How to use the Pigeonhole Principle Calculator
Given n boxes, how many items guarantee that at least one box holds k of them? The answer is n(k-1)+1. With 10 boxes and a target of 3, you need 21 items; up to 20 the items can sit 2 per box and nothing is guaranteed. Running the other direction, however you spread m items across n boxes, some box must hold at least the ceiling of m/n, and the page also shows the even-split base and how many boxes take one extra.
These results are certainties, not probabilities. The principle never identifies which box is the crowded one, and it assumes items and boxes are whole, indivisible units, which is why decimal entries are refused. The same reasoning explains why more people than available dates forces a shared birthday.
Probability questions, such as the chance of a shared birthday when there are fewer people than dates, are not covered here, and neither is distribution with per-box capacity limits. Treat the output as a reference. Current as of October 2026.
Frequently asked questions
Putting k-1 items in every box keeps all n(k-1) of them below k. Add one more item and wherever it lands, that box reaches k, so the guarantee holds from that point on.
No. The pigeonhole principle only promises that such a box exists; it never says which one. What makes it unusual is that this is a certainty rather than a probability.
With more people than available dates, two people must share a date, and that is the pigeonhole principle. Working out the chance of a shared date with fewer people than dates is a probability calculation and is not covered here.