Catching falling characters

In my experience, whenever a character falls because of a pit trap or a botched climb check there is always at least one player that that tries to catch them. I can’t seem to find any specific rulings to this.

I’m not the sort of GM that likes to say, ‘no, you can’t do that’ so I at least let an adjacent player make a reflex save if they ask (using the same DC as the trap save seems like a good idea).

Then perhaps a Strength check to pull them up (or avoid being pulled in themselves).

Does anyone have an official rule to this, or at least a solid house rule that can be used consistently?

Edit I’ve done a bit of snooping around and found SRD rules for catching character whilst climbing, which covers part of my question.

Catching a Falling Character While Climbing

If someone climbing above you or adjacent to you falls, you can attempt to catch the falling character if he or she is within your reach. Doing so requires a successful melee touch attack against the falling character (though he or she can voluntarily forego any Dexterity bonus to AC if desired). If you hit, you must immediately attempt a Climb check (DC=wall’s DC + 10). Success indicates that you catch the falling character, but his or her total weight, including equipment, cannot exceed your heavy load limit or you automatically fall. If you fail your Climb check by 4 or less, you fail to stop the character’s fall but don’t lose your grip on the wall. If you fail by 5 or more, you fail to stop the character’s fall and begin falling as well.

It would make sense that catching a character who is falling into a pit would use a similar mechanic, though probably replacing the climb check with something else.

Catching ball – finding the maximum number of caught balls

I’m attempting to solve the following problem:

Balls are falling from the sky. We know at which location (on a straight line) will each ball drop, and we know the time (in seconds) at which will the ball reach the ground. We are trying to catch them into a ball net, which we can move left or right, but each movement costs 1 second. The initial position of a ballnet is always on the left (position 0). We are allowed to > drop (not catch) $ k$ number of balls.

What is the highest score we can achieve?

My first attempt at solving this was a greedy algorithm:

if the |next ball position - current position of the ball net| > (time of the next ball - current time)       then attempts++      if attempts>$  k$              print game over else      current ball net position = next ball position      current time = time of the next ball      score++ 

however my algorithm doesn’t take into consideration that sometimes it’s better to sacrifice some number of balls in order to reach a higher score in the long run. This needs an approach via dynamic programming, I think.

Is this problem a known one so I can find some help? Could you help me with this problem? I can solve this in a greedy way, however I am failing to do it dynamically.

Thanks!

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