# Reconstructing the relationship of ancient Egyptian mummies

You are an archeologist and you have found a tomb with ancient Egyptian mummies. You know that each person has two types of genes, where one type is inherited from the first parent and one type is inherited from the second parent. For example if the first parent has XY genes and the second parent has WZ genes, then their children can be any of the following: XW, XZ, YW and YZ. Note that the order doesn't matter, so XY is equivalent to YX. You have found 10 mummies in the tomb with the following genetic makeup:

AA, AB, AC, AD, BC, CC, CD, CE, DD, DE


You know that they all come from the same family tree. You also know that each person has either none or both parents present. What can you establish about their relationship to each other?

• I came up with this puzzle after watching a show on the top 10 ancient Egyptian mummies. There are still many mysteries surrounding their identities. Aug 22 at 12:18
• Do all ten members come under the same family tree?
– Sid
Aug 22 at 12:59
• @Sid yes. I will clarify this. Aug 22 at 12:59
• "Each person is either a parent or a child". Does that mean that a person cannot be both a parent and a child? That does not seem to be compatible with all being from the same family tree.
– wimi
Aug 22 at 13:04
• @wimi sorry that wording was confusing. I tried to fix it. Aug 22 at 13:20

This is one possible solution:

 AA --- DD
|
|
/|\
/ | \
AB AC CD --- CE
/\
/  \
CC  DE


and this is another possible solution:

 AA --- CC   DD --- CE
\         /
\       /
AC --- DE
/\
/  \
|
|
BC


• Nice work, you get my tick. I just realised that my intended solution is not unique. Aug 22 at 13:48
• By the way, there is at least one more solution that I know of. Aug 23 at 2:33

If you acknowledge that ancient Egyptian political relationships were sometime quite messy, there are even more possible solutions. Here's one I found (in two parts for legibility):

BC --- AA --- DE
/ |      |  |
/  |    --|---
/   |   /  |   \
AB   AC-/   |    CE




|      |
CC     DD


• Looks like you have CD twice Aug 24 at 5:23
• @DmitryKamenetsky fixed the typo (one was supposed to be CC) Aug 24 at 17:24

Indeed there are many, many solutions to this. I decided to make a little game out of this. What is the maximum number of generations that can be represented?

If we discount interbreeding, each new generation will require a new mate, so with only 10 mummies present the maximum will be 5 generations:

CC --- AA
|
AC --- DD
/\
/  \
|
CE --- AB
|
CB


It cannot be only 1 generation as then all 10 mummies would just be strangers and not related, so the minimum has to be 2:

AB --- AD --- CD --- BC --- DE
|     /\      |      |
AA   AC DD   CC      CE


Ok, what if we allow interbreeding?

Well, each sexual encounter results in a child. There can obviously not be 10 generations because we only have 10 mummies, and 2 of them have to start the ball rolling, that brings the maximum down to 9. Also since 2 people can have a maximum of 4 traits, and there are 5 traits represented, at some point down the line an unrelated parent has to be introduced into the mix, brining down the maximum number of generations to 8:

  AD ------ BC
|    |   |
--- AB   |
|        |
AA -------
|
AC ------- DE ---
|     |     |   |
-----CE     |   |
|           |   |
CC ----------   |
|   |
CD --
|
DD


The last family tree is a bit unrealistic, although technically possible, so I decided to see if I can come up with a more realistic incestuous family - really twisted stuff here:

     AD -------------- BC
|       /|\       |
|      / | \      |--CC
|     /  |  \  ___|
|    AB-AC   CD_______CE
|      |          |
|     AA         DE
|                 |
-------------------
|
DD


• This is very cool! Thanks for investigating this. Ancient Egyptian royal families were quite incestuous, so I think some of these are possible :) Aug 25 at 23:55
• How about one mummy is the king, and the rest are his wives? Still fulfill the requirement that "either none or both parents present", and "they come from the same family tree" (loosely), I suppose? It's a bit stretching the family tree part, tho. Aug 26 at 3:47