How to Find Pythagorean Dates for Any Year


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Pythagorean Dates in 2013 – click for larger view: Image by Mike DeHaan

Upon reading Patrick Honner’s article on Right Triangle Day, I asked myself the math question, “How can I find Pythagorean dates for any year?”

Perhaps the more obvious questions are “Will 2013 have other Pythagorean dates?” and “What are Pythagorean dates?”

What are Pythagorean Dates for 2013?

Honner reported May 12, 2013 as a Pythagorean date.

Obviously the paired Pythagorean date for this year will be Dec. 5th.

Let’s define what these dates have in common.

What are Pythagorean Dates?

A “Pythagorean Date”, or “right triangle date”, is one in which the values for the day, month and year form a solution of the Pythagorean theorem for right angle triangles.

Taking May 12, 2013 as an example, we have:

  • Day = 12
  • Month = 5
  • Year = 13

The sum of the squares of ’12’ and ‘5’ equals the square of ’13’.

  • 122 = 12 x 12 = 144
  • 52 = 5 x 5 = 25
  • 132 = 13 x 13 = 169 = 144 + 25

Therefore {5, 12, 13} is a Pythagorean triple: these integers solve the Pythagorean theorem formula.

What is the Pythagorean Theorem?

The Pythagorean Theorem states that, for any right angle triangle, the square on the hypoteneuse equals the sum of the squares on the other two sides.The hypoteneuse, opposite the right angle, is the longest side of the triangle.

The “ancient” Greek mathematician and philosopher, Pythagoras, was interested in geometry that could be described as the ratio of integers. When he said, “The square on a side of a triangle”, he literally meant “The area of the square with sides equal to the length of the specific side of the triangle in question.” Modern mathematics “square” a number simply by multiplying it by itself, but Pythagoras relied upon geometry.

The most well-known Pythagorean triple may be {3, 4, 5}.

5-12-13 Right Angle Triangle via Microsoft Mathematics Triangle Solver : Image by Mike DeHaan

What is a Right Triangle?

A “right triangle” – or “right angle triangle” – has one 90o angle where two sides are perpendicular to each other.

How many degrees in a right triangle? Every triangle has 180o in total.

This 5-12-13 right angle triangle shows the 90o angle at vertex ‘C’. It represents both the May 5 and Dec. 12, 2013 Pythagorean dates.

Use a Pythagorean Theorem Worksheet to Find Pythagorean Dates for Any Year

My solution to find Pythagorean dates involves building a spreadsheet, as shown in the first image.

The two-digit year is in cell B3. The months go across the top, from ‘1’ to ’12’ in F2:Q2. The days of the month are in column B6:B36. The light grey cells increment the months and days, saving some data entry.

To calculate the squares of the year, months and days, I simply multiplied a cell by itself.

The ‘IF’ function is the trickiest part because it has to be as general as possible. The day, month or year value might be the largest value, so it’s important to test each as the hypoteneuse. That led to the ‘OR’ portion. The net result of the ‘IF’ function is to display “Yes” when any one Pythagorean relationship is satisfied.

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