Understanding the Unique Date Property

The question describes a unique mathematical property found in three specific dates when written in Date-Month-Year format (D/M/Y). We are given the dates: 3rd April, 2005; 6th August, 2010; and 5th December, 2013. Let's write these dates numerically and look for the pattern.

• 3rd April, 2005 is written as 3/4/2005.
• 6th August, 2010 is written as 6/8/2010.
• 5th December, 2013 is written as 5/12/2013.

For problems involving dates and years, the property often relates the Date (D), Month (M), and the last two digits of the Year (YY). Let's extract these numbers for each date:

Now let's examine the sets of numbers (D, M, YY): (3, 4, 5), (6, 8, 10), and (5, 12, 13).

Consider the relationships between these numbers. Notice that these sets of numbers are well-known groups called Pythagorean triples. A Pythagorean triple is a set of three positive integers a, b, and c, such that \(a^2 + b^2 = c^2\). The largest number in the triple is always 'c', the hypotenuse in a right-angled triangle context.

• For (3, 4, 5): \(3^2 + 4^2 = 9 + 16 = 25\), and \(5^2 = 25\). So, \(3^2 + 4^2 = 5^2\). This fits the pattern.
• For (6, 8, 10): \(6^2 + 8^2 = 36 + 64 = 100\), and \(10^2 = 100\). So, \(6^2 + 8^2 = 10^2\). This fits the pattern. Note that (6, 8, 10) is a multiple of (3, 4, 5) by 2.
• For (5, 12, 13): \(5^2 + 12^2 = 25 + 144 = 169\), and \(13^2 = 169\). So, \(5^2 + 12^2 = 13^2\). This fits the pattern.

In each case, the numbers representing the Date (D), Month (M), and the last two digits of the Year (YY) form a Pythagorean triple, with the last two digits of the Year (YY) being the hypotenuse (the largest number).

So, the unique mathematical property is: \(D^2 + M^2 = YY^2\), where D is the date, M is the month number, and YY are the last two digits of the year.

Applying the Property to Indian Independence Day

We need to find which year among the options has the same property for Indian Independence Day. Indian Independence Day is on August 15th. So, the Date (D) is 15, and the Month (M) is 8.

• Date (D) = 15
• Month (M) = 8

According to the property \(D^2 + M^2 = YY^2\), we substitute the values for D and M:

\(15^2 + 8^2 = YY^2\)

Calculate the squares:

\(225 + 64 = YY^2\)

\(289 = YY^2\)

To find YY, we need to take the square root of 289:

\(YY = \sqrt{289}\)

\(YY = 17\)

This means that for Indian Independence Day (August 15th) to have the same mathematical property, the last two digits of the year (YY) must be 17.

Checking the Options for the Year

Now let's look at the last two digits of the years given in the options:

• Option 1: 2017 > YY = 17
• Option 2: 2016 > YY = 16
• Option 3: 2018 > YY = 18
• Option 4: 2015 > YY = 15

The year whose last two digits are 17 is 2017.

Verification

Let's verify if August 15th, 2017 satisfies the property:

• Date (D) = 15
• Month (M) = 8
• Year = 2017
• YY = 17

Check if \(D^2 + M^2 = YY^2\):

\(15^2 + 8^2 = 225 + 64 = 289\)

\(YY^2 = 17^2 = 289\)

Since \(289 = 289\), the property \(D^2 + M^2 = YY^2\) holds true for August 15th, 2017.

Thus, the Indian Independence Day of the year 2017 has the same unique mathematical property as the given dates.

Revision Table: Key Steps in Finding the Date Property

Additional Information: Pythagorean Triples in Math

A Pythagorean triple consists of three positive integers a, b, and c, such that \(a^2 + b^2 = c^2\). These triples are fundamental in geometry because they represent the side lengths of a right-angled triangle, where 'c' is the length of the hypotenuse (the side opposite the right angle).

The most famous Pythagorean triple is (3, 4, 5). Others include (5, 12, 13), (8, 15, 17), (7, 24, 25), etc. Any positive integer multiple of a Pythagorean triple is also a Pythagorean triple (e.g., (6, 8, 10) is 2 times (3, 4, 5)).

These triples appear in various mathematical puzzles and problems, often disguised in different contexts, like the date property seen in this question.