Kirkman's Tetrahedron and the Fifteen Schoolgirl Problem

Falcone, Giovanni; Pavone, Marco

doi:10.4169/amer.math.monthly.118.10.887

Cited by 12 publications

(31 citation statements)

References 7 publications

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“…(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3,4) For c 6 then 6, 7, 8, 9, (because c 5 = 5). Each sequence in H 05 is characterized by an ordered pair [(c 2 , c 3 , c 4 ), c 6 ] with a count of 4•4 = 16 according to the product rule.…”

Section: Methodology Of Research and Discussionmentioning

confidence: 99%

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Teaching Combinatorial Principles Using Relations through the Placemat Method

et al. 2021

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The presented paper is devoted to an innovative way of teaching mathematics, specifically the subject combinatorics in high schools. This is because combinatorics is closely connected with the beginnings of informatics and several other scientific disciplines such as graph theory and complexity theory. It is important in solving many practical tasks that require the compilation of an object with certain properties, proves the existence or non-existence of some properties, or specifies the number of objects of certain properties. This paper examines the basic combinatorial structures and presents their use and learning using relations through the Placemat method in teaching process. The effectiveness of the presented innovative way of teaching combinatorics was also verified experimentally at a selected high school in the Slovak Republic. Our experiment has confirmed that teaching combinatorics through relationships among talented children in mathematics is more effective than teaching by a standard algorithmic approach.

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Section: Methodology Of Research and Discussionmentioning

confidence: 99%

“…79 on the papyrus was a combinatorial one [1,2]. Thomas Kirkman was one of the pioneers of modern combinatorics in the 19th century, and he became famous for the "15 schoolgirls" [3] combinatorial problem. With his original works, Kirkman also contributed to discrete geometry and group theory.…”

Section: Introductionmentioning

confidence: 99%

Teaching Combinatorial Principles Using Relations through the Placemat Method

et al. 2021

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“…It is known that, up to isomorphism, there exist exactly seven KTS (15), i.e., there are seven non-isomorphic solutions to the well-known Kirkman fifteen schoolgirls problem. It is possible to show, applying the proposition on page 894 of [27], that the solution above is necessarily isomorphic to the original solution given by Kirkman, that is the solution denoted by 1a in [17] Table 1.28, p. 30.…”

Section: A 3-pyramidal Kts(15)mentioning

confidence: 99%

The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Bonvicini

Buratti

Garonzi

et al. 2021

Des. Codes Cryptogr.

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Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We partially fill this gap by proving that whenever $$v \equiv 39$$ v ≡ 39 (mod 72), or $$v \equiv 4^e48 + 3$$ v ≡ 4 e 48 + 3 (mod $$4^e96$$ 4 e 96 ) and $$e \ge 0$$ e ≥ 0 , there exists a KTS on v points having at least $$v-3$$ v - 3 automorphisms. This is only one of the consequences of an investigation on the KTSs with an automorphism group G acting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms of G, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves.

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“…In [10] and [11] it is shown that symmetric and affine 2-designs D = (P, B) can be embedded in a finite commutative group in such a way that the blocks are exactly the ksets of elements of P that sum up to zero, whereas the only Steiner triple systems with this property are the point-line designs of AG(n, 3) and PG(n, 2) (see also [15], for a visual representation of the case of PG (3,2)). Furthermore, the only known Steiner 2-design over a finite field, found by Braun et al [5] and revisited in [8], can be seen as a 2-(8191, 7, 1) design with the property that the points on each block sum up to zero.…”

Section: Introductionmentioning

confidence: 99%

Binary Hamming codes and Boolean designs

Falcone

Pavone

2021

Des. Codes Cryptogr.

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In this paper we consider a finite-dimensional vector space $${\mathcal {P}}$$ P over the Galois field $${\text {GF}}(2),$$ GF ( 2 ) , and the family $${\mathcal {B}}_k$$ B k (respectively, $${\mathcal {B}}_k^*$$ B k ∗ ) of all the k-sets of elements of $$\mathcal {P}$$ P (respectively, of $${\mathcal {P}}^*= {\mathcal {P}} \setminus \{0\}$$ P ∗ = P \ { 0 } ) summing up to zero. We compute the parameters of the 3-design $$({\mathcal {P}},{\mathcal {B}}_k)$$ ( P , B k ) for any (necessarily even) k, and of the 2-design $$({\mathcal {P}}^{*},{\mathcal {B}}_k^{*})$$ ( P ∗ , B k ∗ ) for any k. Also, we find a new proof for the weight distribution of the binary Hamming code. Moreover, we find the automorphism groups of the above designs by characterizing the permutations of $${\mathcal {P}}$$ P , respectively of $${\mathcal {P}}^*$$ P ∗ , that induce permutations of $${\mathcal {B}}_k$$ B k , respectively of $${\mathcal {B}}_k^*.$$ B k ∗ . In particular, this allows one to relax the definitions of the permutation automorphism groups of the binary Hamming code and of the extended binary Hamming code as the groups of permutations that preserve just the codewords of a given Hamming weight.

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Kirkman's Tetrahedron and the Fifteen Schoolgirl Problem

Cited by 12 publications

References 7 publications

Teaching Combinatorial Principles Using Relations through the Placemat Method

Teaching Combinatorial Principles Using Relations through the Placemat Method

The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Binary Hamming codes and Boolean designs

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