Quick guide
How to use this calculator
- Choose the question you need answered.
- Enter the number of boxes and either objects or target occupancy.
- Read the guaranteed minimum.
Calculation method
Use the generalized pigeonhole bound
Distributing n objects among k boxes guarantees one box contains at least ceiling(n/k). To guarantee at least m objects in one box, k(m−1)+1 objects are sufficient and necessary.
Worked example
Thirteen objects in four boxes
At least one box must contain four objects.
⌈13/4⌉ = 4
Supported inputs
Precision and limits
Discrete domain
Inputs are whole counts. Negative, fractional, grouped, and scientific-notation inputs are rejected.
Exact integers
Results use arbitrary-precision integers and are never rounded. Inputs and output size are bounded to keep the page responsive.