# The genes of Gilgamesh

Can you work out the parentage of the ancient hero Gilgamesh?

If three of my grandparents were French and one Russian then I would be said to be one quarter Russian and three quarters French (RFFF). If half of my great-grandparents were English and half other nationalities then I would be said to be half English (EX)

The king of Sumeria around 2600 BC was a great warrior called Gilgamesh . The tale of Gilgamesh the hero stated that Gilgamesh was "Two Thirds God and One Third Man".

Assuming normal reproductive behaviour between a set of ancestors of type pure G and a set of anscestors of type pure M, could you create an offspring of type two-thirds G and one-third M?

How many generations would it take to create a genetic stock to within 1% of (GGM)?

* Do you think that it is possible to make (GGM)?

* What other ratios (e.g. GGGGMM) would be equivalent to (GGM)?

* What possible fractions would be within 1% of (GGM)?

Can a fraction of one third ever be equal to a power of one half?

* What other ratios (e.g. GGGGMM) would be equivalent to (GGM)?

* What possible fractions would be within 1% of (GGM)?

Can a fraction of one third ever be equal to a power of one half?

It is important to realise in this question that the cross is not like a normal Mendelian cross, but that all previously accumulated genetic infortmation is inherited.

It would be impossible to create an offspring who was two-thirds G and one-third M. This can be explained as follows: each successive generation doubles the number of possible 'parts' that it can be made out of. For example the first generation is purely G or M, whereas the second generation has two 'parts' - it can be GG, MM or GM. Additionally, the third generation has four parts and can be GGGG, GGGM, GGMM, GMMM or MMMM. Thus, the number of 'parts' is clearly $2^n$ where n is an integer.

In order to be able to be composed on one-third M, we are essentially asking if there is a value of n such that $\frac{2^n}{3}$ is an integer. It can be seen that there is no value of n to make this true because $2^n$ generates numbers which are divisible only by the prime number 2, but by no other primes. Because 3 is a prime number, this means that the expression can never yield an integer.

We are looking to find a composition which is within 1% of $\frac{1}{3}$. which is equivalent to the range $\frac{99}{300}$ - $\frac{101}{300}$ which is $0.33 - 0.33\dot{6}$.

Using trial and error:

$\frac{3}{8} = 0.375$

$\frac{5}{16} = 0.313$

$\frac{11}{32} = 0.348$

$\frac{21}{64} = 0.328$

$\mathbf{\frac{43}{128} = 0.3359}$

Thus, 128 = 2$^7$, and so there need to be a minimum of 8 generations. Can you draw out the crossing scheme to create this final progeny?

### Why do this problem?

This is an interesting, quick question for use whilst studying
genetics and will draw the learners in to the mathematical issues
surround inheritance and fractions.

### Possible approach

Discuss the question before attempting a detailed solution.
Some students might see it as possible to make (GGM), others might
see it as impossible. The challenge will be in explaining clearly
why this is or is not the case. Do not allow students to get away
with saying 'it is obvious' one way or the other. A full
explanation will require clear reasoning, and this should be
encouraged. Help by asking 'why is it obvious?' Part of the
subtlety will be that the parents, grandparents and other
generations could be made from a variety of mixes of (GM). Students
will need to be clear that they have considered this point.

### Key questions

- Do you think that it is possible to make (GGM)?
- What other ratios (e.g. GGGGMM) would be equivalent to (GGM)?
- What possible fractions would be within 1% of (GGM)?

### Possible extension

- Generally speaking, starting from a stock of pure G and pure M, can you determine thestock of all possible descendants?
- Can students invent a similar question of their own?

### Possible support

Students struggling to get started could be asked directly to
work out possible great-grandparents for (GM), (GGGM),
(GGGGGGGM)