# Group 32 items into 32 groups of 4 with no 2 items appearing in the same group more than once?

I'm not a mathematician, I'm a programmer. A client wants a spreadsheet to do the following. I would just Google it, but I don't know what maths terms to use to look it up. I think it's not possible, but would love someone to either a) prove it can't be done, or b) tell me it can be done, and how. Thanks!

Take 32 people and group them into 32 groups of 4, such that no 2 people ever meet more than once.

The original spec said that there are 4 months, with 8 meetings of 4 in each month. Each person goes to 1 meeting a month, but should never see anyone they have been to a meeting with before.

I'm sure I've tagged this wrong, and I'm sure someone out there will find this trivial. Thanks for all input Andy

• Related: Kirkman's Schoolgirl Problem and Steiner Systems – JMoravitz Apr 10 '17 at 19:31
• As each person only sees $24$ others out of $31$ there should be a lot of slack in the system. You can probably just do it by hand and not have a problem. A Howell movement from bridge would do what you want but the ones I find online top out at $28$ players. – Ross Millikan Apr 10 '17 at 19:44

Let each block represent a month, and each person have the same coordinates within the block for each month. The $8$ colors each represent a meeting group for that month. As an example, the highlighted person (in tan, in that black box) will end up meeting, over $4$ months, with those $12$ highlighted in light grey. In the first month they are in the orange group, in the second the light red group, the third also in the light red, and in the fourth month in the yellow meeting group. The colors are only labels to split up the $32$ people each month; it does not matter at all that a person is in the same color group in different months.
• Note that the colors in each month form a line in a specific direction. I'm sure this could be generalized quite a bit more. I was inspired by how you might fit a much tighter problem, $16$ people, $5$ months, $4$ groups of $4$, which (I think...) involves the $1$ dimensional subspaces of the two dimensional vector space over the field of $4$ elements. – Josh B. Apr 11 '17 at 1:25
• Yes, I did notice the slopes were $0, \pm 1$, and "$\infty$". That's an interesting tighter version; I briefly considered the "right" way to find a smaller version of the problem, but gave up pretty quickly. At any rate, I was just curious. Cheers! – pjs36 Apr 11 '17 at 1:33