Relationship between Henry VII and Richard III

Henry VII and Richard III are second cousins once removed through Richard's mother. 

Henry VII and Richard III and third cousins once removed through Richard's paternal grandfather. 

They are fourth cousins once removed through Richards paternal grandmother. 

Henry VII and Richard III relationship through the Beaufort line.  Richard III's mother descended from a daughter of John of Gaunt and Henry VIII's mother descended through a son of John of Gaunt.  Henry VII's mother and Richard III were second cousins.  Henry VII and Richard III were thus second cousins once removed.

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Yea, you’re right. They are 2nd cousins 1xs removed, at least through the line that they share with their common ancestor, John of Gaunt, on Richard III’s mother’s side through Cecilly Neville up through the Beauforts.

And although they are also cousins on two other separate lines going through their common ancestor, King Edward III, they’re still not 3rd cousins.

On Richard III’s mother’s side, they share the same common ancestor, John of Gaunt, who is Richard III’s great grandfather and Henry VII’s 2nd great grandfather. This makes them 2nd cousins, 1xs removed on this line.

Since Edward III is father of John of Gaunt, this makes Edward III Henry VII's 3rd great grandfather.

In addition to the same line for Richard III going up to King Edward III through their common and closest ancestor John of Gaunt, Richard III also has two more separate lines that go up to Edward III through two separate brothers of John of Gaunt as well.

On Richard III's father side on his line going through paternal grandfather, it goes up to Edward of Langley, Duke of York, and on this line, King Edward III is Richard III's 2nd Great Grandfather. And since Edward III is Henry VII’s 3rd great grandfather through John of Gaunt, this also makes them 3rd cousins 1xs removed on this line of Richard III’s through Edward of York.

However, Richard III also has yet another line through his paternal grandmother, Anne Mortimer, daughter of Roger Mortimer V, son of Philippa Plantagenet, daughter of Lionel of Antwerp, Duke of Clarence, second oldest son after the Black Prince to Edward III, and through this line, King Edward III is Richard III's 4th great grandfather on that line, which makes them 4th cousins 1xs removed on this line since Edward III is Henry VII’s 3rd great grandfather.

So yea, they’re cousins 3 times over, but none of those lines are straight up 3rd cousins. Either 2nd cousins, 1xs removed via John Beaufort and Richard III’s mother’s line, or they are 3rd cousins 1xs removed or 4th cousins 1xs removed if going through their connection to Edward III on the two other lines Richard III has up to Edward III on his father’s sides