For questions about Combinatorial design theory, a part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry. The theory has applications in the area of the design ...

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How do you create projective plane out of a finite field?

I have heard and read unclear mentions of links between projective planes and finite fields. Is it possible to construct a projective plane (or a Steiner system) starting out with a field? Could you, ...
9
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3answers
267 views

Question about members in sets

Let $A_1,A_2,...,A_n$ be sets with $k$ members in $A_i$ for every $1\le i\le n$. Suppose that the $A_i$ satisfy: 1) $|A_i\cap A_j| = 1$ for all $i\ne j$, 2) $A_1\cap A_2\cdots\cap A_n =\emptyset$. ...
8
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3answers
337 views

Symmetries of combinatorial structures.

Studying the automorphism groups of graphs/finite geometries/designs has been quite useful and important for both group theory and combinatorics. I know of the following books which cover the ideas ...
7
votes
1answer
229 views

What is the minimum number of guesses in order to guarantee to win the prize?

Your friend will pick a $4$-letter word and you will make guesses in order to find it. -A word can contain only the letters $A, B, C,\:\text {and} \:D$, and they can be used more than once. ...
7
votes
1answer
116 views

How many different ways can the signs be chosen so that $\pm 1\pm 2\pm 3 … \pm (n-1) \pm n = n+1$?

How many different ways can the signs be chosen so that $\pm 1\pm 2\pm 3 ... \pm (n-1) \pm n = n+1$? This is an extension of this question: For what $n$ can $\pm 1\pm 2\pm 3 ... \pm (n-1) \pm n = ...
7
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0answers
124 views

Minimal number of animals in a matching card game

I saw a card game designed for small children. Each card has a picture of 6 animals on it, and there are 31 cards. When any two cards are compared to each other, they share exactly one animal. The ...
5
votes
2answers
855 views

Is there a memorable solution to Kirkman's School Girl Problem?

Given a solution to Kirkman's School Girl Problem, it is of course easy enough to check that it actually is a solution. But how could you reconstruct it if you lost it? Is there a method or algorithm ...
5
votes
3answers
529 views

Orthogonal Latin Square 6*6

I need to make remarks about Tarry's Proof for the nonexistence of 6x6 Latin Squares as part of my final exam for a class I'm in. Problem is, I can't find it ANYWHERE on the internet. I can only find ...
5
votes
2answers
414 views

Applications of design theory

I have recently started reading up on design theory, with the ultimate purpose of doing some original research in that area. I understand the mathematics fairly well, but am not understanding the ...
5
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1answer
208 views

Number of combinations such that each pair of combinations has at most x elements in common?

I am doing research on the sense of smell and have a combinatorics problem: I have 128 different odors (n) and I mix them in mixtures of 10 (r). There are 2.26846154e+14 different mixtures. What I ...
5
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1answer
367 views

A generalization of Kirkman's schoolgirl problem

A friend of mine asked me this question. "I have $3n$ elements, and I want to know which is the maximum number of triplets $(a,b,c)$ so that no two triplets have more than one element in common". The ...
4
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1answer
248 views

Finding a system of sets resembling the projective plane

For every natural number $r$ I wish to construct a family of sets $\mathcal{F}$ such that Every set in $\mathcal{F}$ has cardinality $r$ and is a subset of $\{1,2,\ldots,r^2-r+1\},$ For every ...
4
votes
2answers
506 views

Minimum number of X-subsets needed to cover all K-subsets

Assume I have a universe of N elements. The question is: How many sets of size $X$ are needed to assure that every set of K elements is a subset of (at least) one of these sets (where $K \ll X \lt ...
4
votes
1answer
259 views

the table at the end of Theoretical Computer Science Cheat Sheet

Theoretical Computer Science Cheat Sheet, created by Steve Seiden, is a hodgepodge of well-known mathematical theorems and notions. I can understand (or guess at least) many of them, but I'm not sure ...
4
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1answer
83 views

Steiner Triple Systems block clique

Given a Steiner Triple System (STS) of order $v$, one can build its graph in the following way: each vertex is a block, and two verticies are adjacent if their blocks have nonempty intersection. Thr ...
4
votes
1answer
98 views

Diner Combinations, Each Pair Sits Together Exactly Once

There are $N^2$ guests at a party. How can we seat these guests at $N$ tables, in a number of rounds, so that each guest sits with every other guest exactly once? I've come up with an algorithm that ...
4
votes
1answer
52 views

Steiner triple system with $\lambda \le 1$

What's the maximum number of 3-sized subsets of $[n]$ that can exist such that no two subsets contain more than one common element? When $n \equiv 1,3 \mod 6$ then this is equivalent to a Steiner ...
3
votes
2answers
136 views

How large can a set of pairwise disjoint 2-(7,3,1) designs (Fano planes) be?

As wikipedia defines well, the Fano plane is a small symmetric block design, specifically a 2-(7,3,1)-design. The points of the design are the points of the plane, and the blocks of the design are the ...
3
votes
2answers
178 views

Geometric solution to classic committee problem

Most people know the classic committe style problems. I read this solution to one of the basic version of the committe problem and was impressed, but not sure why it works. I was hoping someone ...
3
votes
1answer
72 views

how to make a $3$-$(10,4,1)$ design using graphs

A $t$-$(v,k,\lambda)$ design is defined this way : We have a set with $v$ elements (called the points). We also have a collection of distinguished subsets each having $k$ elements, we call each of ...
3
votes
3answers
56 views

Points necessary to intersect all lines in finite projective geometry

I'm reading about finite geometries, projective and affine. I wonder what the smallest set of points is, given a geometry $PG(d,q)$, that intersects all lines. (or hyperplanes.) For example in the ...
3
votes
2answers
128 views

combinatorics question from the real world

This is a real world scenario - please help. My brain hurts and I can't figure it out on my own. Suppose I host an event with the following constraints There will be exactly 5 lectures There will ...
3
votes
1answer
102 views

Finding symmetric commuting matrices $A,B,C,D \in M_n(1,-1)$ such that $ A^2+B^2+C^2+D^2=4nI_n $

I am trying to construct a Hadamard matrix of order 28 using Williamson's construction. But I am unable able to construct the necessary symmetric and commuting matrices. Definition: $H_n \in ...
3
votes
1answer
54 views

Forming clubs with an odd number of members, with each pair of clubs having an even number of members in common

Suppose we have a town with $n$ residents who love forming groups. To limit the number of groups, the town head decided: 1) Every club must have an odd number of members, and 2) Any two clubs must ...
3
votes
1answer
38 views

Trying to find a combinatorial design which describes my problem

A group of 8 golfers would like to play in teams, split into two teams of 4, with a different arrangement of teams on each of 5 consecutive days; they would like each pair of players to be on the same ...
2
votes
2answers
42 views

Known classes of Hadamard matrices

In the book Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices by Wallis et al., Appendix A of the chapter on Hadamard matrices gives a list of known classes of Hadamard matrices. However, ...
2
votes
1answer
69 views

Special subspace in vector space over $\mathbb F_5$

Let $\mathbb F_5=\{0,1,2,3,4\}$ is finite field of size $5$. I am trying to find minimal $n$ so that vector space of dimension $n$ over $\mathbb F_5$ contains $2$ linearly independent vectors so that ...
2
votes
1answer
94 views

Hypergraph terminology

Suppose I have a hypergraph with vertices V and hyperedges H, where each hyperedge is a subset of V. I want to form a normal graph with vertex set V, where two vertices are adjacent if they lie in ...
2
votes
1answer
150 views

Determining the size of an automorphism group for a given design

I'm trying to wrap my head around the idea of automorphisms, and I'm having a lot of issues. One of the questions I've been given as an exercise is thus; Let $\mathbb{V} = \{1, 2, 3, 4, 5, 6\}$ ...
2
votes
1answer
46 views

Choosing a committee from two people who are not sitting beside each other.

Assume that $10$ people are sitting around a table. Determine the number of ways to choose a committee, where the committee is made up of two people who are NOT sitting next to each other. Take ...
2
votes
1answer
108 views

Pairwise balanced designs

Let $X$ be a finite set containing $v$ elements and $\lambda$ be a positive integer. Let $K$ be a set of positive integers. Further let there be a multiset $\mathcal{B}$ containing subsets of $X$ ...
2
votes
1answer
33 views

Balanced incomplete Block design for testing an experiment

I am reading something balanced incomplete block design from a book. I don't understand why is it easy to see that in this design Each vehicle is evaluated 8 times, each test driver evaluates 4 ...
2
votes
1answer
102 views

unbalancing lights

I'm reading the following notes on unbalancing lights, http://www.cs.berkeley.edu/~sinclair/cs271/n5.pdf. The question i have is regarding the first page. Where it says Consider a square $n ...
2
votes
2answers
64 views

Terminology - variant of a hypergraph

In a hypergraph, we have vertices $V$ and hyperedges $H$, where each hyperedge is a subset of $V$. Suppose that we would like the hyperedges to be (ordered) tuples, rather than subsets. Does this ...
2
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0answers
61 views

Is Einstein's riddle an example of a combinatorial design?

I have just learned a bit about combinatorial designs (BIBDs, constructing a ($b,v,r,k, \lambda$)-design, necessary conditions for a design, cyclic designs) and it reminded me a lot of Einstein's ...
2
votes
1answer
88 views

How does the multiplicative group of a finite field, considered as a vector space, act on subspaces?

Given that a finite vector space $V = \operatorname{GF}(p)^n$ corresponds to the finite field $F = \operatorname{GF}(p^n)$, I'm wondering about how the multiplicative subgroup of $F$, $F^*$, acts on ...
2
votes
0answers
87 views

$24$ people in groups of $3$ where everyone meets exactly once at the end of some number of rounds

I was presented with this problem at work. Say you have $24$ people and $8$ tables in a room. You want to set people at these tables in groups of three such that during each new round (where people ...
2
votes
0answers
97 views

Out of all combinations (n,k), largest set such that each combination overlaps with others by d or less.

This problem is relevant to determining the number of discriminable combinations of components in a sensory perception task. Suppose that there are N components to choose from, and we are only ...
2
votes
0answers
87 views

Configuration analogues of projective spaces?

In a configuration, each point is incident to the same number of lines and each line is incident to the same number of points. The Fano plane is a configuration, with 3 points on each line, and 3 ...
2
votes
0answers
48 views

Question related to designs where t= 2

1)Let D be a 2−(v,k,λ) design with b blocks and r blocks through every point. Let B be any block. How to show that the number of blocks that meet B is at least $k(r−1)^2/[(k−1)(λ−1)+(r−1)]$ 2) How to ...
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vote
2answers
240 views

France Olympiad Team Selection Test 2005

In an international meeting of n ≥ 3 participants, 14 languages are spoken. We know that: - Any 3 participants speak a common language. - No language is spoken by more than half of the participants. ...
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2answers
82 views

Scheduling gym class

My cousin came to me with this problem yesterday: She has 8 students in her gym class. In tomorrows class she has planned 4 different activities to rotate them through, each of which requires ...
1
vote
2answers
205 views

Grouping Problem

Suppose there are 9 strangers. We will assign them into 3 groups and each group has exactly 3 people. For each grouping, the strangers who were assigned into the same group will get to know each other ...
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1answer
32 views

How can I create 14 unique team rosters from 24 students in groups of 2?

I have a class of 24 students, and they have 14 labs in which I have students partner up. I am trying to find a way to create a roster automatically such that none of my students has a repeat partner ...
1
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1answer
75 views

Combining kindergardeners in 'fair' cookie-baking groups. Kirkman's schoolgirl problem extended version

I am coordinating cookie-baking events with kindergarten kids. This turns out to be a challenging problem, and I could use a little help: We would like a general way of creating 'fair' cookie-baking ...
1
vote
1answer
46 views

Show that it is possible to guarantee a win by buying $14$ tickets.

You enter a lottery by picking a subset of three numbers from $\{1,2,3,4 \dots 14\}$ . You win a prize if you match at least two of the numbers on the winning ticket. Show that it is ...
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2answers
58 views

Choosing sets of vectors on a complex sphere

Consider a complex $t$ dimensional unit sphere. Can we have $t$ sets of $2^t$ vectors $v_{ij}\in \Bbb C^t$ on the sphere where $i=1$ to $t$ and $j=1$ to $2^t$ on this with inner products satisfying ...
1
vote
1answer
385 views

perfect binary e-error correcting code

let C be a perfect binary e-error correcting code of length n. assume 0 is a symbol and that 0 vector is a codeword.show that P={1,2,...,n} together with supports of codewords of weight 2e+1form an ...
1
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1answer
21 views

Minimum length $m$ of $n$ string with pairwise Hamming distance $m/2$

I want to construct $n$ binary strings, each of the same length $m$ (to be determined), such that each pair of string has Hamming distance exactly $m/2$ (i.e. the strings disagree on $m/2$ positions). ...
1
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1answer
29 views

a problem with tasters-combinatorial design theory

I have a collection of flavors being sampled in batches of $3$. I know that each pair of flavors occur together in exactly one batch. Also each flavor appears in the same number of batches. How can ...