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Topic: problem of arrangement
Replies: 1   Last Post: Jan 2, 1997 1:31 PM

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Henry W

Posts: 3
Registered: 12/8/04
problem of arrangement
Posted: Dec 29, 1996 8:09 AM
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Hello to all mathematicians,

i have a problem and hope you guys can help...

let say i have 16 objects (A,B,C,D,E....P) or (A1,A2...A16) what ever
you want to call them, only 8 objects is allowed in one row, and each
object appear 5 times, that is 5 * 16 gives 80 objs all together, right,
that means 10 rows of 8....
each object can only appear once in each row... arrange them in any
order,...
eg.
A-B-C-D-E-F-G-H (eight in one row)
B-C-D-F-G-I-J-K (another eight ...)
.
.
.
C-F-G-H-I-L-O-P (total of 10 rows)

now, it may seem pretty easy, but there are some conditions...
i) each object must be in a row with the others at lease 2 times
ie. A will have at least two Bs followed in any of the 5 rows containing
A, two Cs, two Ds,... so on
ii) no object will have 4 or above of another object in the same rows
eg. A-B-...F.....; B-C....-F....; B...F.....; ..B..F....; is not allowed
(max 3 B-F combinations)

at close investigation you will see that each object only appear 5
times, let say K, each row will have only 8 objects, that is only (8
objects * 5 rows) = 40, take out 5 Ks, 35 more objects are needed to
fill the rows of Ks. only 16 objects, again take K out, 15 objects
available to fill 35 spaces, thus 5 of the 15 objects must with K 3
times to fill all the rows of K. the same conditions apply to all of the
objects, so, say P, can only appear 5 times, P will overlap with all the
15 objects (including K) twice, 5 of those are to be with P 3 times....
so on.

1) K-. . . . . . need 7 more to complete
2) K-. . . . . . all together need 35 objects to fill rows with K
3) K-. . . . . . 15 object leave to choose (A-P, except K)
4) K-. . P . . . (15 * 2 = 30 only, 15 * 3 = 45 not enough spaces)
5) K-. . . P . . so 5 of them must be in the A rows 3 times
6) P-. . . . . . say P is only with K twice, 3 Ps are left.
7) P-. . . . . . also 5 of the other object must be with P 3 times, but
not K
8) P-. . . . . . now only 14 objects to choose from (A-P, excpet K & P)
9) ............. HOW?
10) ............ HELP!!

Can this be done??
If so, what if we always have more objects than we can arrange in one
row.
eg. 12 objects, arrange in rows of 6, each has to overlap with all the
others, say each appear 4 times, 12 * 4 = 48, we need 8 rows of 6 to
allocate all objects. (4 rows for each obj) * (6 in one row) = 24, 24
- itself = 20 places to fill, and (12 object - 1 object)=11, thus 2
objects are to be with that particular once and 9 objects are to be with
it twice, 9 * 2 + 2 * 1 = 20. Or any problem alike. Is there any general
rules?

i don't know if you can understand what i mean,... i have spent weeks to
solve that problem, it just didn't come right (maybe i am too dump),...
see if you can help or you may forward this to any mathematicians you
know, thanks very much.

HAPPY NEW YEAR........

regards,

HENRY.





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