Michel Planat
email: michel.planat@femto-st.fr

A following up on the FQXi Essay Contest -It from Bit or Bit from

In the standard formulation of quantum mechanics, there exists an inherent feedback of the measurement setting on the elementary object under scrutiny. Thus one cannot assume that an 'element of reality' prexists to the measurement and, it is even more intriguing that unperformed/counterfactual observables enter the game. This is called quantum contextuality. Simple finite projective geometries are a good way to picture the commutation relations of quantum observables entering the context, at least for systems with two or three parties. In the essay, it is further discovered a mathematical mechanism for 'drawing' the contexts. The so-called 'dessins d'enfants' of the celebrated mathematician Alexandre Grothendieck feature group, graph, topological, geometric and algebraic properties of the quantum contexts that would otherwise have been 'hidden' in the apparent randomness of measurement outcomes.

Introduction

The motivation for being interested in the topic of quantum contextuality dates back the celebrated Bohr-Einstein dialogue about the fundamental nature of quantum reality. The first sentence of the Einstein-Podolski-Rosen (EPR) paper [START_REF] Einstein | Can quantum-mechanical description of physical reality be considered complete?[END_REF] is as follows If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to that physical quantity. and the last sentence in Bohr's [START_REF] Bohr | Quantum mechanics and physical reality[END_REF] reply is I should like to point out, however, that the named criterion contains an essential ambiguity when it is applied to problems of quantum mechanics. It is true that in the measurements under consideration any direct mechanical interaction of the system and the measuring agencies is excluded, but a closer examination reveals that the procedure of measurements has an essential influence on the conditions on which the very definition of the physical quantities in question rests. Since these conditions must be considered as an inherent element of any phenomenon to which the term "physical reality" can be unambiguously applied, the conclusion of the above mentioned authors would not appear to be justified.

In a recent essay, taking into account the work of Bell about non-locality [START_REF] Bell | On the Einstein Podolsky Rosen paradox[END_REF] and further important papers by Gleason, Kochen-Specker [START_REF] Peres | Quantum theory: Concepts and Methods[END_REF][START_REF] Kochen | The problem of hidden variables in quantum mechanics[END_REF] and Mermin [START_REF] Mermin | What's wrong with these elements of reality[END_REF], I arrived at the conclusion that further progress about the elusive elements of reality, or rather the elements of knowledge, can be performed by resorting to Grothendieck's dessins d'enfants as summarized in the note [START_REF] Planat | It from qubit: how to draw quantum contextuality, Fourth price in the 2013 FQXi contest "It From Bit or Bit From It?[END_REF] and the paper [START_REF] Planat | Quantum contextual finite geometries from dessins d'enfants[END_REF] intended to illustrate Wheeler's it from bit perspective [9]. In Grothendieck's words [10, (a), Vol. 1] The demands of university teaching, addressed to students (including those said to be advanced) with a modest (and frequently less than modest) mathematical baggage, led me to a Draconian renewal of the themes of reflection I proposed to my students, and gradually to myself as well. It seemed important to me to start from an intuitive baggage common to everyone, independent of any technical language used to express it, and anterior to any such language it turned out that the geometric and topological intuition of shapes, particularly two-dimensional shapes, formed such a common ground. This consists of themes which can be grouped under the general name of topology of surfaces or geometry of surfaces, it being understood in this last expression that the main emphasis is on the topological properties of the surfaces, or the combinatorial aspects which form the most down-to-earth technical expression of them, and not on the differential, conformal, Riemannian, holomorphic aspects, and (from there) on to complex algebraic curves. Once this last step is taken, however, algebraic geometry (my former love!) suddenly bursts forth once again, and this via the objects which we can consider as the basic building blocks for all other algebraic varieties. Whereas in my research before 1970, my attention was systematically directed towards objects of maximal generality, in order to uncover a general language adequate for the world of algebraic geometry, and I never restricted myself to algebraic curves except when strictly necessary (notably in etale cohomology), preferring to develop pass-key techniques and statements valid in all dimensions and in every place (I mean, over all base schemes, or even base ringed topoi...), here I was brought back, via objects so simple that a child learns them while playing, to the beginnings and origins of algebraic geometry, familiar to Riemann and his followers! Dessins d'enfants (also known as bicolored maps) are bipartite graphs drawn on a smooth surface but they also possess manifold aspects. They are at the same time group theoretical, topological and algebraic objects and, as revealed by the author, they allow to stabilize the finite geometries attached to quantum contexts. There seems to exist a remarkable confluence between the so-called 'magic' configurations of quantum observables found to illustrate the no-go theorems à la Kochen-Specker and the symmetries obeyed by the algebraic extensions over the field of rational numbers, a subject briefly advocated by Grothendieck as In the form in which Belyi states it, his result essentially says that every algebraic curve defined over a number field can be obtained as a covering of the projective line ramified only over the points 0, 1 and ∞. The result seems to have remained more or less unobserved. Yet it appears to me to have considerable importance. To me, its essential message is that there is a profound identity between the combinatorics of finite In Sec. 2, a brief account of the 'technology' of dessins d'enfants is provided. Sec. 3 addresses the relation between Bell's theorem and some dessins 'living' in the extension field Q( √ 2). Sec. 4 and 5 deal about dessins attached to the two-and threequbit Kochen-Specker theorem about contextuality. Then, in Sec. 6, it is shown that generalized polygons GQ(2, 2) and GH(2, 2), and their corresponding driving dessins, encode the commutation relations of two-and three-qubit sustems, respectively. As our last example, in Sec. 7, a dessin related to the contextuality of the six-qudit system, described in [START_REF] Lisonek | Kochen-Specker theorem with seven contexts[END_REF], is displayed.

The manifolds traits of a 'dessin d'enfant'

I will explain that quantum contexts can be drawn as Grothendieck's 'dessins d'enfants'. A 'true' dessin d'enfant is shown in Fig. 1. This section accounts for the mathematics of 'false' 'dessins d'enfants'. Of course, there are constraints that a child does not take care about: the 'dessins' in question are connected, they are bipartite with black and white points and they are also chosen to be 'clean', meaning that the valency of white vertices is ≤ 2. The last constraint can easily be removed -the valency of white vertices can be made arbitrary to correspond to an hypermap-although this is not necessary for our quantum topic. Doing this, a 'dessin' acquires a topological genus g (which quantifies the number of holes on the smooth surface where it is drawn) such that 2 -2g = B + W + F -n, where B, W , F and n stands for the number of black vertices, the number of white vertices, the number of faces and the number of edges, respectively.

Given a dessin D with n edges labeled from 1 to n, one can recover the combinatorial information by associating with it a two-generator permutation group P = α, β on the set of labels such that a cycle of α (resp. β) contains the labels of the edges incident to a black vertex (resp. white vertex) and by computing a passport [START_REF] Lando | Graphs on surfaces and their applications[END_REF] in the form [C α , C β , C γ ], where the entry C i , i ∈ {α, β, γ} has factors l n i i , with l i denoting the length of the cycle and n i the number of cycles of length l i .

Another observation made by Grothendieck is of utmost importance. The dessins are in one-to-one correspondence with conjugacy classes of subgroups of finite index of the triangle group

C + 2 = ρ 0 , ρ 1 , ρ 2 |ρ 2 1 = ρ 0 ρ 1 ρ 2 = 1 .
The existence of dessins with prescribed properties can thus be straightforwardly checked from a systematic enumeration of conjugacy classses of C + 2 . Note that enumeration becomes tedius for large dessins since the number of dessins grows exponentially with the number of their edges. To proceed with the effective calculations of a dessin, one counts the cosets of a subgroup of C + 2 and determine the corresponding permutation representation P by means of the Todd-Coxeter algorithm implemented in an algebra software such as Magma.

Then, according to Belyi's theorem, a dessin may be seen as an algebraic curve over the rationals. Technically, the Belyi function corresponding to a dessin D is a rational function f (x) of the complex variable x, of degree n, such that (i) the black vertices are the roots of the equation f (x) = 0 with the multiplicity of each root being equal to the degree of the corresponding (black) vertex, (ii) the white vertices are the roots of the equation f (x) = 1 with the multiplicity of each root being equal to the degree of the corresponding (white) vertex, (iii) the bicolored graph is the preimage of the segment [0, 1], that is D = f -1 ([0, 1]), (iv) there exists a single pole of f (x), i. e. a root of the equation f (x) = ∞, at each face, the multiplicity of the pole being equal to the degree of the face, and, finally, (v) besides 0, 1 and ∞, there are no other critical values of f [START_REF] Planat | Quantum contextual finite geometries from dessins d'enfants[END_REF][START_REF] Lando | Graphs on surfaces and their applications[END_REF]. This construction works well for small dessins D but it becomes intractable for those with a high index n; however a complex algebraic curve is associated to every D.

Last but not least, in many cases, one may establish a bijection between notable point/line incidence geometries G i D to a dessin D, i = 1, • • • , m with m being the number of non-isomorphic subgroups S of the permutation group P of the dessin that stabilize a pair of elements. We ask that every pair of points on a line shares the same stabilizer in P . Then, given a subgroup S of P which stabilizes a pair of points, we define the pointline relation on G D such that two points will be adjacent if their stabilizer is isomorphic to S. A catalog of small finite geometries is given as tables 1 and 2 of [START_REF] Planat | Quantum contextual finite geometries from dessins d'enfants[END_REF]. Remarkably, most geometries derived so far have been found to rely on quantum contextuality, that is, the points of a G D correspond to quantum observables of multiple qubits and the lines are mutually commuting subsets of them.

Bell's theorem with 'dessins d'enfants'

John Bell: First, and those of us who are inspired by Einstein would like this best, quantum mechanics may be wrong in sufficiently critical situations. Perhaps nature is not so queer as quantum mechanics. But the experimental situation is not very encouraging from this point of view. . . Secondly, it may be that it is not permissible to regard the experimental settings a and b in the analyzers as independent variables, as we did. We supposed them in particular to be independent of the supplementary variables λ, in that a and b could be changed without changing the probability distribution ρ(λ). . . Apparently seperate parts of the world would be deeply and conspirationnaly entangled, and our apparent free will would be entangled with them. Thirdly, it may be that we have to admit causal influences do go faster than light. . . Fourthly and finally, it may be that Bohr's intuition was right -in that there is no reality below some 'classical' 'macroscopic' level. Then fundamental physical theory would remain fundamentally vague, until concepts like 'macroscopic' could be made sharper than they are today. [13, p. 142] Bell's theorem is generally considered as as a proof of nonlocality, as in the third item of Bell's quote. But, as Bell's theorem is encompassed by Kochen-Specker theorem about contextuality, the second item of Bell's quote appears to be the most relevant, this alternative rules out the introduction of exophysical automatons-with a random behaviorlet alone observers endowed with free will. If you are willing to accept that option, then it is the entire universe which is an indivisible, nonlocal entity. [4, p. 173].

Bell's theorem consists of an inequality that is obeyed by dichotomic classical variables but is violated by the (dichotomic) eigenvalues of a set of quantum operators. The simplest form of Bell's arguments [4, p. 174] makes use of four observables σ i , i = 1, 2, 3, 4, taking values in {-1, 1}, of which Bob can measure (σ 1 , σ 3 ) and Alice (σ 2 , σ 4 ). One introduces the number

C = σ 1 σ 2 + σ 2 σ 3 + σ 3 σ 4 -σ 4 σ 1 = ±2
and observes the (so-called Bell/Clauser-Horne-Shimony-Holt (CHSH)/Cirel'son's) inequality [4, p. 164]

| σ 1 σ 2 + σ 2 σ 3 + σ 3 σ 4 -σ 4 σ 1 | ≤ 2,
where here means that we are taking averages over many experiments. This inequality holds for any dichotomic random variables σ i that are governed by a joint probability distribution. Bell's theorem states that the aforementioned inequality is violated if one considers quantum observables with dichotomic eigenvalues. An illustrative example is the following set of two-qubit observables: σ 1 = IX, σ 2 = XI, σ 3 = IZ, and σ 4 = ZI, where X, Y and Z are the ordinary Pauli spin matrices and where, e. g., IX is a shorthand for I ⊗ X (used also in the sequel).

The norm ||C|| of C (see [START_REF] Peres | Quantum theory: Concepts and Methods[END_REF] or [START_REF] Planat | Quantum contextual finite geometries from dessins d'enfants[END_REF] for details) is found to obey ||C|| = 2 √ 2 > 2, a maximal violation of the aforementioned inequality. The point-line incidence geometry associated with our four observables is one of the simplest, that of a square -Fig. 1a; each observable is represented by a point and two points are joined by a segment if the corresponding observables commute. It is worth mentioning here that there are altogether 90 distinct squares among two-qubit observables and as many as 30240 when three-qubit labeling is employed, each yielding a maximal violation of the Bell-CHSH inequality.

Dessins d'enfants for the square and their Belyi functions

The methodology described at Sec. 2 was used to arrive at the result that the geometry of the square/quadrangle can be generated by four different dessins, (b 1 ), • • • , (b 4 ), associated with permutations groups P isomorphic to the dihedral group D 4 of order 8. Two of them, (b 1 ) and (b 2 ) are tree-like and the other two, (b 3 ) and (b 4 ), contain loops.

The first dessin (b 1 ) has the signature s = (B, W, F, g) = (3, 2, 1, 0) and the symmetry group P = (2, 3), (1, 2)(3, 4) whose cycle structure reads [2 1 1 2 , 2 2 , 4 1 ], i.e. one black vertex is of degree two, two black vertices have degree one, the two white vertices have degree two and the face has degree four. The corresponding Belyi function reads f (x) = x 2 (2 -x 2 ), see [START_REF] Planat | Quantum contextual finite geometries from dessins d'enfants[END_REF] for details. The Belyi functions for the other cases (b 2 ) to (b 4 ) are f (x) = (x 2 -1) 2 , f (x) = (x-1) 4 4x(x-2) and f (x) = (x-1) 4 16x 2 , respectively. Just observe that (critical) points of the dessin, where the derivative f ′ (x) vanishes, correspond to black points where the valency is larger than one, that black point coordinates correspond to solutions of the equation f (x) = 0, that white point coordinates correspond to the solution of the equation f (x) = 1 and that the number of loops reflects in the number of poles of the corresponding Belyi function.

It is intriguing to see that all coordinates of a dessin live in the extension field Q( √ 2) of the rational field Q. Hence, a better understanding of the properties of the group of automorphisms of this field may lead to fresh insights into the nature of this important theorem of quantum physics. Bell's theorem is a no-go theorem that forbids local hidden variable theories. Kochen-Specker theorem [START_REF] Kochen | The problem of hidden variables in quantum mechanics[END_REF] is stronger by placing new constraints in the permissible types of hidden variable theories. Kochen-Specker theorem forbids the simultaneous validity of the two statements (i), that all hidden variables have definite values at a given time (value definiteness) and (ii), that those variables are independent of the setting used to measure them (non-contextuality). Thus, quantum observables cannot represent the 'elements of reality' of EPR paper [START_REF] Einstein | Can quantum-mechanical description of physical reality be considered complete?[END_REF].

Kochen-Specker theorem establishes that even for compatible/commuting observables A and B with values v(A) and v(B), the equations v(aA + bB) = av(A) + bv(B) (a, b ∈ R) or v(A)v(B) may be violated. The authors restricted the observables to a special class, viz. so-called yes-no observables, having only values 0 and 1, corresponding to projection operators on the eigenvectors of certain orthogonal bases of a Hilbert space.

One of the simplest types of violation is a set of nine two-qubit operators arranged in a 3 × 3-grid [START_REF] Mermin | What's wrong with these elements of reality[END_REF]. This grid is a remarkable one: all triples of observables located in a row or a column are mutually commuting and have their product equal to +II except for the middle column, where XX.Y Y.ZZ = -II. Mermin was the first to observe that this is a Kochen-Specker (parity) type contradiction since the product of all triples yields the matrix -II, while the product of corresponding eigenvalues is +1 (since each of the latter occurs twice, once in a row and once in a column) [6, (b)]. Note that the Mermin square comprises a set of nine elementary squares/quadrangles that themselves constitute a proof of Bell's theorem, as shown at the preceeding section.

The Mermin 'magic' square may be used to provide many contextuality proofs from the vectors shared by the maximal bases corresponding to a row/column of the diagram. The simplest, a so-called (18, 9) proof, (18 vectors and 9 bases) has, remarkably, the orthogonality diagram which is itself a Mermin square (9 vertices for the bases and 18 edges for the vectors) [15, (c), eq. ( 6)].

A 'dessin d'enfant' for the Mermin square

Mermin square is shown Fig. 3a. One can recover this geometry with a genus one dessin, with signature (2, 5, 2, 1), as shown in Fig. 3b. The corresponding permutation group is P = (1, 2, 4, 8, 7, 3)(5, 9, 6), (2, 5)(3, 6)(4, 7)(8, 9) ∼ = Z 2 3 ⋊ Z 2 2 , having the cycle structure [6 1

. This dessin lies on a Riemann surface that is a torus (not a sphere Ĉ), being thus represented by an elliptic curve. The topic is far more advanced and we shall not pursue it in this paper (see, e. g., [START_REF] Girondo | Introduction to Riemann surfaces and dessins d'enfants[END_REF] for details). The stabilizer of a pair of edges of the dessin is either the group Z 2 , yielding Mermin's square M 1 shown in Fig 3a, or the group Z 1 , giving rise to a different square M 2 from the maximum sets of mutually non-collinear pairs of points of M 1 . The union of M 1 and M 2 is the Hesse configuration.

Kochen-Specker theorem with 'dessins d'enfants': three qubits

Poincaré wrote: Perceptual space is only an image of geometric space, an image altered by a sort of perspective [17, p. 342] and Weyl wrote: In this sense the projective plane and the color continuum are isomorphic with one another [17, p. 343].

Color experience through our eyes to our mind relies on the real projective plane RP 2 [START_REF] Flanagan | Are perceptual fields quantum fields?[END_REF]. Three-qubit contextuality also relies on RP 2 thanks to a Mermin 'magic' pentagram, that for reasons explained below in (i) we denote P (by abuse of language because we are at first more interested to see the pentagram as a geometrical configuration than as a graph). One such a pentagram is displayed in Fig. 4a. It consists of a set of five lines, each hosting four mutually commuting operators and any two sharing a single operator. The product of operators on each of the lines is -III, where I is the 2 × 2 identity matrix. It is impossible to assign the dichotomic truth values ±1 to eigenvalues while keeping the multiplicative properties of operators so that the Mermin pentagram is, like its two-qubit sibling, 'magic', and so contextual [6, a],[15, (b)

and c)].

Let us enumerate a few remarkable facts about a pentagram. (i) The graph P of a pentagram is the complement of that of the celebrated Petersen graph P. One noticeable property of P is to be the smallest bridgeless cubic graph with no three-edge-coloring. The Petersen graph is thus not planar, but it can be embedded without crossings on RP 2 (one of the simplest non-orientable surfaces), as illustrated in Fig 4b.

(ii) The Petersen graph may also be seens as the complement of the Desargues configuration 10 3 (a celebrated projective geometry that has ten lines with three points and ten points each of them incident with three lines), see [START_REF] Planat | Quantum contextual finite geometries from dessins d'enfants[END_REF]Fig. 11].

(iii) There exist altogether 12096 three-qubit Mermin pentagrams, this number being identical to that of automorphisms of the smallest split Cayley hexagon GH(2, 2) -a remarkable configuration of 63 points and 63 lines [15, (b)] pictued in Fig. 7a.

(iv) Now comes an item close to the it from bit perspective. The Shannon capacity of a graph is the maximum number of k-letter messages than can be sent through a channel without a risk of confusion. The Shannon capacity of P is found to be optimal and equal to 4, much larger than that √ 5 of an ordinary pentagon. (v) Finally, the pentagram configuration in Fig. 5a may be generated/stabilized by a 'dessin d'enfant' on the Riemann sphere, having permutation group isomorphic to the alternating group A 5 and cycle structure [3 2 1 1 , 2 4 1 2 , 5 2 ], as shown in Fig. 5b. The stabilizer of a pair of edges of the dessin is either the group Z 1 giving rise to the Mermin's pentagram, or the group Z 2 giving rise to the Petersen graph.

Dessins d'enfants and generalized polygons

Jacques Tits: I would say that mathematics coming from physics is of high quality. Some of the best results we have in mathematics have been discovered by physicists. I am less sure about sociology and human science [START_REF] Raussen | Interview with[END_REF].

Jacques Tits discovered generalized polygons (also called generalized n-gons). A generalized polygon is an incidence structure between a discrete set of points and lines whose incidence graph has diameter n (the maximum eccentricity of any vertex) and girth 2n (the length of a shortest cycle). A generalized polygon of order (s, t) has every line containing s + 1 points and ever point lying on (t + 1) lines. Remarkably, the generalized 4-gon/quadrangle of order (2, 2), namely GQ(2, 2) controls the commutation structure of the 15 two-qubit observables [15, (a)] and the generalized 6-gon/hexagon GH(2, 2) does the job for the 63 three-qubit observables [15, (b)].

An important concept pertaining to generalized polygons is that of a geometric hyperplane. A geometric hyperplane of a generalized polygon is a proper subspace meeting each line at a unique point or containing the whole line. The substructure of a polygon of order (2, t) highly relies on its hyperplanes in the sense that one 'adds' any two of them to form another geometric hyperplane. The 'addition' law in question is nothing but 'the complement of the symmetric difference' of the two sets of points involved in the pair of selected hyperplanes. There exists three kinds of hyperplanes in GQ(2, 2) one of them being the Mermin square described in Sec. 4. The structure of hyperplanes of GH(2, 2) is of utmost importance to describe the three-qubit Kochen-Specker theorem as described exhaustively in [15, (b)].

Next, we find that generalized polygons are induced/stabilized by dessins d'enfants. As for the case of the geometry of the square, a selected geometry G may be induced/stabilized by many dessins D i , i.e. the correspondance f : D i → G is non injective. Moreover, the number of dessins grows exponentially with the number of their edges so that the systematic search of all maps f may become tedious. To simplify the search of a solution f inducing a selected G (such as a generalized polygon) one restricts the search to subgroups of the cartographic group C + 2 (go back to Sec. 2 for the definition).

A dessin d'enfant for the generalized quadrangle GQ(2, 2)

The generalized quadrangle GQ(2, 2) encodes the commutation relations of two-qubit operators as shown in Fig. 6a [15, (a)].

A dessin stabilizing the generalized quadrangle GQ(2, 2) may be obtained by studying the conjugacy classes of the subgroup C + 2 /[ρ 4 2 = 1], whose permutation representation of the cosets is isomorphic to the symmetry group of GQ(2, 2), that is the symmetric group S 6 . This is shown in Fig. 6. One finds that the dessin in Fig. 6b has two types of stabilizers for a pair od edges, one isomorphic to Z 5 2 and inducing GQ(2, 2) and the other one isomorphic to Z 6 and inducing the complement of GQ(2, 2).

A dessin d'enfant for the generalized hexagon GH(2, 2)

The generalized hexagon GH(2, 2) encodes the commutation relations of three-qubit operators as shown in Fig. 7a [15,(b)]. A dessin stabilizing the generalized hexagon GH(2, 2) may be obtained by studying the conjugacy classes of a subgroup of C + 2 whose finite representation is that of the symmetry group of the hexagon. Then one selects the dessins of permutation group P isomorphic to the wreath product S 3 ≀ S 3 (of order 1296). Remarkably, one finds only two dessins satisfy these requirements, one is of genus 0 and induces GH(2, 2) as shown in Fig. 7 and the other one (not shown) is of genus 1 (drawn on a torus) and induces the dual of GH(2, 2). Recently, a remarkable (minimal) Kochen-Specker configuration built from seven contexts and 21 rays, to a six-qudit system, has been built [START_REF] Lisonek | Kochen-Specker theorem with seven contexts[END_REF]. The set in question is an heptagram (see Fig. 8a) in which the seven lines are hexads of mutually orthogonal vectors (they are not quantum observables as was the case at the previous sections). It is straightforward to check (with a computer) that the collinearity graph of this geometry contains two kinds of maximal cliques, the seven hexads just mentioned and, in addition, 35 triangles. The graph of the latter 35-triangle geometry is nothing but the line graph of the complete graph K 7 [the line graph of the complete graph K 5 is the Desargues configuration that occured at Sec. 5, item (ii)]. The symmetry group of the 35-triangle geometry, as that of the heptagram, is the seven-letter symmetric group S 7 . Starting from a finite representation of S 7 that underlies a subgroup of the cartographic group C + 2 , it is not difficult to find a dessin stabilizing the aforementioned 35-triangle geometry, as shown in Fig. 8b. Once again, it has been shown that quantum contexts are intimately related to dessins d'enfants.

A dessin d'enfant for six-qudit contextuality

Conclusion

It was not anticipated by his creator that geometries induced by 'dessins d'enfants' would so nicely fit the drawings/geometries underlying quantum contextuality. I believe that this key observation opens new vistas for the interpretation of quantum measurements in terms of algebraic curves over the rationals. The symmetries of general dessins rely on a fascinating group called the universal (or absolute) Galois group over the rationals, a still rather mysterious object. May be the present work gives some substance to an hidden variable interpretation of the quantum world, as hoped by Einstein, but in a subtle way. We just mentioned in passing (in Sec. 6) that the geometries relevant to quantum contexts have a rich substructure of hyperplanes that also has to be incorporated in the new design. More details will be given at a next stage of our research.

Bibliography

Figure 1 .

 1 Figure 1. A dessin d'enfant drawn at a kindergarten by a 5-year old girl, http://www.parents.com/fun/arts-crafts/kid/decode-child-drawings/

Figure 2 .

 2 Figure 2. A simple observable proof of Bell's theorem is embodied in the geometry of a (properly labeled) square (a) and four associated dessins d'enfants, (b 1 ) to (b 4). For each dessin an explicit labeling of its edges in terms of the four two-qubit observables is given. The (real-valued) coordinates of black and white vertices stem from the corresponding Belyi functions as explained in the main text.

4 .

 4 Kochen-Specker theorem with 'dessins d'enfants': two qubits I am grateful to N. D. Mermin for patiently explaining to me that ref. 11 [A. Peres, Phys. Lett. A 151, 107 (1990); Found. Phys. 22, 357 (1992)]) was a Kochen-Specker argument, not one about locality, as I had wrongly thought [14].

Figure 3 .

 3 Figure 3. A 3 × 3 grid with points labeled by two-qubit observables (aka a Mermin magic square) (a) and a stabilizing dessin drawn on a torus (b).

Figure 4 .

 4 Figure 4. (a) A Mermin pentagram P and (b) the embedding of the associated Petersen graph P on the real projective plane as a hemi-dodecahedron.

Figure 5 .

 5 Figure 5. (a) Mermin pentagram P and (b) a generating dessin.

Figure 6 .

 6 Figure 6. The generalized quadrangle GQ(2, 2) (a) with its points labeled by the elements of the two-qubit Pauli group and a stabilizing dessin (b).

Figure 7 .

 7 Figure 7. (a) The hexagon GH(2, 2) and (b) the dessin for its collinearity graph. To simplify the drawing, white points are not shown but half-edges are labelled.

Figure 8 .

 8 Figure 8. (a) The seven-context geometry of six-qudit contextuality and (b) a dessin stabilizing the 35-triangle geometry lying in (a). To simplify the drawing, white points are not shown but half-edges are labelled.


Rosa Maria Bottino

Michela Ott

Mauro Tavella

Serious Gaming at School: Reflections on Students' Performance, Engagement and Motivation

Keywords: Serious Gaming, Game-enhanced learning, Primary education, Motivation, School Performance, Attention, Mind Games

The concept of Serious Gaming refers to the adoption of classical entertainment games for purposes other than entertainment, including learning and instruction. In this paper we report on a Serious Gaming field experiment where typical board games (such as battleship, master mind and domino) were employed with the shifted purpose of triggering and sustaining primary school students' reasoning and logical abilities. The results of the field experiment showed that: 1) there is a strong correlation between school achievement and the ability to play and solve this kind of games and that 2) motivation and engagement in game-based learning tasks is very high, irrespective of the level of achievement of the subjects. Final considerations are drawn about the potential and the opportunity of adopting the considered games to support those reasoning skills that are widely recognized as transversal to any kind of learning and thus deeply affecting overall school performance.

INTRODUCTION

This paper presents the results of a research study in the field of game-based learning. More precisely, it refers to an experiment conducted in primary schools where digital versions of traditional board games were employed with the aim of sustaining the students' logical and reasoning abilities.

The games adopted during the experiment deeply require the enactment of thinking skills. They can be considered "cognitively responsible" [START_REF] Martinovic | Critic-proofing of the Cognitive Aspects of Simple Games[END_REF] and are usually referred to as mind games [START_REF] Bottino | Digital mind games: experience-based reflections on design and interface features supporting the development of reasoning skills[END_REF], brainteasers or puzzles [START_REF] Kebritchi | The effects of modern mathematics computer games on mathematics achievement and class motivation[END_REF][START_REF] Milovanović | Best Practices for the Knowledge Society. Knowledge, Learning, Development and Technology for All[END_REF].

If we follow the classification proposed by [START_REF] Djaouti | Classifying Serious Games: The G/P/S Model[END_REF] we can set the reported experience in the area of Serious Gaming.

According to these authors, in fact, Serious Gaming is a label that refers to the use of "any video game for serious purposes, whether the serious dimension is or is not designed within the software" while "Serious Games is a label that refers to applications featuring both a serious and a game dimension within the software".

Actually, the games adopted, were not Serious Games in that they had not been originally designed having in mind an instructional (serious) purpose but rather for entertainment [START_REF] Kirriemuir | Literature Review in Games and Learning[END_REF]. Nevertheless, they were used by the authors in a school context by shifting their original purpose to serve a serious, instructional purpose and this was done without operating software modifications, in a genuine Serious Gaming perspective. This paper, based on concrete data from the above mentioned experiment, explores the links between the possession of the reasoning abilities involved in the use of a set of digital mind games [START_REF] Rohde | Predicting academic achievement with cognitive ability[END_REF][START_REF] Bottino | The impact of mind game playing on children's reasoning abilities: reflections from an experience[END_REF] and school achievement.

To date, few scientific studies report experimental data confirming the possible link between school achievement and the ability to solve mind games, despite the fact that research studies in the field have evidenced that the use of digital games can:  offer a variety of educational benefits [START_REF] Sandford | Teaching with Games[END_REF][START_REF] Prensky | Digital game-based learning[END_REF][START_REF] Hong | Assessing the educational values of digital games[END_REF][START_REF] De Freitas | How can exploratory learning with games and simulations within the curriculum be most effectively evaluated?[END_REF][START_REF] Pivec | Play and learn: potentials of game-based learning[END_REF];  promote the development of cognitive and complex problem solving skills (Felicia, 2009a);  contribute to the enhancement of school performance [START_REF] Robertson | Learning gains from using games consoles in primary classrooms: a randomized controlled study[END_REF][START_REF] Franco | Exploring the Effects of a Mindfulness Program for Students of Secondary School[END_REF].

The research study this paper refers to confirms such results and shows that there is a strong correlation between school achievement and the ability to play and solve digital mind games and that, however, motivation and engagement in game-based learning tasks is very high, irrespective of the level of achievement of the students involved. In the following, the context of the study is briefly presented together with the tools adopted; subsequently, the methodology of the experiment is detailed and the main results are presented as to students' performance; further reflections are proposed regarding some affective aspects of the experiment including students' engagement and motivation. This allows drawing some final considerations on the potential of mind games to support young children's reasoning and logical abilities.

SETTING THE SCENE

In the following, before presenting the methodology underpinning our study and its key results, we briefly summarize some basic aspects of the experiment including the description of the target population involved and of the tools adopted.

Target population

The experiment was conducted in twenty Italian primary schools classes in the Lombardia region: it involved 60 children in 4th and 5th grades. A further group of 20 students with certified cognitive impairment (and consequent deep, generalized learning difficulties) from a school in Genoa was also considered. This was done with the specific objective of further verifying the games suitability and potential effectiveness in respect of this target population and, also, to understand the personal attitudes of these children towards the educational use of games.

Tools adopted

The main tool adopted in the experiment was the LOGIVALI (LOGIcal thinking eVALuatIon) test, a game-based norm referenced test explicitly aimed at assessing the reasoning abilities of primary school pupils [START_REF] Bottino | Can digital Mind Games be Used to Investigate Children's Reasoning Abilities[END_REF]. The LOGIVALI test follows a custom set-up, specific methodology to investigate and assess the possession of some specific logical and reasoning abilities; in doing so it adopts a blended approach in that it foresees the use of both digital and non-digital tools (actually playing sessions with digital games and assessment sessions with paper and pencil sheets).

In the following, we briefly illustrate the abilities encompassed by the LOGIVALI test and provide a short description of the adopted digital games.

The abilities considered by the LOGIVALI test

The LOGIVALI test takes into account a limited number (six) of specific reasoning abilities identified as crucial [START_REF] Bottino | Mind games, reasoning skills, and the primary school curriculum: hints from a field experiment[END_REF] although they are, obviously, only a subset of the abilities required to deal with the games at hand.

The six abilities investigated through the LOGIVALI Test are:  Ability 1 "knowing and remembering the rules of the game": to know the rules underlying a

given game and to be able to apply them in concrete game situations.  Ability 2 "first level reasoning": to be able to make an inference taking into consideration a single given datum.  Ability 3 "second level reasoning": to be able to make an inference taking into consideration two given information or game constraints.  Ability 4 "third level reasoning": to be able to make an inference taking into consideration more than two given information and game constraints.  Ability 5 "managing uncertainty": to be able to establish whether the data available at a given moment of the game are sufficient to decide whether a certain guess is correct or not.  Ability 6 "operatively applying reasoning abilities": to be able to implement into the game the results of own reasoning (actions should follow thoughts).

In the LOGIVALI test, the ability of making inferences is not considered as a unique skill but it is regarded and treated as an ability varying in difficulty according to the number of different data, information and constraints to be dealt with (abilities 2, 3 and 4). On the basis of the results obtained with the involved sample population (around 500 primary school students) the test was standardized and the reference norms were defined on a percentile ranking basis. In [START_REF] Bottino | Can digital Mind Games be Used to Investigate Children's Reasoning Abilities[END_REF] the whole LOGIVALI test validation and standardization procedure is presented. During such procedure, it emerged that the number of correct answers progressively decreased in the three stages: Ability 2 (80% of correct answers); Ability 3 (63% of correct answers); Ability 4 (46% of correct answers). This confirmed that the three abilities should be considered as increasingly complex abilities for the target population.

The digital games employed

As explained above, the test makes use of five digital mind games [START_REF] Bottino | Can digital Mind Games be Used to Investigate Children's Reasoning Abilities[END_REF] that fall into the category of mini-games, that is games that take less than an hour to complete [START_REF] Prensky | Educational Games Complexity Matters: Mini-games are Trivial -but "Complex" Games Are Not. An important Way for Teachers, Parents and Others to Look At Educational Computer and Video Games[END_REF].

The five games adopted in the LOGIVALI test do not require specific prerequisites in curricular school subjects, beyond very basic literacy and, most importantly, do not imply the possession of specific arithmetical skills.

Two of the games are the computerized versions of well-known board games: mastermind (Figure 1) and domino (Figure 2). The goal of MasterMind is to guess the sequence of a number of colored pegs that the computer has selected at random. The degree of difficulty is determined by the number of different colours that can make up the sequence (in the example, the five colours on the left-hand side) and the number of pegs that comprises it. The player starts making guesses by filling the holes at the bottom row of the left column. When a row is completed, the program shows, in the right column, how many pegs have been placed in the correct position (black pegs) and how many are of the right colour but are placed in the wrong position (white pegs).

Mastermind is strongly based on the capacity of the child to understand and use the feedback provided by the program to infer what next move might prove effective.

The game requires the player to build strategies that help to reduce the number of possible sequences, for example, trying a sequence of pellets of the same colour at the beginning, making guesses by altering the position of only one pellet, reasoning only on a subset of positions, etc..

Figure 2. Tetravex (http://live.gnome.org/GnomeGames)

Tetravex is a domino-like game whose goal is to drag in the left hand board the tiles given in the right hand board so that any two adjacent numbers are the same. Each tile is divided into four triangles marked with four numbers from 0 to 9. Tetravex has a range of difficulty levels that go from the simple (two-by-two boards) up to the very difficult (six-by-six boards).

Tetravex is a game that, at the simple levels (e.g. two-bytwo and three-by-three boards), can be solved through a trial and error strategy. When the difficulty increases (that is the number of tiles is higher), this approach appears inadequate and it becomes necessary to conceive and enact an effective solution strategy.

Two of the games are board games which resemble the traditional battleship game but actually introduce further interesting features genuinely eliciting logical thinking processes (Figure 3, Figure 4). TreeTent is a single-player puzzle game in which the goal is to work out where the computer has positioned a certain number of tents on a board where a certain number of trees appears. A tent can only be found in a cell horizontally or vertically adjacent to a tree, tents are never placed adjacent to each other, and for each tree only one tent can be positioned on the board. The numbers along the right and lower sides provide a clue as to the number of tents in that row or column.

Solving the game entails reasoning on available data and matching all the available information and constraints (e.g. only one tent for each tree, tents never placed adjacent to each other, etc.) in order to infer the content of the cells. This game also emphasises the need to find a strategy to reduce the complexity of the task. For instance, the task becomes much easier if the player first identifies the "peculiar case" of lines marked with 0 (lines where there are no tents). Hexip is a battleship-type game whose aim is to find the position of ships hidden on a hexagon-shaped board. The game provides information on the number of boxes occupied by ships along both the horizontal and diagonal rows of the board (numbers outside the hexagon).

Hexip requires that students are aware of the importance of analysing given data such as the numbers outside the hexagons, and using this information effectively. Like TreeTent, Hexip requires to find ways to reduce the complexity of the task. For example, it is useful to identify first the empty rows, and, subsequently, to recognize the situations where the number outside a row corresponds exactly to the available positions. One important peculiar aspect of this game is that when the difficulty increases, the solution can be reached only by making inferences on the basis of a hypothesis/validation process.

The fifth game used in LOGIVALI test is a game named Pathological. It introduces an interesting variable in that it supports the real-time ability to adapt the ongoing logical processes to different situations according to the changing of data and related constraints (Figure 5). The Pathological game is a puzzle game consisting of marbles and wheels. The marbles are of different colours and roll along paths. Each wheel has four slots that are to be filled in with marbles of the same colour: to do so, the user clicks on the wheel to rotate it. When the wheel is completed, the marbles vanish and the wheel turns dark. Pathological is organized in progressive difficulty levels that correspond to different schemas. In order to complete each schema it is necessary to take into consideration not only the colour of the ball that is coming but also the colours of the next ones (left-hand side of the screenshot).

Since in Pathological the configuration of the board changes at each level, the game requires the students to adapt to different situations and constraints. When playing the game, the user not only needs to consider changes in board configuration, but must also take into account the colour of upcoming marbles, which can be seen in advance in the right-hand column of the interface. Anticipation therefore plays a crucial role in devising a solution strategy

The LOGIVALI test uses the selected games to verify the actual possession of the above mentioned reasoning abilities by children of the target age (4th and 5th grade students). The test encompasses five different subtests, one for each of the games employed. The test foresees an administration methodology that implies that: a) teachers explain the selected games to the students; b) students play individually with the five selected digital mind games according to fixed game sessions; c) teachers monitor students at work during the playing sessions, but abstain from intervening with suggestions and help; d) after completing the playing sessions with each game, each student individually takes a specific test on that game (subtest); test are administered by the teachers who also are in charge of making students aware of the fact that no curricular evaluation is foreseen for the test.

METHODOLOGY

As mentioned above, for the purpose of the present study 60 pupils were involved out of twenty Italian primary schools classes (4th and 5th grades) in the Lombardia region (3 for each class). They were selected following specific criteria, monitored while individually playing with digital mind games and finally the LOGIVALI test was administered. The students were selected on the basis of their school performance and were classified in three groups according to their level of school achievement (school actual results based on the scores obtained, plus teachers' judgments) as follows:

 group A-high achievers;  group B-medium achievers;  group C-low achievers.

This target group of 60 students comprised three students per class: one for each group A, B and C.

In each class, the selected students were individually monitored by the teachers. The LOGIVALI test was administered during school hours individually to all the students in each classroom.

To favor the monitoring of students at work, teachers were provided with made on-purpose monitoring sheets where they were asked to record relevant data on each student's behavior, in particular on engagement and motivation.

Detailed results of the LOGIVALI test were recorded (score obtained by each student and consequent positioning of the subject in the percentile scale) and carefully analyzed.

The available data for each student thus consisted in:  Scores at the LOGIVALI test (global score and individual scores in each single sub-test)  Percentile analysis of performance  Qualitative data on: autonomy, engagement and attention For the scope of this study, all the data available through the LOGIVALI test were analyzed and compared with each student's data on school performance. It was thus possible to make some inferences on the possible correlations between students' performance in playing/solving mind games and that attributed in disciplinary tasks. In order to obtain a complete view of each student's profile, behavioral aspects (derived from monitoring sheets analysis) were also considered and the correlations with the above mentioned performances were further studied.

The research approach adopted in this study basically followed the paradigms of mixed research methodology [START_REF] Burke | Toward a Definition of Mixed Methods Research[END_REF][START_REF] Burke | Mixed Methods Research: A research paradigm whose time has come[END_REF]; in fact, a combined use was made of both quantitative and qualitative approaches.

Actually, as said above, the performance of each student at the LOGIVALI test was assessed according to the percentile ranks determined by the test; subsequently, the obtained quantitative data on performance at the LOGIVALI test were compared with available data on each student's level of school achievement.

Data recorded by teachers in the monitoring sheets and referring to students' engagement and attention were, instead, analyzed by employing qualitative research methods, in particular, the direct observation research method (Herbert, 1995) The work with the additional group of students from Genoa (those with severe certified learning disabilities) followed a different procedure and each students was constantly and individually assisted by one member of the research team during all the phases of the study (introduction, game play, test…).

In the following, while examining the results of this study we mainly focus on the results of the former group (60 students) and only use the Genoa group to make some lateral considerations on the acceptance and suitability of the proposed educational activity.

RESULTS

In the following we present the results of the students as to: 1) their performance at the LOGIVALI Test; 2) key behavioral aspects identified during game play.

As to the performance, as said above, we report data from the 60 subjects and do not consider those data coming from the 20 students with ascertained and severe learning disabilities from Genoa, mainly because data on their performance were deeply influenced by the assistants' continuous help and advice.

Performance

Performance was evaluated based on the percentile level reached. Four categories of performance were then identified:  Poor -25 th centile and below  Fair -from 25 th up to 50 th centile  Good -from 50 th centile up to 75 th  Very good -beyond 75 th centile Global performance Figure 6 details the results of the global performance of the three groups at the LOGIVALI test: dark parts stand for low performances (actually the 2 levels of low performances: poor and fair) and light parts for high performances (actually the 2 levels of high performances: very good and good).

Figure 6. Performances of the three groups of students at the LOGIVALI Test

The results show that:

 Only students in group A (high achievers) have prevailing high performances at the LOGIVALI test (light parts in the figure) in respect to the other two categories where dark parts are prevailing; more than half of the students of this group perform at the two highest levels.  The very majority of the students in group C (low achievers) perform at low levels (fair and poor); most performances of these students are at the lowest level.  Performances of students in group B (medium achievers) are comparable to those of students in group C if we consider dark and light parts but performances at the fair level prevail on performance at poor level. Actually, we see that performances of high and low achievers are very different (in favour of high achievers) while the medium group is situated half way between the two.

As a general observation we thus can affirm that the global results of the students' performance at the LOGIVALI Test show that a substantial consistency exists between school achievement and the ability of playing mind games.

In the following we present observations as to the specific abilities considered in the test.

Ability 1

As to the ability of recalling the game rules (Ability 1), the performance of the three groups are scaled.

As shown in Figure 7, very good performances decrease sensibly from Group A to group C. If we look at the two big blocks (dark and light parts) we see that high achievers perform better in respect to the other two groups, whose performance is still comparable.

Ability 1

Figure 7. Performances of the three groups at the ability of recalling the game rules Ability 2, 3 and 4-inferential abilities

As to the inferential abilities (core abilities 2, 3 and 4 of the LOGIVALI test), we see that performances of high, medium and low achievers are always scaled (with the prevalence of good performances for high achievers and of poor performances for low achievers).

This emerges from data reported in Figure 8 where the results at the three abilities of the test related to the capacity to make inferences on the basis of given data (namely Ability 2, 3 and 4) are presented. Figure 8 shows that in all the three abilities poor performances prevail in group C and viceversa good performances prevail in group A while medium achievers are always in between the two. While looking at Figure 8 nevertheless we must consider that the three graphs should be regarded singularly and no direct comparison among the three graphs representing the three different abilities can be done.

The structure of the available data doesn't allow us to directly compare the performances of the three groups of students in the different abilities: our data are, in fact, based on the calculation of the percentiles which varies in the different abilities (treated as independent) as a consequence of the different number of correct answers given by the sample population.

Despite this, if we look singularly at the considered reasoning abilities, we see that high achievers always perform better than the other two groups.

The emerging trend is the same for all the six considered abilities, despite reasonable differences can be found due to the intrinsic level of difficulty of each of them, as illustrated below.

Ability 5

As to the ability of managing uncertainty (Ability 5), Group C students show genuinely poor performances while high and medium achievers performances are comparable.

Ability 5, namely the ability to be able to establish whether the data available at a given moment of the game are sufficient to decide whether a certain guess is correct or not, clearly discriminates among the three groups: it appears to be very difficult for low achievers (Figure 9), so that only very few of them perform well. Both the other two groups, instead, appear to be more confident with the task. As to the ability of operatively applying reasoning abilities (Ability 6), performances of students in Group A are higher with respect to those of students in the other two groups. Differences between group B and Group C are found as well, in favor of Group B.

Ability 6 implies to "operatively applying reasoning abilities", namely to be able to implement into the game the results of own reasoning (in the logic that actions should follow thoughts). Group A performance is far higher with respect to the one of the other two groups (Figure 10): this could allow arguing that students in this group are more able to "put into practice what they have learnt" and this attitude/ability could be responsible for their good achievement in curricular activities.

Ability 6

Figure 10. Performances of the three groups at the ability of operatively applying reasoning abilities

Attitudes

During the experiment, issues related to children's behavior such as engagement, attention and motivation were also investigated. This was done by means of made on-purpose monitoring sheets where the teachers and the members of the research group, while observing students at work, carefully recorded data on their attitudes considering a number of common specific indicators, to guarantee comparability.

Autonomy

The first attitude considered was autonomy, i.e. the capacity of children to find and enact solution strategies on their own. As shown in Figure 11, it emerged that while students in the high and medium achievers groups were substantially autonomous most of those in the low achievers group frequently required help and advice, although only few of them were considered non-autonomous at all.

Figure 11. Level of autonomy shown by pupils in the three groups Engagement and Motivation

The monitoring sheets where personal notes of the observers and answers to questions by the involved subjects were annotated showed that all pupils were fundamentally attentive and engaged in the task (Figure 12). Their level of motivation resulted to be quite high during the whole experiment; this was true also for a consistent number of low achievers, irrespective of their actual gaming ability and capacity to reach the solution.

Figure12. Levels of engagement and motivation shown by pupils in the three groups

These data confirm that a considerable amount of children of the target age are attentive and engaged in the interaction with mind games, despite their level of achievement, even those that need adequate support by educators to reach the game solution.

The findings on students' attitudes towards gaming activities are confirmed by the results of the parallel investigation conducted on the sample of 20 pupils with certified learning disabilities from the Genoa school [START_REF] Dagnino | Reasoning abilities in primary school: A pilot study on poor achievers vs normal achievers in computer game tasks[END_REF]).

If we consider also this group (Group E) we see (Figure 13) that engagement and motivation while playing of these students is to some extent comparable with that of low achievers of the other considered sample (Group C). Of course, while reflecting on these data we need to take into account the fact that these students, during all the phases of the experiment, were individually monitored and assisted (i.e. they could count on the constant presence/assistance/advice of one person of the research team).

An interesting consideration regards the fact that motivation in Group E students appears to be higher with respect to those in group C; this is probably due to the fact that these students, being constantly assisted by adults, were less impressed by personal failures [START_REF] Ott | A contribution to the understanding of what makes young students genuinely engaged in computer-based learning tasks[END_REF] Fig. 13. Level of engagement and motivation shown by pupils in the three groups plus a group of 20 students with severe learning disabilities

DISCUSSION

The final aim of the described experiment was to investigate whether digital mind games can be used successfully to elicit and support the development of those "transversal" reasoning abilities that affect global school achievement.

To this aim, evidence was found that:  A substantial consistency exists between primary school students' school achievement and the ability of playing digital mind games.  The great majority of students, independently from their level of school performance, are basically very attentive and engaged in game-based learning tasks.

As a matter of fact, from the conducted experiment it emerged also that:  The performances at games are scaled and reflect the results of high, medium and low achievers.  The ability itself of recalling games rules is more present in high achievers and progressively at a lesser extent in the other groups of students.  Inference tasks are well performed by high achievers but represent a sensible obstacle for low achievers (mainly when inferences need processing more than one datum).  The ability of verifying the work done by also managing uncertainty and establishing whether the data available at a given moment of the game are sufficient to decide whether a certain guess is correct or not is peripheral for the group of low achievers.  The ability to put into practice what learnt during exercises is almost a prerogative of high achievers.

 The level of autonomy in playing games is good for both high and medium achievers while low achievers need some external help.  Digital games attract all students' attention and stimulate their engagement and motivation.

Since the experiment was based on games that do not require knowledge/competences specific to curricular school subjects (e.g. arithmetic, linguistic), the obtained results suggests that playing with mind games can profitably exercise transversal skills that are involved also in curricular tasks and activities (e.g. making inferences, verifying the work done, recalling rules, etc.).

Digital games are suitable tools to exercise such skills in that they are well accepted by students and able to stimulate their positive attitude towards learning.

The actual effectiveness of game-based training programs in the field of logical abilities is substantiated also by the findings of a previous research experience conducted by the authors [START_REF] Bottino | Developing strategic and reasoning abilities with computer games at primary school level[END_REF] that showed a positive impact of the gaming activity on both school performance and learning behavior.

The experiment, in fact, had evidenced that a group of students who had undergone a gamebased reasoning and logical training with games for a long period (three school years) performed better at a national math test with respect to a control group. The teachers involved in the experiment had also reported a positive impact of the gaming activities on the general learning behavior of the students involved in the experiment.

Based on all these results, we could reasonably draw the conclusion that mind games could be adopted profitably in early educational programs targeting the development and the enhancement of "transversal" reasoning skills and that they may have, in turn, a positive impact on the students' global school achievement.

The experiment conducted with severe low achievers and the comparison of their levels of engagement and motivation with those of students with different levels of school achievement shows that they could also benefit of specific targeted, personalized interventions [START_REF] Pozzi | Supporting Group and Individual Processes in Web-Based Collaborative Learning[END_REF] based on a specific users' profiling (Felicia, 2009b) and a carefully planned use of mind games.

CONCLUSIONS

The results of the reported experience, together with previous findings in the field, support the idea that digital mind games can effectively be used to sustain logical and reasoning abilities in 8-10 years old primary school students and that this could influence global school achievement as well. The specific case of students with severe ascertained learning disabilities was also investigated and it emerged that, if properly assisted and tutored, also these students find such an exercise motivating and engaging and, if appropriately followed by teachers, can almost reach the level of performance of students who are low achievers but who have not ascertained disabilities. According with such finding, it seems possible to argue that also cognitive disabled students can benefit from activities with mind games and that their reasoning abilities can to some extent be enhanced.

The target population of the experiment, as said above were students in the age range from 8 to 10, nevertheless, the hypothesis (to be verified with further in-depth studies) can also be put forward that also children in different age groups could benefit from playing with mind games.

In the case of younger children (e.g. 6 years old students), for instance, a specific game-based training could help them to start their school career with the right foot. Moreover, mind games playing could be included among the activities of a remedial program for older students who have evidenced some learning difficulties.

Further studies in both these directions are in progress.

To conclude a set of general indications based on our experience can be given to support those teachers and educators interested in using digital mind games with their students. In particular, we see that there is a need for reflecting on the importance of:  Making an accurate choice of the specific games to be adopted by carefully paying attention to the entailed abilities and to games usability.  Choosing games that are relatively independent from specific prerequisites in curricular school subjects.  Accurately planning the sequence of the mind game based exercises by tuning the progression of the difficulty level.  Appropriately calibrate the level of assistance that teachers offer to each single student so as to trigger logical thinking and adequately support students' motivation and engagement.  Consider that the use of mind games can prove useful if it is not sporadic and planned on a short term basis but if it is incorporated in activities that cover a medium/long term period of time (e.g. one hour a week for a 5/6 months).

It is worth underlying that, in general, the educational effectiveness of games in school activities is strictly linked to the way they are used and that the role of the teacher is crucial to this end. The teacher has to define the specific aim, context and modalities of games use, also including, if needed, specific customizations (e.g. setting difficulty levels, deciding on help available to students, configuring interface constraints).

Figure 1

 1 Figure 1 Mastermind (http://gmoerth.freeservers.com/mm/index.htm)

Figure 3 .

 3 Figure 3. Tree Tent (http://yoogi.com)

Figure 4 .

 4 Figure 4. Hexip (http://yoogi.com)

Figure 5

 5 Figure 5. Pathological (http://pathological.sourceforge.net)

Figure 8 .

 8 Figure 8. Performances of the three groups at the three "core" abilities related to inference

Ability 5 Figure 9 .

 59 Figure 9. Performances of the three groups at the ability of managing uncertainty Ability 6As to the ability of operatively applying reasoning abilities (Ability 6), performances of students in Group A are higher with respect to those of students in the other two groups. Differences between group B and Group C are found as well, in favor of Group B.Ability 6 implies to "operatively applying reasoning abilities", namely to be able to implement into the game the results of own reasoning (in the logic that actions should follow thoughts). Group A performance is far higher with respect to the one of the other two groups (Figure10): this could allow arguing that students in this group are more able to "put into