In one of my earlier post, I had posted a question about uniqueness of the squares between 1600 to 3600.. To recap, let's look at these squares
1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364, 3481
As you we could seem as the first two digits increase by 1, the last 2 digits are perfect squares in decreasing order till we hit 2500 and again goes in increasing order till we reach 3600!
Now the fun doesn't just stop there!
Let's look at the squares preceding these
Listing the ones before 1600
1521, 1444, 1369, 1296, 1225, 1156, 1089, 1024, 961, 900, 841, 784, 729, 676, 625, 576, 529, 484, 441, 400, 361, 324, 289, 256, 225, 196, 169, 144, 121, 100, 81, 64, 49, 36, 25, 16, 9, 4, 1
Listing the ones after 3600
3721, 3844, 3969, 4096, 4225, 4356, 4489, 4624, 4761, 4900, 5041, 5184, 5329, 5476, 5625, 5776, 5929, 6084, 6241, 6400, 6561, 6724, 6889, 7056, 7225, 7396, 7569, 7744, 7921, 8100, 8281, 8464, 8649, 8836, 9025, 9216, 9409, 9600, 9801
The pattern is very much clear..Isn't it!
a) Last 2 digits are same in both the series
b) Add the first 2 digits and you will see a pattern - 52 for the first few numbers and then 54 and the increases by 2 till 98 (1 can be written as 0001)
One of the reason behind this is the fact that these squares fit in the formulae a^2 - b ^2 if you closely observe it. Though it is not obvious the resulting patterns are indeed very interesting!
1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364, 3481
As you we could seem as the first two digits increase by 1, the last 2 digits are perfect squares in decreasing order till we hit 2500 and again goes in increasing order till we reach 3600!
Now the fun doesn't just stop there!
Let's look at the squares preceding these
Listing the ones before 1600
1521, 1444, 1369, 1296, 1225, 1156, 1089, 1024, 961, 900, 841, 784, 729, 676, 625, 576, 529, 484, 441, 400, 361, 324, 289, 256, 225, 196, 169, 144, 121, 100, 81, 64, 49, 36, 25, 16, 9, 4, 1
Listing the ones after 3600
3721, 3844, 3969, 4096, 4225, 4356, 4489, 4624, 4761, 4900, 5041, 5184, 5329, 5476, 5625, 5776, 5929, 6084, 6241, 6400, 6561, 6724, 6889, 7056, 7225, 7396, 7569, 7744, 7921, 8100, 8281, 8464, 8649, 8836, 9025, 9216, 9409, 9600, 9801
The pattern is very much clear..Isn't it!
a) Last 2 digits are same in both the series
b) Add the first 2 digits and you will see a pattern - 52 for the first few numbers and then 54 and the increases by 2 till 98 (1 can be written as 0001)
One of the reason behind this is the fact that these squares fit in the formulae a^2 - b ^2 if you closely observe it. Though it is not obvious the resulting patterns are indeed very interesting!
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