Birthday Problem for 3 people I know that, in a room of 23 people, there is a 50-50 chance that two people have the same birthday. However, what I want to know is: How many people do you need to have a 50-50 chance that 3 people share the same birthday?
Note: Assume for this question, that birthdays are equally distributed

Empirically the answer seems to be $87$ or $88$.
More precisely, the probability of at least three people sharing a birthday is very close to $0.50$ if there are $87$ people in total (making the standard assumption of an i.i.d. uniform distribution over $365$ days). 
Using the following R code to test a million cases for $87$ people in a room 
days   <- 365
people <- 87
cases  <- 1000000
set.seed(1)
maxhits <- function(x){ max(table(x)) }
eg <- matrix(sample(days, people*cases, replace = TRUE), nrow=cases)
table(apply(eg, 1, maxhits) )

the sample distribution for the most people sharing a single birthday was
 1      2      3      4      5      6      7 
13 500212 461918  36116   1688     50      3  

