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Suppose a magazine editor wishes to obtain a comparison of 25 automobiles by assessing the opinions of a certain number of test-drivers after each of the test-drivers evaluates 3 of the vehicles. Construct a block design for this comparison. List the values of the parameters $(v,b,r,k,\lambda)$ for the design, and state what each parameter represents.

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Your way of phrasing the question makes it look as if you're passing on to us something written by someone else for use in a context different from this forum, specifically in a homework assignment. It's perfectly OK to ask for homework help here, but writing "Construct a block design..." and "List the values..." are the kind of language used when giving someone instructions. It's not so much that you're doing that that is irritating; it's that merely copying rather than asking your own questions makes it look as if you haven't though about it enough to know what to ask. –  Michael Hardy Jan 31 '12 at 0:19
    
I thought this was a forum, where we communicated back and forth. The last question I posted, I kept saying what I was trying to do and I got zero responses. Before I spend 10 minutes explaining what I tried to do, I would like to at least see if someone is actually interested in helping me. Please do not assume that since I did not post what I tried, I am just demanding an answer. –  Jackson Hart Jan 31 '12 at 0:29
    
There are standard algorithms to solve CSPs and you can just apply them once you fit a CSP for the BIBD design you have. Else, you can just use MATLAB or MAPLE to enter the CSP and it will give you out an output. I hope this will help. I am sorry, I might have missed out your comment the last time you requested my help. One of the issue with BIBD is that for any set of parameters, there is no guarantee of uniqueness. So, you might have to do some trick if you using any mathematical tool. I have used that algorithm to generate a BIBD, so I am sure, it works! –  Jalaj Jan 31 '12 at 0:35
    
So my attempt was thus: I know that vr = bk and that (v-1)lambda = r(k-1). From there I just plugged in the numbers and used substitution to solve. And I get (25,100,12,3,1). Is there a way to check this answer? I know some of them do not actually work. –  Jackson Hart Jan 31 '12 at 0:37
    
I am trying to use Maple. –  Jackson Hart Jan 31 '12 at 0:38

1 Answer 1

up vote 1 down vote accepted

Indeed you have the parameters right.

  • The number of points $v=25$.
  • The number of points in a block $k=3$.
  • Every pair of points belongs to a unique block, so $\lambda=1$. (This is somewhat implicit in the question, since it's about BIBDs.)
  • The number of blocks containing a given point $r=12$, since there are $24$ other points, and, given any point, each block contains two "other" points.
  • The number of blocks $b=100$, which can be deduced from the above.

These parameters can also be achieved. A cyclic Steiner triple system on 25 nodes can be constructed from the starter: $$S:=\{\{0,1,6\},\{0,2,10\},\{0,3,12\},\{0,4,11\}\}.$$ These triangles are illustrated below:

Triangle 1

Triangle 2

Triangle 3

Triangle 4

We can see the each node distance occurs exactly once, so by cyclically rotating these triangles, we generate every edge exactly once.

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