## Solution to Puzzle #108: The Double Square Number Problem

Apologies that I could not post the answer last weekend as I was traveling. I was overwhelmed by the number of answers i received for this puzzle – most of them correct. However, many people used spreadsheets or a small software script to figure out the number. Some people used trial and error to derive the answer. Only a few people found the answer with a method – Karan Sharma, my nephew, Pratik Poddar and someone who is posting by the name “Bill Gates” from India – great job! Other people to solve included Sabeer Bhatia, Aman Singla, Adithya Herur, Akshay Joshi, Suman Saraf. If I have missed someone out, my apologies!

A simple way to think about it is as follows:

Let x and y be the two parts of the number. Therefore:

x^2 + y^2 = 100*x + y, which can be rearranged as:

y * (y-1) = x * (100-x)

We know that 1233 is a solution, which means that the solution works for y = 33. In the equation above you will notice that if the solution works for x (which in this case is 12), it will also work for 100-x, therefore 8833 has to be a solution.

Pratik gave a more elaborate answer, reproducing it here:

Let the number be of the form “ab” = 100*a + b where a and b are >0 and <100 and represented as two digit numbers like 07 or 95

100*a + b = a^2 + b^2
a*(100-a) = b*(b-1)

One of b and b-1 is even
So, At least one of a and 100-a have to be even
So, a is even

Lets say a = 2*x

4*x*(50-x) = b*(b-1)

Max value of LHS is 2500
So, Max value of b is 50

One of b and b-1 is odd. So, whatever is even, is of the form 4*y

Case 1: b = 4*y and y<13 and a = 2*x and x Will explore later
If x is even, y is a multiple of 4 or one of 4,8,12 => No solution

Case 2: b-1 = 4*y and y<13 and a = 2*x and x Will explore later
If x is even, y is a multiple of 4 or one of 4,8,12 => y=8,x=44,6 is a solution

We have a solution.
b=33
a = 12 or 88

so 8833 and 1233 are valid numbers

Finally, Adithya Herur gave an interesting twist, which is that what if x and y ( or a and b) are 3 digits or 4 digits each. Here are answers for these, not sure though if these are the only solutions:

for an N with 6 digits : 990100
for an N with 8 digits : 94122353

Thanks to Karan, “Bill Gates”, Pratik and Adithya!

Hope you all enjoyed the puzzle!

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