Roland Dürre
Monday April 17th, 2017

Solution

Two weeks ago, I formulated a Logelei I very much like. I also considered it extremely hard to solve. Well, one email I received had the correct answer.

Here is the solution to the problem I gave you on April, 2rd, 2017. I copied the formal part of the solution from the winner: Jörg.
The question was:

How can the criminals make sure that they all survive?

And the solution is surprisingly simple!

Jörg – the winner!

The 10 gangsters have to assign a number between 0 and 9 to every one of them, for instance by counting them down!

As soon as a gangster will see the nine images of all the other nine gangster, he will do the sum of all the numbers on these images and then add “his” number to the sum.

Then he applies the operation modulo 10 to the result and calls the resulting number. Every gangster does this during her or his interview.

This is how you can make sure that exactly one of the gangsters will give the number on his/her picture. The others will, of necessity, say a wrong number – but that is irrelevant, since it will suffice if one of them gives his correct number. That means they all will survive.

Well, as you see, you must never give up hope – once in a while, even mathematics can help.
Here is the formal description of the solution (after Dr. Rothermel).

• Let the number of gangsters be: N
• Let zi be the number assigned to the gangster I (not known to him). It is not necessarily unique and it is part of the set {0, 1, …, N-1} of which a minimum of one needs to be told in the end..
• Let S be the sum of all pre-defined numerals S = Σ zi
The gangsters agree upon the following procedure:
1. Initially, each of them gets a personal, unique number i (known to him/her) assigned to her/his picture from the set {0, 1, …, N-1}.
2. During the interview, every gangster builds the sum of all the numbers he/she can see – that is the (definite) total sum S minus his own (not known to him) numeral zi , i.e. S – zi . That is the only information at his disposal.
3. Since the gangsters are only interested in the numbers in the range {0, 1, …, N-1}, they will modulo N or the congruency relation ≡ N. Now each gangster will calculate an integer x such that:
x ≡ N i – (S – zi ) or
x = ( i – (S – zi )) mod N (I)
With this procedure, exactly one gangster will get his correct zi!

Proof:

S is congruent with a number s from the set {0, 1, …, N-1} or S ≡ N s, consequently, you can also write (I) as:
x ≡ N i – (s – zi )
Since no two N i’s are identical, one of them equal s, consequently, we have for one gangster:
x ≡ N zi.
both x and zi belong to the set {0, 1, …, N-1} which means they are not only congruent, but identical:
y = zi,
and that means this gangster will have the correct number for himself.
(Solution and proof by Dr. Jörg Rothermel)
Now I would recommend that you read the problem  again and ponder it a little bit.

RMD
(Translated by Evelyn)

Roland Dürre
Wednesday April 12th, 2017

Another Logelei!

Since my last Logelei gave many people some joy (and brought others close to desperation), here is another one. However, it is
Basically very simple!

A house owner has a square-shaped patio paved with 64 simple square-shaped tiles (each measuring 30 cm x 30 cm). He wants to put 31 domino stones (he already bought them) over them. They are twice as big as the tiles, that is, their width is twice their length, i.e. 60 cm x 30 cm.

His special desire is that the two opposing corners should be left untouched by the domino stones. He wants cubic tiles with a very special symbol at those corners. He already bought the two of them (just as big as the original ones, i.e. 30 cm x 30 cm). Naturally, for aesthetic reasons, the domino stones cannot be halved.

He assigns the task to a craftsman. The craftsman is happy to do the job and starts his work. He leaves a corner open and starts positioning the domino stones.

Now here are the questions:
Will the craftsman manage to do it?

If yes, how?
or
If not, why not?0

I will publish the solution two weeks from now – and until then, I look forward to many  emails  suggesting a solution!

Here is some help: I, too, have a square-shaped patio 🙂

In 1968, Werner gave me this question while we were watching a boring soccer match in the Rosenau Stadium. He was five years my senior and already studied mathematics at TUM at the time. I was still a pupil at the Jakob-Fugger-Gymnasium. Unfortunately, I do not know the original source of this logelei, either.

RMD
(Translated by Evelyn)