The Pigeonhole Principle: Bridging Mathematics and SQL

The other day I was reading Wikipedia, and I ended up on the Pigeonhole Principle. I find it very intriguing about this and I decide to see if I can convert it to SQL Server simulation. Let us see how I do it. If you have a better method, I am eager to learn from you.

Decoding the Pigeonhole Principle

Also known as Dirichlet’s box principle, the Pigeonhole Principle states that if you have more items (or pigeons) than containers (or pigeonholes), then at least one container must contain more than one item.

In mathematical terms, if “n” items are put into “m” containers, with “n” being larger than “m”, then at least one container must contain more than one item. Despite its simplicity, this principle often leads to unexpected and insightful results.

An In-Depth SQL Example

The Pigeonhole Principle isn’t just confined to mathematics. It can be found in areas such as programming and data analysis. SQL, a standard language for managing data in relational databases, provides an excellent platform for implementing this principle.

Let’s consider a scenario where we have a number of items that we want to assign randomly to a smaller number of boxes. We can represent this situation with an `Assignments` table in SQL, where `ItemId` represents the item and `BoxId` represents the box to which the item is assigned.

CREATE TABLE Assignments 
(
  ItemId int,
  BoxId int
);

Next, let’s create a stored procedure that generates assignments for `@NumItems` items across `@NumBoxes` boxes. Each item is assigned to a random box.

CREATE PROCEDURE GenerateAssignments
  @NumItems int, 
  @NumBoxes int
AS
BEGIN
  DECLARE @ItemId int, @BoxId int

  WHILE @NumItems > 0
  BEGIN
    SET @BoxId = CAST(RAND() * @NumBoxes AS int) + 1
  
    INSERT INTO Assignments VALUES (@NumItems, @BoxId)
    
    SET @NumItems = @NumItems - 1
  END
END;

Let’s execute the procedure to assign 10 items across 5 boxes:

EXEC GenerateAssignments 10, 5;

Now, according to the Pigeonhole Principle, if we have more items than boxes, at least one box must contain more than one item. We can confirm this with the following SQL query:

SELECT 
  BoxId, 
  COUNT(*) AS ItemsPerBox
FROM 
  Assignments
GROUP BY 
  BoxId
HAVING 
  COUNT(*) > 1;

This statement groups the `Assignments` table by the `BoxId` column and counts the number of items in each box. The `HAVING` clause then filters out boxes that contain only one item.

Result:
| BoxId | ItemsPerBox |
|-------|-------------|
|  2    |     3       |
|  4    |     2       |

This output would indicate that Box 2 contains 3 items and Box 4 contains 2 items.

To view the final assignments:

SELECT * FROM Assignments;

Result:
| ItemId | BoxId |
|--------|-------|
|   1    |   5   |
|   2    |   3   |
|   3    |   2   |
|   4    |   2   |
|   5    |   1   |
|   6    |   4   |
|   7    |   2   |
|   8    |   4   |
|   9    |   5   |
|  10    |   3   |

This output indicates the specific BoxId each ItemId has been assigned to.

Conclusion

The Pigeonhole Principle, while simple, is a powerful tool in various fields, including mathematics, data analysis, and programming. The principle’s application in SQL, as demonstrated by our example, shows how fundamental mathematical concepts can extend into different areas of technology and practical problem-solving.

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Reference: Pinal Dave (https://blog.sqlauthority.com)