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.