Your 4 coinflips (denoted here by X) will yield an expected value of 2 times head with a very, very approximated normal distribution of the values 0-4 for this event X. However, the sample sice is very small so that I would not use this approximation. However, if you take the variance of this random variable you get the sum of (X -2)^2 which is indeed chi-squared distributed if you claim your approximation to be valid. Considering your degrees of freedom you should be able to compute your theoretical distribution by chi-squared.
PS: Did he really make you flip 20 coins 100 times = 2000 coinflips?