“How can three be one and yet be distinct at the same time?”
Three Persons, One God.
It is an ineffable mystery, but it is not a case of 3 equaling 1 and not equaling 1 at the same time.
The Trinity has three Persons. We know this by Christ’s revelation of God.
But we state that God has more than one person each and every time we say that “God is Love”.
For if God was/contained only one person then who would He love while being Himself, whole and entire for the whole immensity of eternity that existed before He created angels and men?
Only the doctrine of the Trinity captures this mysterious dance at the center of all things - that God is Love and also the unselfish object and subject of that Love.
I believe one of the best ways to describe the Trinity is from a Chemistry persepective: Water! Water can be solid (ice), liquid (water), or gas (steam or water vapor). Each of these forms of water is different (solid, liquid, or gas) but each are exactly the same! Water! Even Christ is sometimes described as the Living Water.
These are the three phases of nature, and each pure substance can exist in the three phases. Maybe a little reminder for us in the works of God in His universe.