After writing my blog post on the mystery of the number 89, I read this article on the Fibonacci sequence entitled "89: Why 89 is the real deal? The 11th Fibonacci number" by Karan Singla. He wrote:
Squaring the digits of a number and adding them
Start with 89. Square the digits and add them. This gives us a new number. Let's keep doing that and see what happens.
89 --> 8^2 + 9^2 = 64 + 81 = 145 --> 1^2 + 4^2 + 5^2 = 42 --> .....
If we keep doing this, what we get is the following :
89 --> 145 -> 42 --> 20 --> 4 --> 16 --> 37 --> 58 --> 89
Wow. We come back to 89.
What if we start at a different number, and repeat the process. Would we arrive at the initial number always. Lets check that out with say...43.
43 --> 25 --> 29 --> 85 --> 89
And it's 89 again.
What's really cool is that, you could start with any number, with any number of digits, and you would always end at 89 or 1. Isn't this just amazing? Try that out for a few numbers...
So I did. I found out that numbers like 0,1,10,13 and 100 go back to 1 and all th eothers can go back to 89.
13> 1^2 +3^2 = 1+9=10 >1^2+0^2 = 1> 1^2 = 1
14> 1^2 +4^2 = 1+16 = 17 > 1^2+7^2 = 1+49=50 >5^2+0^2 = 25+0 = 25 >2^2+5^2 = 4+25 = 29 >2^2+9^2 = 4+81 = 85 > 8^2+5^2 = 64+25 = 89.
12> 1^2+2^2 = 1+4 = 5 >5^2 = 25 >2^2+5^2 = 4+25 = 29 >2^2+9^2 = 4+81 = 85 > 8^2+5^2 = 64+25 = 89.
11> 1^2+1^2 = 1+1=2> 2^2 = 4> 4^2 = 16 > 1^2 +6^2 = 1+36 = 37 > 3^2+7^2 = 9+49 = 58 > 5^2+8^2 = 25+64 = 89
10> 1^2+0^2 = 1+0 = 1
9>9^2 = 81>8^2+1^2 = 64+1 = 65 >6^2+5^2 = 36+25 = 61 >6^2+1^2 = 36+1 = 37> 3^2+7^2 = 9+49 = 58 > 5^2+8^2 = 25+64 = 89
8>8^2 = 64 >6^2+4^2 = 36+16 = 52 >5^2+2^2 = 25+4 = 29 >2^2+9^2 = 4+81 = 85 > 8^2+5^2 = 64+25 = 89.
177>1^2+7^2+7^2 = 1+49+49 = 99>9^2+9^2 = 81>8^2+1^2 = 64+1 = 65>6^2+5^2 = 36+25 = 61 >6^2+1^2 = 36+1 = 37> 3^2+7^2 = 9+49 = 58 > 5^2+8^2 = 25+64 = 89
353>3^2+5^2+3^2 = 9+25+9 = 43>4^2+3^2 =16+9 = 25>2^2+5^2 = 4+25 = 29 >2^2+9^2 = 4+81 = 85 > 8^2+5^2 = 64+25 = 89.