Balance
Posted: Wed 28 Jun, 2023 7:22 pm
So after that Puzzle GP round, it's left me wondering how I'm supposed to be solving Balance puzzles. It can't be the trial and error approach I've been doing so far. Any tips please?
A couple of things I found helpful were a) the total weight can be used as a constraint, and b) you can sometimes cut down on options if some of the balances imply factors. 2023 round 5 dropped out very nicely for me with these - if you label the total weights hanging off each pivot by A, B, C and D from left to right then you get
A + B + D + weight on top level = 45 (sum of all weights)
3 (A + B) = 3D + 2*weight on top level (moments about pivot C)
Rearranging gives
3(A+B-D) = 2*weight on top level
A+B+D = 45-weight on top level.
The first of these implies that the weight on the top level must be a multiple of 3. But by also using the "total weight" constraint, you can also get
3(A+B+D)=135-3*weight on top level
And then combine this with the torque balance to get
6D = 135-5×weight on top level
This only works if the weight on the top level is 3 or 9. If its 9, then D=15, and by doing a similar analysis of the weights around pivot C, you can figure out that none work.
I dont know if there's any clever way of picking which pivots to use and combine - I guess as a rule of thumb quick progress is possible if large factors are forced. There's also a global constraint that you can't reuse weights, which often helps cut down options or makes some options mutually exclusive. I'd also suggest caution on the total weight constraint - I went in circles for a long time last weekend before rediscovering that the sum from 1 to 10 is not 45...