[UCSD白板题] Primitive Calculator

Problem Introduction

You are given a primitive calculator that can perform the following three operations with the current number \(x\): multiply \(x\) by 2, multiply \(x\) by 3, or add 1 to \(x\). Your goal is given a positive integer \(n\), find the minimum number of operations needed to obtain the number \(n\) starting from the number 1.

Problem Description

Task.Given an integer \(n\), compute the minimum number of operations needed to obtain the number \(n\) starting from the number 1.

Input Format.The input consists of a single integer \(1 \leq n \leq 10^6\).

Output Format.In the first line, output the minimum number \(k\) of operations needed to get \(n\) from 1. In the second line output a sequence of intermediate numbers. That is, the second line should contain positive integers \(a_0, a_2, \cdots, a_{k-1}\) such that \(a_0=1, a_{k-1}=n\) and for all \(0 \leq i < k-1\), \(a_{i+1}\) is equal to either \(a_i+1, 2a_i\), or \(3a_i\). If there are many such sequences, output any one of them.

Sample 1.
Input:

1

Output:

0
1

Sample 2.
Input:

5

Output:

3
1 2 4 5

Sample 3.
Input:

96234

Output:

14
1 3 9 10 11 22 66 198 594 1782 5346 16038 16039 32078 96234

Solution