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My knees aren’t in the best shape, one in particular. I had a rather devastating 13 hour up-and-down hike on day two of a three day hike, and the last day my knees were hurting every step. What are some ways to take some load off the knees, or other ways to prevent this? Not including ‘taking more time for the hike’. Assume already decent footwear and some kind of stick to balance and spread weight.

Carrying less weight makes a big difference. My packing follows the "be prepared" style more than the ultralight one, but even I admit it's easy to carry too much.

Stepping cautiously especially on descents and trying to avoid dropping too much on one step are both important.

A pair of trekking poles is very helpful, and you might need to lengthen them for descents/shorten them for ascents.

Assuming you are not opposed to it, preemptively taking an anti-inflammatory such as ibuprofen before your knees start hurting and then re-dosing later if it is a long hike can help to keep swelling / inflammation down.

Another thing that can help when going down a particularly steep downhill; if the trail is wide enough try zigzagging your path. This will slow you down some, although probably less than if you were trying to descend while in pain. Zigzagging essentially decreases the slope that you are descending, thereby decreasing the impact on your joints with each step.

Also, make sure that your hiking poles, or stick, are the correct height for you. If you have to reach your arm downward significantly to bear weight or stabilizing force on the pole it will not do very much to help.
ETA: two proper hiking poles of correct length will serve you much better than just one stick or pole.

Well, I would say taking more time is a very valid option. Also if you know you have knee trouble you could reduce distances to an extent where you do not have to walk for hours with exhausted muscles. From my experience, pushing yourself over the point where the joints lose sufficient muscle support is when you start to have fun the next days.

Therefore, preparation and training well in advance of the trip is very important, and you should not go on a long trip if you are out of shape. I did ignore this a few times in the past, and it was not a big issue, but I had to take the consequences during the first, let's say 3 or 4 days.

Honestly speaking, I would also say that after a 13 h walk almost everybody will feel the strain in one way or another, especially if you carry extra weight as usually on a multi-day trip, and knee pain is extremely common. So you might also have to face the fact that there is no reasonably healthy way to avoid it, unless you have a lot of time for getting used to the strain by training.

Another option is to analyze if there are any aspects in your walking technique which cause your knee problems. I have no idea how that is exactly done, maybe other people can comment on that.