Interesting Number Pattern - 24 - Question!

Find this 4 digit number - call it as ABCD

- It is a perfect square
- If you reverse the number, it is also a perfect square (DCBA)
- The sum of the first two digits and the last two digits of the number is the square root of DCBA

Please don't use Google or don't write a  program :)

More than the answer just examine ABCD, DCBA after you are through and see how simple but beautiful the pattern is

Calculating the n-th root of a number - Question! - Interesting pattern - 23

How to calculate (217)^ 1/1025 -  217 to the power 1/1025

Well, everyone has a mobile or a PC which will do this but the intent is not to see how quickly a machine can calculate - rather what's the logic which we humans can use to do this calculation

I am opening this up as a question for others to share their view and then I will share some details as to how we can go about this

Clue: Start off with smaller numbers like 2nd, 3rd and 5th root :)

Sum of two consecutive Squares - Interesting Pattern ----> 22

Take a look at this patten

1 ^ 2 + 2 ^ 2 = 5
2 ^ 2 + 3 ^ 2 = 13
3 ^ 2 + 4 ^ 2 = 25
4 ^ 2 + 5 ^ 2 = 41
5 ^ 2 + 6 ^ 2 = 61 --------------------> Pattern breaks here
6 ^ 2 + 7 ^ 2 = 85
7 ^ 2 + 8 ^ 2 = 113
8 ^ 2 + 9 ^ 2 = 145
9 ^ 2 + 10 ^ 2 = 181
10 ^ 2 + 11 ^ 2 = 221 ------------------> Pattern breaks here

Continue this sequence - what do you notice?

Alternate results are either multiples of 5 or a Prime number except for situations when we encounter 5, 10, 15, 20.... (as the first number on the left) the pattern breaks for a brief while

Check this....

14 ^ 2 + 15 ^ 2 = 421
15 ^ 2 + 16 ^ 2 = 481 --------------> Pattern breaks here
16 ^ 2 + 17 ^ 2 = 545

Similarly...

19 ^ 2 + 20 ^ 2 = 761
20 ^ 2 + 21 ^ 2 = 841 --------------> Pattern breaks here
21 ^ 2 + 22 ^ 2 = 925

Squares ending with 5 - Interesting Number Pattern ---> 21

Notice this pattern for squares ending with 5

05 * 05 = 0025 ---> 00 + 25 = 25 ----> Subtract the original number 5  = 25 - 5 = 20
15 * 15 = 0225 ---> 02 + 25 = 27 ----> 27 - 15 = 12
25 * 25 = 625  ----> 06 + 25 = 31 ----> 31 - 25 = 6
35 * 35 = 1225 ----> 12 + 25 = 37 -----> 37 - 35 = 2

The sequence starts froom 20 and comes down to 2. Now notice this

45 * 45 = 2025 -----> 20 + 25 = 45 ------> 45 - 45 = 0
55 * 55 = 3025 -----> 30 + 25 = 55 ------> 55 - 55 = 0

So two zeroes in between as the difference... Now...

65 * 65 = 4225 -----> 42 + 25 = 67 ------> 67 - 65 = 2
75 * 75 = 5625 -----> 56 + 25 = 81 ------> 81 - 75 = 6
85 * 85 = 7225 -----> 72 + 25 = 97 ------> 97 - 85 = 12
95 * 95 = 9025 -----> 90 + 25 = 115 -----> 115 = 95 = 20

It shows the increasing pattern from 2 to 20 :)

Completes a full circle!