## Puzzle #151: Journey from 458 to 14

This is a beautiful teaser contributed by Pooja Singh, who tried this with her 9 year old. So will encourage everyone to try this, it could be a nice, 5-minute puzzle!

You begin with the number 458. You are only allowed two operations, each of which can be repeated any number of times:

– You can multiply the number by 2

– Remove the last digit, i.e. the one in the unit’s place (e.g. 8 in the number 458)

Your task is to get to the number 14, beginning with 458. Do this in the minimum number of steps.

As always, please send your answers as comments within the blog (preferred), or send an e-mail to alokgoyal_2001@yahoo.com. Please do share the puzzle with others if you like, and please also send puzzles that you have come across that you think I can share in this blog.

Happy number crunching!

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### 7 Responses to Puzzle #151: Journey from 458 to 14

1. 458,45,90,9,18,36,72,144,14. Is there a shorter method?

2. Prakhar Prakash says:

458*2 =916
916*2=1832
183
18
18*2=36
36*2=72
72*2=144
14

3. 1- Remove units digit to get 45
2- Multiply by 2 to get 90
3- Remove zero to get 9
4- 9*2=18
5- 18*2=36
6- 36*2=72
7-72*2=144
8- Remove 4 to get 14

4. 458 -> 916 -> 91 -> 182 -> 366 -> 732 -> 73 -> 146 -> 14

Intuitively, this seems like the smallest chain (along with many others). Not sure how to “prove” this is the smallest.

5. karan says:

458>916>1832>3664>7328>14656>1465>146>14

6. Ram says:

458->45->90->9->18->36->72->144->14

7. Amit Mittal says:

One can start with one operation of each of the allowed two and compare which one is faster.

One should not try for larger numbers( in my solution) as the answer is fairly smaller than the root – 458

if we take reduction first then the sequence should be:

458 > 45 > 90 > 9 > 180 > 360 > 720 > 72 > 7 >14

or

458 > 45 > 90 >180 > 18 > 36 >72 >7 >14 which reduces one step and matches other routes traversed in solutions

On the other hand

458 > 916 > 91 > 182 > 18 > 36 > 72 > 144 > 14 which is also 8 steps

seems to be because 458/14 = 33X times so we need at least 3 reductions and more than 32 X times which is 5 powers of 2. That however is definitely just an improbability unless induction shows this is a possibility