In this chapter we enumerate the main attributes and rules of the Poly-Universe Game Family from mathematical approach. The selected inspection method is the combinatorics, since this branch of the discrete mathematics is suitable for discovering the number of possibilities inherent in the product family in the most comprehensible and effective way.
This chapter is meant for every enquirer who wants to contemplate knowingly the mathematical and logical correlations of the product family. All considerations described here are plausible, but the number of the join possibilities explained below is many times surprising. The present chapter is not intended to examine the Poly-Universe Game Family closed and from every direction. This goal is hindered by the limits of its extension and by the high number of approaches. At the same time we undertake to feature the fundamental correlations of the game family we find important.
The foundations – of what do we build?
The colours of the product family are: red, yellow, green, blue.
The basic shapes of the product family are: triangle, circle, square. In fact, the game is built of these three “families”, and all of the three basic shape families have 24 members, as we will see it in the next chapters. Beyond the above forms and colours the game is founded on scale-shifting symmetry. The main point of this principle is that bisecting the length of the sides (or the diameter of the circle) we get smaller and smaller shapes of different colour inside the basic shapes.
The inventor developed the packages containing the game elements by using the above formal and colour building elements so that all of the elements are different. Therefore the number of the elements in each package presents itself through the simple permutation without repetition of the four colours in all of the three cases, i.e. each package contains 4 * 3 * 2 * 1= 4! = 24 elements.
The rules – how do we build?
Basic position, inverse (mirrored) position
This is an aesthetic and technological consideration, but it has some mathematical consequences too. The orientation changes at the axial mirroring, thus we get a new element regarding the set up opportunities when we mirror, or else, when we turn round an element.
Entire side joins of triangles
We talk about entire side joins when we join triangles so that the corners touch each other, namely the starting and the end point of the two joined edges concur. A triangle can be placed in six different ways. A second triangle can also be set down in 6 different ways, so 6 * 6 = 36 different possibilities turn up by the arrangement of two triangles with entire side joins.
We get the same result, if we examine that two triangles upturned with the given faces can be set in nine different ways related to each other, since we can match the three sides of the first triangle with three different sides of the other triangle. Nevertheless, both triangles can be placed in two ways (basic and inverse position – see above) on the table.
Of course, we get the above result in this way too: 9 * 2 * 2 = 36.
Returning to the original train of thought, let us proceed that one triangle can be set down in six ways on the tabletop. From that follows, for example, that by putting down all of the 24 triangles we can get 624 = 4,74 * 1018 different configurations without changing the place of any triangle within the configuration. As long as we examine how the subsequent triangles relate to each other, the next rule is true: (n+1)! * 3!n-1, where n is the number of the triangles.
Adopting the formula for two triangles, we definitely get the 36 possibilities back already calculated in two ways earlier. But in case of three triangles we get 864 different set-ups in that way. If we use all of the 24 triangles, 1,2 * 1043 different possibilities would already turn up. (It has to be mentioned that at this time we have not calculated with closed configuration possibilities that can reduce the number of solutions.)
Slidings of the triangles
Let us lay down the rule that we only permit the sliding of the triangles’ sides along the so-called stable linkage points. It means that the linkage points have to contact the corners or each other, or rather the contact of only the corners is not allowed – because of the instability of the configuration’s angle.
Nevertheless, the possibilities of the slidings are a bit more complicated than it would seem at first sight, since the number of the possibilities depends on the sides to join. The triangles can be slid along the different sides as shown in the above table.
(“R” means the short-, “K” means the middle- and “H” means the long sided internal triangle). This is a symmetric matrix by the reason of the axial symmetry of the reverse elements.
One of the maximums explained on an illustration, for example the 12 sliding possibilities of the H-R – K-R sides:
One of the minimums explained on on an illustration, for example the 7 sliding possibilities of the H-R – R-H sides:
These slidings increase the above calculated big number of the different configuration possibilities remarkably.
Entire circle joins
We talk about entire circle joins when circles of the same size are connected with each other. In this way two circles can be set in four ways, and this is true for all of the three different sized semicircles. This results in 3 * 4 = 12 ways of placement of entire circle joins related to each other.
If we set a circle on the table, the second circle can be set next to it in six different ways with entire circle join, since all of the three sized semicircles are free, and the next circle can be set in two different ways on all of the three places. If we go on combining so that the next circle will be joined to the circle just put down , we only have two free semicircles, and the subsequent circle can be joined to them in two ways. That means that the set-up sequence can always be continued in four ways starting from the third circle. That also means that we get 6 * 422 = 1014 possibilities by placing the 24 elements set in this way – i.e. as it were in one row.
Let us make two comments on this result. On the one hand, if we compare this – although still a remarkably great number – with the result got for the triangle in the previous chapter, we can see that it is by four orders of magnitude smaller. Apparently, the reason is that while in case of triangles an optional side can be joined to an optional side, here we only allowed entire circle joins. In the next chapter we allow special slidings of the circles. On the other hand, we note that by this quasi-linear allocation we have not calculated with the formation of some closed configuration. First it can happen when connecting six circles, and to some extent, it decreases the number of possibilities.
Partial circle joins
By partial circle join we mean when a larger circle is joined to a smaller circle so that the smaller circle is inside the larger one. Practically speaking, this opportunity is the sliding of the circles, similarly to the slidings of triangles and squares along the stable side linkage points, and in this way it increases the solution possibilities significantly. By this method it is possible to join two circles in 15 ways, and this number is multiplied by four because of the difference between the basic and inverse positions. So we have 60 different possibilities by this time, i.e. we can set up so many different configurations in case of two circles with partial join.
Entire side joins of squares
Two squares can be set in 64 ways next to each other. This can be easily seen with the extension of the logic used by the triangle so that we consider the different number of the sides, since a square can be put on the table in eight ways. It follows that in case of 24 squares we have 824 = 4,72 * 1021 placing opportunities if we do not examine the location of each element in the configuration, plus if we set aside the closed arrangements.
Slidings of squares
The slidings of the squares are analogous to the slidings of the triangles along the stable side linkage points. The difference is that the number of the sliding possibilities varies between 4 and 9. Stuck by the previous example we set two squares next to each other, and even if we calculate just with the minimal sliding possibilities by the side-couples, we get 64 * 4 = 256 different configurations.
Objectives and opportunities – what do we build?
Let us analyse the closed forms among the many opportunities.
Closed configuration from 6 elements by the triangle
First of all we select 6 optional elements from the 24 pieces set for the construction of closed configuration. This means the six combination of the 24 elements, and can be calculated as follows: C624 = 134596. After the selection we have to consider in how many different ways we can arrange the next triangle according to the already placed triangles in the course of our attempt to get a closed configuration. (The numbers under the factors below mean the ordinal number of the placed triangle.)
so we get a different configuration of an order of magnitude of 100 billion.
Closed configuration from 6 elements by the circle
The formula above alters in case of circles in such a way that instead of the factor 66 that represents the arrangement possibilities of each element we have to consider 26 or106 possibilities depending whether we allow entire circle joins exclusively or partial circle joins as well. In the latter case the above product increases the order of magnitude by one, whereby we step into the range of 1012 (trillion).
Closed configuration from 4 elements by the square
In case of four squares we consider the square arrangement next to each other as closed configuration. If we require only the entire side join, then the number of the different possibilities can be given by the following formula: C424 * 4! * 84 = 10626 * 24 * 4096 = 109
This means the order of magnitude is the billion.
As far as we would like to assemble from the four squares for example a configuration with a cross-shaped hole in the middle, the 84 factor changes to 24, since a square can only be put down in two ways so that the cut-off corner falls in the middle. According to this, the number of the differing possibilities reduces the order of magnitude to a million ( ~ 4 * 106 ), which is – despite the constraints – still a considerable number.
Based upon a further idea of the inventor, packaging happens as follows. Each element of the 24-pcs packages will be placed randomly above each other, and will be covered with trasparent foil. This method means the following numbers of different packages concerning each shape:
Triangle: 624 * 24! = 2,9 * 1043
Explanation: a triangle can be set in 6 ways, so 24 triangles above each other 624 ways. The different sequences of putting the 24 different elements on each other have also to be counted, which means the permutation without repetition of the 24 elements, that is 24(!).
Square: 824 * 24! = 2,9 * 1045
Explanation: the difference from the triangle is that a square can be set in 8 ways. The other considerations are equal to those written about the triangle.
Circle: 24! = 6,2 * 1023
Explanation: in case of the circle we only allow the exact arrangements, which means we count neither with the axial rotation nor with the reverse – mirrored – position. So the formula describing the number of the different columns reduces to the simple permutation without repetition.
Humanity consists of nearly seven billion, i.e. 7 * 109 men at present. One can see that the number of the different packages producible even from the circles has an order of magnitude that exceeds the human population by16(!).
On the basis of the above mentioned we can see that despite the simple set-up of the game family – three forms and four colours – the number of the combination possibilities is extremely high. It is very unlikely for two game users to set up the same configuration from the elements accidentally. In the course of building with the game we have always stayed within a given form in the previous chapters, but the form and the size of each shape allows to combine them. This opportunity results from the main artwork representing the Poly-Universe Game Family, since it can be seen in this first picture that the triangle, the square and the circle can be matched. Considering these facts, we can easily see that merely the closed description of the combination possibilities is an extremely complex task.
We note furthermore, that – although we took into account the different solution possibilities in this part – in the course of the playing people usually refer patterns fitting certain rules as the primary object. This is not a constraint dictated by the creator, but a typical human approach to the game. One of this rules is to aspire to a closed configuration analysed deeper in the previous chapters. It is easy to comprehend, that though the set of this ad hoc rules is finite, but it is not possible to define them closed.
Gábor Kis physicist