Balance
Re: Balance
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...
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...