The game Inverted Dice™

Finally, I have finished my work on this: www.inverteddice.com.
The worlds first inverted dice game is now available online.
If you have read my previous post, The Principle of Inverted Dice™, you know what this is about.

Try the game: www.inverteddice.com. This game was previously called Three Hundred and Sixty™, but some wise people told me to change the name, and so I did.

Play Inverted Dice™ online for free!

The answer is 47

I have been working with a mathematical proof for a whole day … just to finally give up. The technical details are quite abstract, so I’ve created a PDF file with all the information you need (see below).

I have “discovered” a new arithmetic function which has a funny property. I just got the sequence for this function approved on OEIS (sequence A224914). But no one has proved the conjecture, so far. It’s quite thrilling!

Are you a math student or an expert on number theory? In that case, you might want to take a look at this:

The answer is 47 (open PDF in a new window)

Can you prove this new conjecture?

The Principle of Inverted Dice™

Addition (April 2014):
The game Inverted Dice™ has been released online, please visit www.inverteddice.com.
Rules for Inverted Dice™ are available HERE.
Included in the rules are some examples of inverted dice which are not included in the text below.

This text concerns standard 6-sided dice. One might think of various extensions of the concept that follows and then deal with other types of dice. But for the moment, we restrict ourselves to normal dice with 6 faces, numbered from 1 to 6.

When rolling two dice, it is common (in hundreds of different games) to look at the sum of the numbers shown on the upper surfaces. As you know, there are 11 possible results (2 to 12) from a roll with 2 dice. My idea is to look at the sum of the numbers not shown on any dice. In that way we get 11 different possible results (namely the numbers from 10 to 20, i.e. with two dice).

I have chosen the name inverted dice for this way of interpreting a roll of dice.
Some examples are required.

Playing with 1 inverted die
This case is the least interesting, but it can without doubt be useful:
Playing with 1 inverted die
When playing with 1 inverted die, you can achieve any result from 15 to 20 (that is 6 possible results).
You have a chance of 1/6 for each of these results, just as you have in a roll the traditional way.

This can be used in several new (not yet invented) types of games, or in modifications of existing games.
For instance, you might imagine a game in which you move your token after rolling a die, and sometimes (in specific situations) you get to move after an inverted die roll (which is a significantly larger number).
Maybe even games like Ludo might be made stimulating in the future.

Playing with 2 inverted dice
Now it gets much more interesting. That is because we find a radical change of the probabilities.
But first, let’s see some examples:
Playing with 2 inverted dice
When playing with 2 inverted dice, you can achieve any result from 10 to 20 (that is 11 possible results).

We realise that there is only one way of getting the result 20, and that is when both the dice show the number 1. However, we also see that for instance the result 18 can be achieved in several different ways.
It is time to make a complete table to get an overview. We have 36 possible outcomes to look at:
Possible outcomes with 2 dice
We can now without difficulty create two tables showing the odds of all possible results, in the case of both two normal dice and two inverted dice:
Probabilities of dice rolls (2 dice)
One interesting characteristic of two inverted dice are that the two highest numbers are also those most difficult to achieve. This is usually not the case, since it’s normally just as hard to roll 2 as 12 with two dice.
I can think of several uses for this. For example, try playing Monopoly with inverted dice instead of regular dice (or change the rules and add the possibility of a player having to use inverted dice in certain situations). Feel free to invent new games of your own.

Playing with 3 inverted dice
Here it gets more interesting. This is caused by the fact that with 3 inverted dice you don’t have the same number of possible results as with 3 normal dice.

With 3 normal dice, the sum is always between 3 and 18 (i.e. 16 possible results), but with 3 inverted dice, the sum is always between 6 and 20 (i.e. only 15 possible results).

That this is the case, we can easily deduct, since we get the lowest result when only the three largest dice values, being 4, 5 and 6, are shown. The sum is then 1+2+3 = 6. In the same way we see, that we get the largest result when only the number 1 is shown, i.e. 2+3+4+5+6 = 20.

We could create a table showing all possibilities of achieving the results between 6 and 20 using three inverted dice, but to do so would be a rather tedious task.

As an example, I can demonstrate that the result 11 can be achieved with the following combinations:
Playing with 3 inverted dice
I’ll leave it to my readers to calculate all the odds for dice rolls with 3 inverted dice. This can be done manually with the help a table similar to Table 1, but this is quite a lot of work (some programming skills would be useful). There are 6 times 6 times 6 possible outcomes, so the table consists of 216 rows.

Playing with 4 inverted dice
Normally you will have 21 different possible results (from 4 to 24) from a roll using 4 dice, but with an inverted roll we will only have 18 possible results (from 3 to 20). I shall leave to my reader to look at this case (and come up with new games).

Playing with 5 inverted dice
This case is extremely interesting. Firstly, there are exactly 20 possible results from a roll using 5 inverted dice, and the results are between 1 and 20. In other words:

Five 6-sided dice can together form one 20-sided die
(with a very peculiar probability distribution).

Secondly, 5 is also the exact number of dice used in a regular game of Yahtzee. I have attempted to develop a kind Inverted Yahtzee, but since Yahtzee is not based on sums, but rather patterns and numbers, it is hard to invert this game.

Following are a few examples of rolls using 5 inverted dice (keep in mind that the result is the sum of the numbers not being shown):
Playing with 5 inverted dice
When calculating the result of an inverted dice roll, it can be useful knowing that the sum of the numbers from 1 to 6 is 21. For instance, if you only rolled threes and fives, you know that the inverted sum is 13, since 3 plus 5 equals 8, and 21 minus 8 equals 13. It is sometimes easier to count that way.

Playing with 6 inverted dice
This case is also interesting. It is the first case which has 21 different possible results and where it’s also possible to achieve the result zero. I will leave all details for the reader. Perhaps some student of mathematics will like to delve further into this.

Playing with many inverted dice
Before finishing this text, I would like to present a small table showing the number of possible results of rolls with n dice, both regular and inverted.
Possible results for dice rolls

Happening against greed

Here’s a suggestion for a political happening. The aim is to bring focus to what I consider to be one of the greatest threats to the continued existence of our civilisation (to be explained): greed.

I cannot take credit for this idea since I have a faint recollection that someone may have mentioned it to me at a meeting ten years ago. Furthermore, I believe that it has already been implemented in various cities, in different countries.

It’s a fun idea, which does not cost much to carry out. Hopefully, it will open people’s eyes, make some people ashamed, bring joy to others, and finally give those performing the happening a new insight into certain sociological and psychological mechanisms.

Here’s how to do it:

On a pedestrian street, a wide sidewalk or any other kind of passageway with a large flow of walking people stands a person with her pockets full of one pound coins (or one dollar bills, single euro coins, or another appropriate currency). This person has a single task: to give away as many coins as possible to the people walking by.

− Hello, here’s a pound from me. Have a nice day.

Fifty metres further along is another person, who has a single task: to get as many one pound coins as possible from people walking by.

− Excuse me, could spare a pound, please?

What happens is exciting. The pedestrians who first meet the giver would possibly have a tendency to more easily give away a coin shortly after. They become motivated − and a little amused by the world’s random encounters. But those who walk in the opposite direction, and thus meet the beggar first, will hopefully get a little ashamed when they shortly after receive money themselves (well, a few may not need to be ashamed as they actually did give some money to the beggar).

If necessary, there can be two or more givers and as many beggars to better reach out to all the pedestrians. Of course there will be people who only meet one of the “activists” (the beggar or the giver), but that is not important. What’s interesting here, are the two categories mentioned above. How much money will the beggar get? Well, it’s hard to say. But at least, some of the pedestrians will find themselves in a situation where they will feel joy if they give away a pound or suffer from bad conscience or embarrassment if they do not. And that is quite unique in today’s late capitalist society.

Goodbye

I believe this melody is one of the best I have ever written. Hopefully, I will record it in a studio with a band soon, and publish it here. But for now, all you get are two computer-generated samples (from MIDI files).
Play this tune on your own instrument. Use it! (But don’t forget to mention the source).

Click here to listen to the chords and the bass line!
Click here to listen to the melody!
Here comes the sheet music (click on the pictures to open larger versions for printing).

Goodbye (page 1)

… and page 2 …

Goodbye (page 2)

Song of Peace

Last year, my father wrote a text to this one. Hopefully, we will someday present both the music and the English text here (just need to visit the studio first), but for now, you are welcome to listen to the computer-generated version below.

Click here to listen! (MIDI-verion)

Click here to read the text (in Danish)!

Here comes the sheet music (click on the picture to open a larger version).

Song of Peace

Senses

This melody came to me a couple of months ago. When I started to write it down, I discovered that it is in 31/8 which is quite odd. The tune is nice when you play it on the flute.

The guitarist Mats Götherskjöld (from Advokatbyrån Götherskjöld & Jensen) created some wonderful chords to the tune.

But for now, all you get is a fast version that we recorded in the studio …

… and a computer-generated version (i.e. from a MIDI file), with a “piano” instead of the flute.

Click here to listen to the MIDI file!

Click here to see a live version (rather bad sound quality)!

Here comes the sheet music (click on the picture to open a large version which is easier to read).

Good exercise for all flute students!

Senses

Something with Cheese

On the album All You Can Eat from 2005 you’ll find the tune Something with Cheese.
Peter Nilsson made the bass-line and I wrote the melody. The thought was to make it sound like one of those spy-car chase-detective-movies (à la Mancini) from the 1960s. We suceeded, I think:

Here comes the sheet music (click on the picture to open a larger version):

Something with Cheese

Blue Glass

I remember clearly how I, at the age of fifteen, was hit by the force of the movie Koyaanisqatsi by Godfrey Reggio. This movie is the first in the so called Qatsi trilogy: Koyaanisqatsi: Life out of balance (1983), Powaqqatsi: Life in transformation (1988) and Naqoyqatsi: Life as war (2002). Read more at www.koyaanisqatsi.org. Everyone should see these movies!

Koyaanisqatsi was also my first experience of the composer Philip Glass, since the music is playing a vital role in the movie. If you do not know this brilliant minimalist, please do something … start with his masterpiece Solo Piano from 1989. Visit www.philipglass.com for more information.

OK, that was a couple of links for you.

The music of Philip Glass means a lot to me. So five years ago I named a tune after him. It was recorded in 2004. Later it was recorded in an updated version and released on the album All You Can Eat in 2005. Click here to listen to the first version from 2004.

Here comes the sheet music (click on the picture to open a larger version):

Blue Glass

Baghdad 2003

In 2003, when USA initiated the bombings of Baghdad, I wrote a tune in frustration because war once again conquered diplomacy.

The tune was recorded (click here to listen to the first version) and later it was included on an album in an extended free version (due to copyright issues, I can’t make that version available to you). Finally, a third version with Carrilho Jensen Nyberg was recorded in 2009:

Anyway, here comes the sheet music (click on the picture to open a large version which is easier to read):

Baghdad 2003

Definitions and axioms – and inconsistency

I have previously studied mathematics, for a period of one and a half year, and I have walked the first steps along the road. In the mathematical conventions, I discovered, from time to time, a lack of consistency and indistinct logical reasoning which seem to have passed by the professors unnoticed, or not? Here comes an example.

Sometimes, when we want to make a definition by the use of a number of axioms, a strange kind of semi-inconsistency arrises.

Let us assume that we want to define the concept C by using a description D (which can refer to one or more previous definitions) and a requirement that D satisfies some axioms (let us name them A1, A2 and A3). Such a definition would most often be stated like this:

Definition: C is a D that satisfies the following axioms:
i) A1
ii) A2
iii) A3

By this we mean that all the axioms (in this case three of them) must be satisfied by D.

But then we could as well formulate the definition this way:

Definition: C is a D that satisfies the one following axiom: A1 ⋀ A2 ⋀ A3

which can be written in several different ways, for instance this:

Definition: C is a D that satisfies the following axiom: A3 ⋀ A2 ⋀ A1

and thus we see that the original definition could as well be written this way:

Definition: C is a D that satisfies the three following axioms:
i) A3
ii) A2
iii) A1

In short, if we want to define something by using a list of axioms (strictly speaking an unordered list), then the order of those axioms should not have any influence on the definition. This means we need to express all the included axioms in such a way that they are independent of one another. But this is not always the case with some of the available mathematical definitions, see the example below.

In several books discussing fundamental abstract algebra, the following definition of a group is given:
Normal definition of group
In this definition Axiom ii) must be presented before Axiom iii), because otherwise the symbol e (the neutral element) has not been described when it is used in Axiom iii).

You could, in principle, imagine very complicated definitions that contain a list with hundreds of axioms. Then it is easy to imagine that these hundreds of axioms depend on each other in a complex way.

I suggest a new convention: In all definitions that use a list of axioms, we should always formulate the listed axioms in such a way that they do not depend on each other.

As an example, I rewrite the group definition mentioned above:
New definition of group
This new definition is better, since the three group axioms now “have the same status” (i.e. they are mutually independent) – and the order in which we list the axioms is unimportant here. We would have to define the term neutral element separately, but that is no problem (actually, it is better).

Points

I will, from time to time, publish some new sheet music on Blog Off. Here comes a tune I wrote at the grand piano I borrowed a couple of hours while visiting a school in 2006.

Originally I thought of it as film music. Then it would probably sound like this (click to download an mp3 file which has been extracted from my notation software – it is my computer “playing” here).

One year later it transformed into a latin/salsa-inspired thing with my former band. It became wild, download here.

And here comes the sheet music (click on the picture to open a large version for printing):

Points

Jensen’s Paradox

Everyone who is interested in mathematics, physics or common philosophy knows the most famous of Zeno’s paradoxes, namely the one with Achilles and the tortoise. Many similar paradoxes exist, and when you understand the first one it is easy to understand them all.

But one day when I was playing with my thoughts I found a “paradox” that actually requires some extra thinking. We can call it Jensen’s Paradox (don’t mix it up with Jensen’s inequality or The Jensen’s Inequality “Paradox”, a paper on economics by Susan Woodward).

Here it comes:

Consider the figure below which contains the points A, B and C:
The points A, B and C
If we want to move ourselves from the point A to the point C we can choose the following way: First we go in a horizontal line to the point B and then we continue in a vertical line upwards to the point C. If the distance is 1 between A and B and the distance is 1 between B and C, then the total distance (let us call it S) will be 1+1, which is 2, when we choose to go this way.

Now, consider the next figure:
Horizontal half the way to B, and then vertical half the way to C
Here we move ourselves horizontal half the way to B, and then vertical half the way to C. We repeat these two steps once and arrive in the point C. The distance S becomes 2 again.

We develop this procedure a bit further:
We have split every step into two new steps
We have now split every “step” from the figure above into two new steps. The distance S is still 2.

We could continue in this way for some time. Below we have 8 steps:
The distance S is still 2

And now 16 steps:
The distance S is still 2

And 32 steps:
The distance S is still 2

One more time. Here we have 64 steps:
The distance S is still 2
The distance between A and C have now been split into 128 small distances (64 horizontal and 64 vertical), and it is easy to see that the complete distance, S, is still 2. With a little mathematics this can be written in the following way:
Sum of 64 steps
Let us now imagine that we split the distance into many horizontal and vertical parts (let us assume we then have n steps). Then it looks like this:
Sum of many steps
The sum of all the parts (the small vertical and horizontal distances) will always be 2, i.e. the total distance is always 2, even if we split the distance into an infinitely large number of steps:
Sum of infitely many steps
We conclude that the distance below equals 2 (since the limit above is 2):
Infinitely many steps
But according to Pythagoras theorem the hypotenuse of the triangle below is equal to the square root of 2:
Pythagoras theorem used here
So here is the paradox: We have now seen that the distance S between the point A and the point C is always 2, and we do know, simultaneously, that it is always the square root of 2 (according to Pythagoras). Hope you like it!

A logical problem

I heard this one many years ago. I have rewritten it, to make it harder and to avoid that my dear readers search the Internet for the answer instead of thinking themselves.

A mathematician, Professor Anderson, visited her colleague Dr. Smith. During the visit Anderson asked: “How old are your three kids actually?”
Dr. Smith thought for a moment, and then he answered: “None of them have the same age, the sum of their ages is less than eleven and the product of their ages is a number you know well, namely the street number of my house”.
This made Professor Anderson think, but after a short moment she said: “It can’t be solved, since we are missing…”
Dr. Smith interrupted, almost laughing: “Sure, we are…” and then he gave Professor Anderson an extra piece of information.
Now they both knew the exact ages of Dr. Smith’s children.

What is the street number of Dr. Smith’s house?

That Blues

We have to rewind history far back if we want to reach the epoch when the music genre blues was considered to be rebellious, dangerous, a threat against society and existing norms and, above all, sinful. Today, we constantly use those blue notes that constituted a threat back then. We use them in hundreds of old and new genres. So our ears and senses are used to the tritone nowadays.

But I continue to play blues, or at least bluesy. Why? Because it’s got something. Still.

Five years ago I did not exactly know what I was looking for in the blues. But then it happened: The blues needs an odd rythm instead of the old common 4/4. Then it becomes real blue.

So why do I write about this now? Partly because I recently read that Freddie Hubbard died five months ago – he is worth remembering – and partly because I have started a new project where the goal is to digitalize all my own old (and new) notes and present free sheet music here on Blog Off.

Freddie Hubbard released the album Blue Spirits in 1965. The first tune Soul Surge is an instrumental soul-blues with 7 beats in each measure. Brilliant!

So the thought that blues can be combined with odd measures is not a new one. But it is a bit forgotten…

I released an album in 2005 with the tune That Blues, a blues in 7/4. Let us dedicate this tune to Freddie!

And here comes the sheet music (click on the picture to open a large version for printing):

That Blues

An exercise in C++

Learning by teaching
(Jean-Pol Martin)

Before I began teaching C++ I was convinced that I had a good command of C++ and that I knew all the theoretical aspects of this extensive programming language.

But unforeseen questions from my bright students woke me up.

Today, hundreds of lectures and lessons later, I dare say that I know C++. Thanks to the fact that I was learning by teaching!

So let me begin this category with a theoretical exercise for all professional programmers:

Consider the following expression in C++:

(1==0)[“0==1″]

Which value has this expression?
What is the data type of the expression?
Why?

A small exercise

When calculations are performed, people often use several arithmetic and algebraic rules without thinking about how these rules really work. The following example illustrates this fact:


So: 1 equals -1.
Apparently at least one of the equal signs above must be wrong. But which one? Or which ones?

I have noticed that many high school teachers find this problem difficult to solve. And that says more about the teachers than the problem!