To my admittedly amateur eyes, it seems like Napoleon's strategy at Waterloo was to split the Prussians and English so he could defeat each in turn (rather than allow them to combine, outnumber, and overwhelm him as they did). The turning point to me seems to hinge on Grouchy's inability to drive off the Prussians, which ultimately allowed Blucher and his forces to engage Napoleon's right flank that afternoon, sealing the French general's fate.
My question: was Napoleon justified in taking this risk based on the intel that he had? I don't mean to turn this into a history-what-if discussion, but based on what Napoleon knew at the time, was his gamble that Grouchy could keep the Prussians out of the fight a reasonable one, or was this a desperate/hubris-inspired mistake with little to back up its assumption(s)? Or is this question coming from a faulty place in that Napoleon would have lost regardless of the Prussian presence on the field?
After the Battle of Ligny, Napoleon believed the Prussians were finished, and would withdraw towards Namur and the Rhine. To briefly address these respective suppositions:
The fact that the Prussians still had plenty of fight in them should have been apparent from the trophies of Ligny. Napoleon had only captured about 27 cannon there, not a significant number when facing an army of 120,000 men with more than 300 guns; as a point of comparison, at Austerlitz, the French had captured 180 pieces. Nor were the numbers of prisoners very great, at most a few thousand. As such, we can say his turn towards Wellington was premature; it's not exactly speculation to say inflicting a second defeat on an army is easier than facing one yet unconquered.
The belief that the Prussians were withdrawing east towards Namur, instead of due north towards Wavre, was founded first on the fact that their base and home country lay in the east, and second on the capture of an isolated artillery battery moving west [and then attempting to escape east] on the road to Namur. Other pockets of Prussians also retreated towards Namur and Liege, but the bulk of the army retreated north through Tilly to Wavre. Napoleon continued to hold this belief on the 18th, when he did order Grouchy's wing to march on Wavre in response to reports of Prussians in that direction, thinking there was only an isolated column there, with the rest of the army withdrawing further east. The fact that a withdraw due north to keep in contact with Wellington was certainly the most dangerous course of action open to the Prussians does not seem to have entered into Napoleon's calculations. No reconnaissance efforts were launched in this direction on the 17th.
On the basis of Napoleon's belief the Prussians were in full retreat away from Wellington, Grouchy was directed to pursue, which it should be noted is a very different task from observing and containing. Napoleon wasn't thinking of Grouchy holding the Prussians back, but of driving them off. To this effect, Grouchy's columns were directed through Gembloux, to the northeast, thus forcing them to grope their way back northwest towards Wavre. This gave Blucher the 'inside track' to the Waterloo battlefield on the day of destiny. Blucher would have had a much harder time linking up with Wellington if Grouchy's wing had gone due north through Tilly up to Wavre, but this was not Napoleon's intention for Grouchy. Grouchy's corps did fail its function in gathering accurate intelligence, though; when Napoleon did receive the reports on the 18th, they indicated the Prusssians were marching all the way north to Brussels, rather than due west towards the battlefield.
As such, we can see that Napoleon's intended role for his right wing on the 17th and 18th was based on wishful thinking and sloppy reconnaissance.