Roland Dürre
Thursday January 17th, 2019

(Deutsch) (Null), Eins, Zwei, Drei, Vier, Fünf. (…)

Sorry, this entry is only available in German.

Hans Bonfigt
Monday August 7th, 2017

(Deutsch) Hans im Glück, Version 2017

Sorry, this entry is only available in German.

Roland Dürre
Saturday April 22nd, 2017

Solution to Problem 2

On April, 12th, I published a second Logelei in the IF

The question was:
Can you position 31 domino stones (each of which is as big as two tiles) in a square area that consists of 64 square tiles such that two opposing corners remain free?

Solution:
The task is easy as soon as we (mentally) colour the area that consists of 64 tiles like a chess board (Auszeit).
Then we know that one domino stone – as big as two fields (tiles) – will always cover a black field and a white field.

But we will also instantly see that the two opposing fields have identical colours.

So what does this tell us?
If you try to solve this problem by trial and error, you quickly run the danger of making a mistake. However, if you use a simple construct to help you (such as colouring the field like a chess board), the solution will immediately be clear.

You only have to have the right idea!

RMD
(Translated by EG)

P.S.
I am taking time out (Auszeit). This article has been written in advance and will be automatically published (two weeks after the problem was formulated).

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
Monday April 3rd, 2017

A Very Special Task!

The Solution will be Supplied Later!

A short time ago, a good friend of mine came up with a brainteaser. He did not know the source, otherwise I would gladly have cited it. My friend was not able to solve the problem, neither was I. But it is a truly exciting scenario. And it has a surprisingly simple solution, including a beautiful mathematical reasoning. It also gives us a nice metaphor for our lives.

Among other things, it shows that mathematics can also, once in a while, be quite useful. Here is the story:

Here is a female criminal. On her card, the number 1 is written. But she does not know this. After all, she only knows the nine other images with their numbers.

A – not dislikeable – gang of 10 persons constantly violates the prevailing moral concepts in an outrageous manner. The gang members are creative and wise – this is how, with great finesse, they remain unmolested by the arm of the law for their abominable activities. That is lucky for them, because the legal penalty for their crime is death by strangulation.

In the public perception, the gang soon has a legendary reputation, and is idolized by quite a few simple people. For the authorities, this development is totally unacceptable. Consequently, the increased manhunt of the authorities, along with a growing arrogance and flippancy among the gang members led to the capture of the group.

All 10 gang members are quickly sentenced to death due to their abominable behaviour in a show trial. However, there is a way for the ten comrades in crime to save their lives – through an appeal for clemency. The head of state who decides upon said appeal is a very prudent and well-meaning woman. She is very wise; there are even some rumours insinuating that she may to some extent sympathise with the gang.

Actually, she works hard to come to a fair decision. She hands down a conditional amnesty (a little like a “Judgement of God”):
Before the verdict is executed, the ten members are permitted to see each other once more. There is a farewell meeting, the ten gangsters can spend the afternoon before their execution together and without supervision.

Here is a male criminal. On his card, the number 2 is written. But he does not know that. After all, he only knows the nine other images with their numbers.

As the meeting starts, the gangsters are told how the amnesty will work. A picture of each of the members is taken (two of them can be seen here). On each of those pictures, a number from the set 0 – 9 {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is drawn. Each number can be used several times. Consequently, it is possible that the same number is written on all the pictures. Or that only some numbers are used, for instance {1, 2, 3)}. Or maybe only the even or uneven numbers. Whatever. But perhaps all numbers have been used. Nothing is impossible.

After the meeting, each of them is taken into solitary confinement until the time of execution. Each of the ten gang members is shown the nine pictures of the other nine members – but not his own one. And then they ask him the number on his own photo. And if even one of the gangsters gives a correct answer for the number on his card – all of them will get the amnesty.

Initially, you will think that the gangsters have quite a good chance to avoid their punishment and enjoy clemency over justice. And there is no doubt that their situation will have improved. After all, chances are not too bad that one of the ten will guess correctly and thus free them all.

But it is nowhere near as easy as that. Matters may turn out poorly. And there is one thing the wise regent forgot (or perhaps not): by applying a simple agreement, the ten gangsters can make sure that one of them will inevitably say the right number, as written on his picture. And this is how he can guarantee that he and his comrades will enjoy the amnesty.

It is a small problem: what agreement makes it possible for the gang to use the meeting that was meant as a farewell to make sure that “their heads” are out of the sling with a 100% chance?

I will publish the solution in a few weeks – and until then, I look forward to having many email solutions sent to me!

RMD
(Translated by EG)

Roland Dürre
Sunday May 3rd, 2015

Numbers & Taxes & Calculations.

On April, 23rd, 2015, the Federal Ministry of Finances (Bundesfinanzministerium) published information on its website about the entire tax revenue (total tax revenue of the country and states without those of the communities per month) having increased by 4.7 % over the same month of last year in March 2015 to now 57.970 billion Euros. Wow. Doesn’t that sound nice! After a balanced federal budget. Because the economy booms. Consequently, we have a high turnover and increased tax revenue. What a brave new world.

However, it is not as easy as it sounds. There is one adjective missing: “nominally”. Because this would clarify that the number is not such a great number, after all.

Here are a few ideas of mine.

In “West Germany”, the retirement money will be increased by 2.1 per cent in 2015, in “East Germany” by 2.5 per cent. The Federal Government decided late in April that this is going to happen on July, 1st, 2015.

The annual increase of retirement money basically takes the development of the total income into account, separately for western and eastern counties. Let me cite fromRenten-Recht.org:

The data relevant for the increase in retirement money have been processed in the spring of 2015. Now we can give the precise number for the retirement money increase. Due to a statistical once-only effect, however, the retirement money increase of 2015 will be one per cent lower than without this effect. We are talking EU regulations requiring a revision of the workforce statistics. For instance, certain groups of low-income employees have to be included in the statistics, which has a negative influence on the central incomes the retirement money increase rate is based on.

Now that, too, is a topic – just because of a new EU regulation, the retirees get less than they would actually be entitled to. But this is not what this article is about. I conclude that the extra tax revenues result from higher incomes (around 3%) and the additional cold progression.

I also read:

Consequently, the increase in retirement money in 2015 is higher than has been predicted by the German Retirement Money Insurance Association. Neither is there an inflation influence, since the last inflation rate was minus 0.1 per cent.

This is also something that makes me thoughtful. Does that mean that the poor predictions actually make the incorrect increase more legitimate? And how is it possible that persons who have to rely on their retirement money are worse and worse off, regardless of life getting cheaper and cheaper and retirement money increasing? No. In fact, it is not correct. In my own environment, I regularly perceive considerable price increases. For good food on the market, at the (real) bakery and butchery, having to pay craftsmen and doctors, for public transportation, for everyday products such as good bikes. Or for real estate and rent. Everything gets more expensive. Beer, ice-cream and pizza. The increase is actually surprisingly high. And with those (not just) felt inflation rates, those 4.7 % more tax revenues are basically no longer such a great thing.

Only junk gets less expensive. But then, every child knows that cheap junk will eventually cost you more. And fuel (temporarily) got cheaper. But then, who needs fuel if they have no money, anyway? Only the well-to-do (I count myself among them) burn a few hundred litres of fuel each month with their luxury limousines (which is not something I do). The savings might then be spent on a luxury item. For instance a beautiful scarf – which today might well be 370.- Euros at Loden-Frey. I saw it last week. And since even the cheap scarves were all around 200,- Euros, as well, I exited the shop empty-handed. Because the prices for luxurious items actually explode – as a side-effect of which again more tax revenues are earned by the Federal Office. Except that if more luxurious items are sold, this is not an indicator that people in the country are generally better off.

For instance the increase in tax revenues for added value tax and other taxes such as real estate transfer tax benefit from an actually happening inflation, even if it is not statistically visible. The shopping cart developed for statistical reasons is a lie.

Regardless, one should rejoice at hearing that the revenues have increased. Except that you should not forget the opposite side: the money spent. And that basically increases to a far higher extent than the aforementioned 4.7 per cent. Just think of the major components of the Federal Budget such as Social Welfare, for instance pensions for former public servants. Or the almost normal cost increases when it comes to public infra-structure projects. For instance when we consider the construction (Bau) of the Ismaning motorway junction (see also my IF Blog article).

And after reading an article on this where a District Administrator said that “a general increase by 15 per cent for these kinds of projects is rather normal”, I think this is realistic.

Consequently, I am afraid the story they tell us about the balanced federal budget, too, is just a lie. Because we are lucky to have these low interest rates. You know, matters might change quite quickly. No reserves are kept for actually threatening losses. Everybody seems to accept securities that lie in the future for loans, knowing full well that there will be a time when you have to pay them back. It will probably happen quite soon that massive cost increases can only be met by strict economizing, which many parties concerned will have to pay for. Those proudly proclaimed 4.7 per cent will not be any help and presumably the low-wage earners will again be the ones who have to pay the price. Their savings are already being downgraded by the zero-interest policy, anyway.

But let me return to taxes and added value tax. Why don’t you try a hilarious experiment and ask (not just) young persons in this country how high the added value tax rate is? You will be surprised to hear the answers. “I have no idea” is among the harmless ones. Here are the currently important numbers on taxes (from Wikipedia): 
Since January, 1st, 2007, the normal rate is 19 per cent and the reduced rate of 7 per cent has been active since July, 1st, 1983.[2]

And if you wish to be even more surprised, then ask all those people – after having informed them about the correct rate – how much of the 370 Euros you would have to pay for the scarf are added value tax. Many will not know. Consequently, you will get quite humorous replies, but not very often the right answer (that you have to divide the 370 Euros by 119 and then multiply the result with 19). And then people look at their mini calculators and see the result with an air of amazement.

Incidentally, I am quite surprised that they did not make 20% the normal added value rate. The calculation would be so easy: you take 1/6 of the total price and add 1/5 to the net price. But then, who knows the difference between “net”, “gross” and “tax weight” today? And who knows in-hundred, of-hundred, or on-hundred calculation?

To make up for the one extra per cent, one might have lowered the income tax a little. Or at least, they might have coordinated the progression intervals with the inflation rate.

And if you want to further annoy your partner, you can ask him about duty on spirits, duty on energy, real estate transfer tax, coffee tax, tobacco tax, (… Branntweinsteuer, Energiesteuer, Grunderwerbssteuer, Kaffeesteuer, Tabaksteuer …) …

RMD
(Translated by EG)