He gave all the gold bars to his three daughters.
He gave Hephzibah one half of all the gold bars, plus half a gold bar.
He gave Hermione one half of all the remaining gold bars, plus half a gold bar.
He gave Hildegard one half of all the remaining gold bars, plus half a gold bar.
King Henry did not cut up any of the gold bars.
How many gold bars did he start with?
King Henry's gold bars
It is possible to form and solve an equation by starting "Let x be the number of gold bars". But the problem can be solved by working backwards.
Each daughter gets half of a number of gold bars, plus half a gold bar, leaving half a number of gold bars minus half a gold bar. So each daughter gets one more bar than the the number left after she has had her share.
Since there are no bars left after Hildegard has had her share, she gets 1 gold bar. This is half a bar more than half the bars remaining from Hermione.
So one bar remains from Hermione, and she gets 2 gold bars. This is half a bar more than half the bars remaining from Hephzibah.
So 3 bars remain from Hephzibah, and she gets 4 gold bars. Hence King Henry started with 7 gold bars