We perform a shooting experiment for the knapsack facets and observe that 1/k-facets are strong for small k; in particular, k dividing 6 or 8.We also observe spikes of the size of 1/k-facets when k = n or when kC1 divides n+1.We discuss the strength of the 1/n-facets introduced by Aráoz et al. (Math Program 96:377– 408, 2003) and the knapsack facets given by Gomory’s homomorphic lifting. A general integer knapsack problem is a knapsack subproblem where a portion, often a significant majority, of the variables are missing from the master knapsack problem. The number of projections of 1/k-facets on a knapsack subproblem of l variables is O.(⌈k/2⌉), note that this is independent of the size of the master problem. Since 1/k-facets are strong for small k, we define the 1/k-inequalities which include the 1/d-facets with d dividing k and fix k to be a small constant such as k / 6 or k = 8. We develop an efficient way of enumerating violated valid 1/k-inequalities. For each violated 1/k-inequality, we determine its validity by solving a small integer programming problem, the size of which depends only on k.