  We study the effects of repulsive four-body interactions of $\alpha$
particles on nuclear $\alpha$-particle condensates in heavy self-conjugate
nuclei using a semi-analytic approach, and find that the repulsive four-body
interactions could decrease the critical number of $\alpha$ particles, beyond
which quasi-stable $\alpha$-particle condensate states can no longer exist,
even if these four-body interactions make only tiny contributions to the total
energy of the Hoyle-like state of $^{16}$O. Explicitly, we study eight
benchmark parameter sets, and find that the critical number $N_\text{cr}$
decreases by $|\Delta N_\text{cr}|\sim1-4$ from $N_\text{cr}\sim11$ with
vanishing four-body interactions. We also discuss the effects of four-body
interactions on energies and radii of $\alpha$-particle condensates. Our study
can be useful for future experiments to study $\alpha$-particle condensates in
heavy self-conjugate nuclei. Also, the experimental determination of
$N_\text{cr}$ will eventually help establish a better understanding on the
$\alpha$-particle interactions, especially the four-body interactions.
