12 people are split into groups of 4....... My friend sent this today as she is putting together a workshop: 
There will be 12 people invited to the workshop.  In the workshop, we will have 4 sessions.  In each of the four sessions, these 12 people will be divided into groups of four.  We want every person to be in a group with each of the other 11 people in at least one of the four sessions.  
Is this possible, and if so, what is the arrangement?  I made a chart, but I'm not having any luck finding a way to arrange so that every person will be with every other person at least once.
 A: We have a $4\times 3$ grid, with four "seats" in each cell.   Each of the twelve people must attend one group in each of the four sessions.   Let's fit in the first person.
$$\sf(A\;\_\;\_\;\_)(A\;\_\;\_\;\_)(A\;\_\;\_\;\_)(A\;\_\;\_\;\_)\\ (\_\;\_\;\_\;\_)(\_\;\_\;\_\;\_)(\_\;\_\;\_\;\_)(\_\;\_\;\_\;\_)\\ (\_\;\_\;\_\;\_)(\_\;\_\;\_\;\_)(\_\;\_\;\_\;\_)(\_\;\_\;\_\;\_)$$
The only way any person can share at least one session with each of the other eleven people is if there is only one other person with which two sessions are shared. Thusly we can fill in people who share sessions with $\sf A$, and let the repeat be $\sf B$.
$$\sf(ABCD)(AEFG)(AHIJ)(AKLB)\\ (\_\;\_\;\_\;\_)(\_\;\_\;\_\;\_)(\_\;\_\;\_\;\_)(\_\;\_\;\_\;\_)\\ (\_\;\_\;\_\;\_)(\_\;\_\;\_\;\_)(\_\;\_\;\_\;\_)(\_\;\_\;\_\;\_)$$
Likewise, the only way that person can share at least one session with each of the other eleven people is if the only person with whom two sessions are shared is the first person.
$$\sf(ABCD)(AEFG)(AHIJ)(AKLB)\\ (\_\;\_\;\_\;\_)(B\_\;\_\;\_)(B\_\;\_\;\_)(\_\;\_\;\_\;\_)\\ (\_\;\_\;\_\;\_)(\_\;\_\;\_\;\_)(\_\;\_\;\_\;\_)(\_\;\_\;\_\;\_)$$
Now $\sf C$ cannot share any session but the first with $\sf A$ or $\sf B$. 
$$\sf(ABCD)(AEFG)(AHIJ)(AKLB)\\ (\_\;\_\;\_\;\_)(B\_\;\_\;\_)(B\_\;\_\;\_)(\_\;\_\;\_\;\_)\\ (\_\;\_\;\_\;\_)(C\_\;\_\;\_)(C\_\;\_\;\_)(C\;\_\;\_\;\_)$$
Do the same for $\sf D$ who cannot share ... oh, dear...
