The Duke of Densmore was killed by an explosion that damaged his castle. His testament, evidentally destroyed, was said to have displeased one of his 7 ex-wives. However, each of these women had come in the castle by invitation shortly before the crime. Each of the 7 ex-wifes swears that the invitation was the only time she went there. All of them could be guilty, but because of the careful preparation of the bomb, which was hidden and adjusted to fit in a knight's armor in the Duke's bedroom, the murderess must have come in the castle more than one time. So the guilty woman lied: she came in the castle several times. The women do not remember exactly the date when they went there, but they remember who they met:

Ann met Betty, Charlotte, Felicia, Georgia.

Betty met Ann, Charlotte, Edith, Felicia, Helen.

Charlotte met Ann, Betty, Edith.

Edith met Betty, Charlotte, Felicia.

Felicia met Ann, Betty, Edith, Helen.

Georgia met Ann, Helen.

Helen met Betty, Felicia, Georgia.

Which one was lying? Who is therefore the murderess?
________________________________________________

The original problem was posed in 1980 by the French mathematician Claude Berge (1926 - 2002) to demonstrate a solution by graph theory.

**The puzzle is more a mathematical problem than a logical one.** We can also see that every statement of an ex-wife confirms the other ones. (I put the problem in the category of 'logic' only because of its flavor.)

CLAUDE BERGE (1980):

"L'énigme policière", Regards sur la théorie des graphes,

Actes du Colloque de Cérisy, Presses polytechniques romandes, Lausanne.

CLAUDE BERGE (1994): "Qui a tué le duc de Densmore?",

Bibliothèque Oulipienne no. 67.

CLAUDE BERGE (2000): "Qui a tué le duc de Densmore?", Une nouvelle policière où le meurtrier est confondu grâce à l'utilisation d'un théorème de combinatoire.

(= Réédition du 1994 par Ed. Castor Astral)

The version of the problem given above is based on the following source:

http://mathafou.free.fr/pbm_en/pb205.html

The sequence {a

_{n}} is defined by
a

_{1} = 1 and a

_{n+1} = a

_{n}+1/(a

_{n}^{2}).

Show that a_{2016} is over 18.

Let x be a positive integer of the form 24n − 1, where n is an integer.

Prove that if a and b are positive integers such that x = ab,

then a + b is a multiple of 24.

Find the smallest positive integer fitting the following description:

i. All its digits, except two, are sixes.

ii. It is prime.

Why is this puzzle D2?

Number 3 can be expressed as the sum of one or more positive integers in 4 distinct ways:

**
3; 2 + 1; 1 + 2; 1 + 1 + 1 **

Number 4 can be expressed as the sum of one or more positive integers in 8 distinct ways:

** 4; 3 + 1; 1+3; 2 + 2; 2 + 1 + 1; 1+2+1; 1+1+2; 1+1+1+1 **

Prove : any positive integer **n **can be so expressed in** 2**^{n - 1 }ways.

There are 3 known numbers that are repunits at least in four bases.

1. The title mentions one of them.

2.** Please find the other two** without referring to Google, OEIS et al.

3. Limit your search up to 9999.

4. Looking for the "fourth" would be *exercise in futility.*

If a set N

_{9} = {1, 2, 3, 4, 5, 6, 7, 8, 9} of 9 numbers is split into two subsets, then at least one of them contains three terms in arithmetic progression.

The statement is not true for a set N

_{8} of 8 integers.

Seems obvious?

Prove it.