On $a$ natural number $n$ you are allowed two operations: (1) multiply $n$ by 2 or (2) subtract 3 from n. For example starting with 8 you can reach 13 as follows: 8 → 16 → 13. You need two steps and you cannot do in less than two steps. Starting from 11, what is the least number of steps required to reach 121?

Step 1: 11 → ……. Some steps → 121 minimum steps

Step 2: Notice that for each step the number is $a$ natural number

Step 3: => Reverse the process

Step 4: Start from 121 and now condition will be

Step 5: Now, 121 cannot be divide by 2 as result will not be natural number and for minimum step it is better to keep

Step 6: dividing till we get on odd number

Step 7: 124 3 62 4

Step 8: => 121 ⎯⎯

Step 9: → (121 + 3 = 124 ) ⎯⎯

Step 10: → = 62 ⎯⎯→ = 31 ⎯⎯→

Step 11: 2 2

Step 12: 34 6 20 8

Step 13: ( 31 + 3 = 34 ) ⎯⎯

Step 14: → ⎯⎯→ (17 + 3 = 20 ) ⎯⎯

Step 15: → = 10 ⎯⎯→

Step 16: 2 2

Step 17: 10 9

Step 18: 2 = 5 ⎯⎯→ ( 5 + 3 = 8 ) ⎯⎯→ ( 8 + 3 ) at it doesn’t make sense to go half as 1 move to reach 11

Step 19: => minimum 10 moves

Step 20: n   $n$ − 172 