:Let's call the nine students A, B, C, ...., I. On the first day student A can walk with B & C, on the second day he can walk with D & E, on the third day F & G , and on the fourth day H & I. Hence after 4 days student A can't be matched with anyone he hasn't already walked with. Therefore 4 days is the maximum number of days this will work. This doesn't prove that it will actually work for 4 days; only that the answer can't be more than 4 days.