when he was 84.

My mom helped me with this puzzle.

Maria said...

Lynnet has done some very impressive calculations here and correctly defined Diaphantus' age to be 84. She only seems to forget X in the 9=3/28 statement
I understand that what she meant is:
25/28X+9=X
9=X-25/28X
9=(28-25)X/28
9=3X/28
3=X/28
X=28 x 3
X=84