Red Hair
Two Russian mathematicians end up on the same plane.
"If I remember correctly, you have three sons," says Ivan. "What are their ages?"
"The product of their ages is thirty-six," says Igor, "and the sum of their ages is exactly today's date."
"I'm sorry," Ivan says after a minute, "but that doesn't tell me the ages of your boys."
"Oh, I forgot to tell you, my youngest son has red hair."
"Ah, now it's clear," says Ivan. "I now know exactly how old your three sons are."
What are the ages ... and how did Ivan determine them?
"If I remember correctly, you have three sons," says Ivan. "What are their ages?"
"The product of their ages is thirty-six," says Igor, "and the sum of their ages is exactly today's date."
"I'm sorry," Ivan says after a minute, "but that doesn't tell me the ages of your boys."
"Oh, I forgot to tell you, my youngest son has red hair."
"Ah, now it's clear," says Ivan. "I now know exactly how old your three sons are."
What are the ages ... and how did Ivan determine them?
posted by Zaux at
9:59 PM
6 Comments:
2, 3 and 6.
The factors of 36 are 2, 2, 3 and 3. So the possible ages of the three sons are 2,2,9; 2,3,6; 3,3,4.
There is just one youngest son (with red hair) so the other two possibilities are eliminated.
I have assumed that 1 is not counted as a factor of 36, otherwise we also have 1,2,18; 1,3,12; 1,4,9; 1,6,6.
I can't see where today's date comes into it.
the youngest son has red hair the older two cannot be twins so the ages are 2,3, and 6, the date is the 11th.
The factors of 36 with (sum of factors) are:
1,1,36 (38)
1,2,19 (22)
1,3,12 (16)
1,4,9 (14)
1,6,6 (13)
2,2,9 (13)
2,3,6 (11)
3,3,4 (10)
The date must be the 13th, otherwise Ivan would know which set of factors as all the others have unique sums. So the ages must be 1,6,6, or 2,2,9. Knowing there is a youngest eliminates the 2,2,9 leaving 1,6,6 as the ages.
Assume ages are integers only.
Factorize 36 into prime factors
36=2^2*3^2
x*y*z=36
youngest son delivers info that youngest son exists i.e. x< y, x<z
1,1,36 S=38 invalid x=y, invalid date
1,2,18 S=21
1,3,12 S=16
1,4,9 S=14
1,6,6 S=13
2,2,9=13 invalid x=y
2,3,6 S=11
3,3, 4 =10 invalid x=y
If the date was 10,11,14,16,21 Ivan would immediately know the ages as those sums are unique. As he require additional info to solve the combo must have a non-unique sum. S=13 has two combos, 2,2,9 and 1,6,6 but only 1,6,6 has a youngest son i.e. x ! = y
Answer:
The sons ages are 1,6,6
Cam
Chris and Cam ... you guys are right again. Finding a puzzle which is challenging for you both is difficult.
Chris... on a previous puzzle "Hole in the Sphere", you questioned a formula for the volume of a spherical cap. Here's a nice explanation:
http://mathworld.wolfram.com/SphericalCap.html
I confess that I posted what turns out to be the same problem on the 15th September ("You're Kidding!").
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