In Russian the middle name is the patronymic. Thus, the middle initial is the first letter of the father’s first name. And, as in many languages, the first initial is the first letter of the first name.

Below are the names of a Russian family's names:
• A.N. Petrov
• B.M. Petrov
• G.K. Petrov
• K.M. Petrov
• K.T. Petrov
• M.M. Petrov
• M.N. Petrov
• N.M. Petrov
• N.K. Petrov
• N.T. Petrov
• T.M. Petrov

Draw the family tree of the Petrovs, given that every father has two sons, the patriarch of the family has four grandsons, and his sons have two grandsons each.

Prove that the solution is unique.

Source: Tanya Khovanovna (sans solution)

There are a total of seven initials represented.  Here is how the men relate to each initial.  Potential fathers at the left, sons to the right:

AN    ----------  A  ------------------------
BM    ----------  B  ------------------------
GK    ----------  G  ------------------------
KM, KT    ------  K  ---------------   GK, NK
MM, MN    ------  M  ---   BM, MM, NM, KM, TM
NM, NK, NT   ---  N  ---------------   AN, MN
TM    -----------  G  ---------------   NT, KT

There is only one possible pair of sons for TM and only one possible father for NT and KT.  Thus, TM is the father of NT and KT.

There are a total of six men without sons - two in the third generation and all four in the fourth generation.  Three of these are AN, BM, and GK.
Of the remaining three, two are from the NM-NK-NT group, and the remaining is from the KM-KT group.

There are an odd number of sons from father M.  Thus, one of these (BM, MM, NM, KM, TM) must be the patriach.

BM cannot be the patriarch since he has no son.
KM cannot be the patriarch because one of his two possible sons (GK) is childless.
TM cannot be the patriarch because both of his sons (KT and NT) must have grandsons.  KT would have to have grandsons through NK, yet only one N father has sons. So TM's son NT would have to also have sons.
NM cannot be the patriarch because AN would have to be his son, but NM could not have a grandson through AN.

Thus, MM is the patriarch.

The six men without sons are fathered by four fathers, three of whose names begin with K, M, and N.  Since there isn't a second K or N father, and since there can't be two M fathers with sonless sons, the name of the fourth father of a sonless son must begin with the letter 'T'.

That means TM has a son that has no son - either KT or NT, or possibly both.

AN's brother, MN, is not in the group of six without sons, thus MN is of the third generation, and AN-MN's father is a son of MM.  That must be NM, making BM as one of his sons.

This cancels out NM as one without sons, making NK and NT sonless.  This makes NK the brother of GK, and makes NT a sonless third generationer and KT his brother, the father of GK and NK.

So TM is the other son of the patriarch, and KM the other son of MN and brother to BM.

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Patriarch = MM

MM sons = TM, NM

    TM sons = KT, MT
    
        KT sons = GK, NK

    NM sons = AN, MN

        MN sons = BM, KM

Posted by hoodat on 2016-08-03 04:47:38