# How to Find Pythagorean Dates for Any Year

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Pythagorean Dates in 2005 – click for larger view: Image by Mike DeHaan

## The Power of this Worksheet to Find Pythagorean Dates in Other Years

As this image shows, simply changing the year to ‘5’ finds the three Pythagorean dates of 2005.

Why are there three Pythagorean dates in 2005 but only two in 2013?

Both 2005 and 2013 have symmetric dates where the month and day, ‘5’ and ’12’, are swapped. However, the Pythagorean triple {5, 12, 13} cannot be swapped because the day, ’13’, is larger than the maximum month.

## One Potential Pitfall for the Spreadsheet Solution for Pythagorean Dates

The most obvious pitfall for this spreadsheet solution is that some months lack days 29-31, so it could display ‘Yes’ on a non-existent date. Of course, a spreadsheet that takes care of that problem would also have to compensate for leap years.

## Computer Programming Solutions to Generate Pythagorean Dates

A better solution for generating all the Pythagorean dates in a century would be to write a computer program with three nested loops.

Again, the programmer should include tests for invalid dates and leap years.

## Pythagorean Dates

Every right angle triangle has one side called the “hypoteneuse” – this side is opposite the 90o right angle. Let’s call the lengths of the sides ‘a’, ‘b’ and call the hypoteneuse ‘h.’

The Pythagorean Theorem formula states that “h2 = a2 + b2” for all right angle triangles. A Pythagorean triple of integers {a, b, h} satisfies that theorem.

A “Pythagorean date”, or “right angle date”, is one in which the integer values of the month, day and two-digit year form a Pythagorean triple. You can identify dates like this for any given year, and solve Pythagorean Theorem problems if you are given three lengths. Geometry and math look complicated, but they’re just shapes and numbers – so don’t feel intimidated!

## References

Honner, Patrick. 5/12/13 — Happy Right Triangle Day! (2013). Mr. Honner Math Appreciation. Referenced May 13, 2013.

Weisstein, Eric W. Pythagorean Theorem. From MathWorld-A Wolfram Web Resource. Referenced May 13, 2013.

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