Explore guaranteed repetition, occupancy limits, and contradiction checks. Compare pigeons, holes, targets, and impossible arrangements. Build stronger proofs using quick results, tables, and exports.
Advanced counting, collision, and proof supportChoose a mode, enter whole numbers, and submit to see the result above this form.
Guaranteed load = ceil(P / H), where P is pigeons and H is holes. This guarantees at least one hole reaches that value.
To guarantee at least k pigeons in one hole, the minimum required count is H × (k - 1) + 1.
Maximum safe pigeons = H × (k - 1). One more pigeon forces some hole to contain at least k.
Minimum holes = ceil(P / (k - 1)). This keeps every hole below k, if enough holes are available.
| Mode | Pigeons | Holes | Target / Cap | Output |
|---|---|---|---|---|
| Guaranteed minimum occupancy | 17 | 5 | 4 | At least one hole has 4 pigeons. |
| Maximum safe without target collision | — | 6 | Target 3 | Maximum safe pigeons = 12. |
| Minimum holes needed | 25 | — | Target 5 | Minimum holes required = 7. |
| Contradiction checker | 22 | 4 | Cap 5 | Impossible, because capacity is only 20. |
It shows guaranteed repetition or crowding when objects are distributed into limited categories. It also checks thresholds, safe limits, required holes, and contradiction-style claims.
The pigeonhole principle counts discrete objects and containers. Fractional pigeons or fractional holes do not fit the combinatorial meaning of the theorem.
It is the least number that some hole must reach, no matter how evenly the pigeons are distributed. The calculator finds it using the ceiling of pigeons divided by holes.
It tests when at least k pigeons must appear in one hole. The forcing threshold is one more than the largest arrangement that still keeps every hole below k.
Use it when someone claims no hole exceeds a chosen cap. The calculator compares total capacity under that claim against the actual pigeon count.
Yes. It gives fast bounds, forcing thresholds, and contradiction checks that often form the key step in contest proofs, counting arguments, and discrete mathematics exercises.
Because the safe arrangement already fills every hole to the maximum allowed level. The next pigeon has nowhere safe to go, so the target occupancy becomes unavoidable.
They export the computed result metrics for the submitted case. This helps you save worked examples, attach notes, or share proof-supporting calculations.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.