Square my chunks and add

The idea is to transform the integer a into b and iterate.

Cut a into the number of chunks you want, square them, add those squares.

Example

a = 1023 can be transformed in b (last column) like this:

1² + 0² + 2² + 3² = 1 + 0 + 4 + 9  = 14 = b

1² + 0² + 23²     = 1 + 0 + 529    = 530 

10² + 2² + 3²     = 100 + 4 + 9    = 113 

10² + 23²         = 100 + 529      = 629 

102² + 3²         = 10404 + 9      = 10413 

1023²                              = 1046529 

Question #1

Starting with 2 and iterating, what is the shortest path ending in 1?

Here are 11 steps (left column) – can you do better?

2    –> 2² = 4

4    –> 4² = 16

16   –> 1² + 6² = 1 + 36 = 37

37   –> 37² = 1369

1369 –> 1² + 3² + 6² + 9² = 1 + 9 + 36 + 81 = 127

127  –> 12² + 7² = 144 + 49 = 193

193  –> 1² + 9² + 3² = 1 + 81 + 9 = 91

91   –> 9² + 1² = 82

82   –> 8² + 2² = 64 + 4 = 68

68   –> 6² + 8² = 36 + 64 = 100

100  –> 1² + 0² + 0² = 1

1    END

Question #2

What is the shortest path starting with 3 and ending on 1 – and, in general, what are the shortest paths for all a < 100?

Question #3

a = 1233 loops, if we want: 12² + 33² = 144 + 1089 = 1233 (this is mentionned here and there, in the OEIS)

For a < 1,000,000 what are the other 1-step looping integers like 1233 (I guess this is known), 2-steps looping, 3-steps, etc. (a few such "cycles" are visible here, in the OEIS).

____________________

Friday update by Jean-Marc Falcoz (great, thanks!-)

      2

     4

    16....1^2 + 6^2

    37....37^2

  1369....1^2 + 369^2

136162....1^2 + 36^2 + 16^2 + 2^2

  1557....1^2 + 5^2 + 5^2 + 7^2

   100....1^2 + 0^2 + 0^2

     1 

     3

     9

    81....81^2

  6561....6^2 + 56^2 + 1^2

  3173....3^2 + 1^2 + 7^2 + 3^2

    68....6^2 + 8^2

   100....1^2 + 0^2 + 0^2

     1

 

     4

    16....1^2 + 6^2

    37....37^2

  1369....1^2 + 369^2

136162....1^2 + 36^2 + 16^2 + 2^2

  1557....1^2 + 5^2 + 5^2 + 7^2

   100....1^2 + 0^2 + 0^2

     1