I've tried to solve the Knapsack problem the other day but without the so called "trick" from Andrew Lucas (https://www.frontiersin.org/articles/10.3389/fphy.2014.00005/full, 2.4. Reducing N to logN Spins in Some Constraints, 5.2. Knapsack with Integer Weights).
I could not solve it so far I'm still trying and just wondering whether it can be solved or not without the "trick"? It should be possible, right?
From Lucas's paper I've converted the formulas and marked the constraints with letters so you can identify them in my approach:
I've commented out the original solution and added my constraints next to the original constraints so you can compare them. They look similar except I don't use the "trick" and 2 of my constraints (a and b) are missing from the original solution.
Is my approach correct at all? Any suggestions?