The Special Ancestor....

60

Putting numbers together, I'd opine that the king was 22 in 484AD and the son was 8 in 512AD, mostly because ages are generally given in counting numbers, and these are the only ones that start about 500AD and fit within the usual 4 score and ten span. So the king was born in 462AD and the son born in 504AD.

So, my guess is the king was 42 years old when the son was born.

Rap

markr, is there really enough information to solve it uniquely?

If his son was 8 in 512, the father was 42;
If his son was 9 in 729, the father was 18;
If his son was 10 in 512, the father was 50;
If his son was 11 in 512, the father was 60;
If his son was 12 in 512, the father was 76

Scratch the second one. That would make the professor of math 18 - seems a bit young.

Okay, should be 10 in 1000
11 in 1331
12 in 1728

Interestingly enough it can be solved because it is not unique. Otherwise he could have solved for x and y.

Exactly!

Yup guys,

Markr is Correct!


Here is the solution with explanation....

Firstly, we can say that the date of birth (DOB) for the ancestor (ANC) is:

Equation {a}: DOB(ANC) = x^2 - x

Similarly, for the son of the Ancestor (or SOANC for short):

Equation {b}: DOB(SOANC) = y^3 - y

ANC's age when SOANC was born was:

Equation {c}: Age(ANC) = DOB(SOANC) - DOB(ANC) = (y^3 - y) - (x^2 - x)

There are unlimited solutions to equation {c} so we need some assumptions and limits.

We know that ANC must have been between, say 10 years old and 100 years old when SOANC was born:

Equation {d}: 10 < Age(ANC) < 100

We also know that both ANC and SOANC were born some time since the year 500 AD:

Equation {e}: 500 < DOB(SOANC) < 2005.

Based on equation {b} and {e} we can clearly see that there are only 5 solutions for y. They are y = 8, 9, 10, 11 or 12. Any other solutions for y are in breach of equation {e}.

For each of these possible solutions for y there is only a limited number of solutions for x that comply with {d}. They are:

x y
--------
21 8
22 8
26 9
27 9
31 10
36 11
41 12

Any other solutions for x and y are in breach of equation {d}.

ANC's age when his son was born can be calculated for each of these possible solutions by using equation {c} as follows:

x y Age(ANC)
----------------------
21 8 84
22 8 42
26 9 70
27 9 18
31 10 60
36 11 60
41 12 76

So there are 7 different solutions, which is why Harry said he couldn't solve for x and y.

Now comes the lateral part of the puzzle: Mat informs Harry that "...he was your age when his son was born". Of course, Harry knows his own age. He should therefore be able to pick the correct solution from the list of 7 possible solutions shown above. However, he can't. That means that Harry (and the ancestor) must be 60 because there are multiple solutions for an age of 60, whereas any other age would yield a unique solution. Any other age and Harry would be able to solve.

A 60 year old father - quite a special ancestor!