Problem:
 Score : 300 points    Problem Statement To make it difficult to withdraw money, a certain bank allows its customers to withdraw only one of the following amounts in one operation:    1 yen (the currency of Japan)    6 yen, 6^2(=36) yen, 6^3(=216) yen, ...    9 yen, 9^2(=81) yen, 9^3(=729) yen, ...    At least how many operations are required to withdraw exactly N yen in total?  It is not allowed to re-deposit the money you withdrew.      Constraints  1 \leq N \leq 100000  N is an integer.         Input Input is given from Standard Input in the following format:  N       Output If at least x operations are required to withdraw exactly N yen in total, print x .        Sample Input 1 127      Sample Output 1 4  By withdrawing 1 yen, 9 yen, 36(=6^2) yen and 81(=9^2) yen, we can withdraw 127 yen in four operations.       Sample Input 2 3      Sample Output 2 3  By withdrawing 1 yen three times, we can withdraw 3 yen in three operations.       Sample Input 3 44852      Sample Output 3 16

Solution (Python):
def main(sample_file = ''):

    """ convenient functions
    
    
    
    
    
    """

    """convenient decorator
    
    
        
        
        
        
        
        
        
        
        
        
    """
    
    
    import sys
    sys.setrecursionlimit(10**7)
    from itertools import accumulate, combinations, permutations, product 
    
    from math import factorial, ceil, floor, sqrt
    def factorize(n):
        """return the factors of the Arg and count of each factor
        
        Args:
            n (long): number to be resolved into factors
        
        Returns:
            list of tuples: factorize(220) returns [(2, 2), (5, 1), (11, 1)]
        """
        fct = []  
        b, e = 2, 0  
        while b * b <= n:
            while n % b == 0:
                n = n // b
                e = e + 1
            if e > 0:
                fct.append((b, e))
            b, e = b + 1, 0
        if n > 1:
            fct.append((n, 1))
        return fct
    def combinations_count(n, r):
        """Return the number of selecting r pieces of items from n kinds of items.
        
        Args:
            n (long): number
            r (long): number
        
        Raises:
            Exception: not defined when n or r is negative
        
        Returns:
            long: number
        """
        
        if n < 0 or r < 0:
            raise Exception('combinations_count(n, r) not defined when n or r is negative')
        if n - r < r: r = n - r
        if r < 0: return 0
        if r == 0: return 1
        if r == 1: return n
        numerator = [n - r + k + 1 for k in range(r)]
        denominator = [k + 1 for k in range(r)]
        for p in range(2,r+1):
            pivot = denominator[p - 1]
            if pivot > 1:
                offset = (n - r) % p
                for k in range(p-1,r,p):
                    numerator[k - offset] /= pivot
                    denominator[k] /= pivot
        result = 1
        for k in range(r):
            if numerator[k] > 1:
                result *= int(numerator[k])
        return result
    def combinations_with_replacement_count(n, r):
        """Return the number of selecting r pieces of items from n kinds of items allowing individual elements to be repeated more than once.
        
        Args:
            n (long): number
            r (long): number
        
        Raises:
            Exception: not defined when n or r is negative
        
        Returns:
            long: number
        """
        if n < 0 or r < 0:
            raise Exception('combinations_with_replacement_count(n, r) not defined when n or r is negative')
        elif n == 0:
            return 1
        else:
            return combinations_count(n + r - 1, r)
    from bisect import bisect_left, bisect_right
    from collections import deque, Counter, defaultdict 
    from heapq import heapify, heappop, heappush, heappushpop, heapreplace,nlargest,nsmallest 
    from copy import deepcopy, copy 
    import operator
    from operator import itemgetter 
    
    
    from functools import reduce, lru_cache
    def chmin(x, y):
        """change minimum
        if x > y, x = y and return (x, True).
        convenient when solving problems of dp[i]
        
        Args:
            x (long): current minimum value
            y (long): potential minimum value
        
        Returns:
            (x, bool): (x, True) when updated, else (x, False)
        """
        if x > y:
            x = y
            return (x, True)
        else:
            return (x, False)
    def chmax(x, y):
        """change maximum
        if x < y, x = y and return (x, True).
        convenient when solving problems of dp[i]
        
        Args:
            x (long): current maximum value
            y (long): potential maximum value
        
        Returns:
            (x, bool): (x, True) when updated, else (x, False)
        """
        if x < y:
            x = y
            return (x, True)
        else:
            return (x, False)

    from fractions import gcd 
    def gcds(numbers):
        return reduce(gcd, numbers)
    def lcm(x, y):
        return (x * y) // gcd(x, y)
    def lcms(numbers):
        return reduce(lcm, numbers, 1)
    def make_divisors(n, reversed=False):
        """create list of divisors
        
        Args:
            number (int): number from which list of divisors is created
            reversed (bool, optional): ascending order if False. Defaults to False.
        
        Returns:
            list: list of divisors
        """
        divisors = set()
        for i in range(1, int(n**0.5)+1):
            if n % i == 0:
                divisors.add(i)
                divisors.add(n//i)
        return sorted(list(divisors),reverse=reversed)
    
    
    
    INF = 10 ** 18
    MOD = 10 ** 9 + 7
    modpow = lambda a, n, p = MOD: pow(a, n, p) 
    def modinv(a, p = MOD):
        
        
        return modpow(a, p-2, p)
    def modinv_list(n, p = MOD):
        if n <= 1:
            return [0,1][:n+1]
        else:
            inv_t = [0,1]
            for i in range(2, n+1):
                inv_t += [inv_t[p % i] * (p - int(p / i)) % p]
            return inv_t
    def modfactorial_list(n, p = MOD):
        if n == 0:
            return [1]
        else:
            l = [0] * (n+1)
            tmp = 1
            for i in range(1, n+1):
                tmp = tmp * i % p
                l[i] = tmp
            return l
    def modcomb(n, k, fac_list = [], p = MOD):
        
        
        from math import factorial
        if n < 0 or k < 0 or n < k: return 0
        if n == 0 or k == 0: return 1
        if len(fac_list) <= n:
            a = factorial(n) % p
            b = factorial(k) % p
            c = factorial(n-k) % p
        else:
            a = fac_list[n]
            b = fac_list[k]
            c = fac_list[n-k]
        return (a * modpow(b, p-2, p) * modpow(c, p-2, p)) % p
    def modadd(a, b, p = MOD):
        return (a + b) % MOD
    def modsub(a, b, p = MOD):
        return (a - b) % p
    def modmul(a, b, p = MOD):
        return ((a % p) * (b % p)) % p
    def moddiv(a, b, p = MOD):
        return modmul(a, modpow(b, p-2, p))

    class UnionFindTree:
        """union find tree class
        TODO: fix this description...
        how to use (example):
        >>  uf = UnionFindTree(N) 
        >>  if uf.find_root(a) == uf.find_root(b):
        >>      do something
        >>  else:
        >>      do something
        >>      uf.unite(a, b)
        """
        def __init__(self, N):
            self.root = [-1] * (N+1)
            self.rank   = [0]  * (N+1)
            self.connected_num = [1] * (N+1)
    
        def find_root(self,x):
            root = self.root
            while root[x] != -1:
                x = root[x]
            return x
    
        def unite(self,x,y):
            root = self.root
            rank = self.rank
            connected_num = self.connected_num
            find_root = self.find_root
    
            rx = find_root(x)
            ry = find_root(y)
            if rx != ry:
                if rank[rx] < rank[ry]:
                    root[rx] = ry
                    rx,ry = ry,rx
                else:
                    if rank[rx] == rank[ry]:
                        rank[rx] += 1
                    root[ry] = rx       
                connected_num[rx] += connected_num[ry]

    
    
    
    
    

    def BellmanFord(N, M, ABC, vertex_start, vertex_end, value_if_inf = -1, find_shortest = False):
        """to calculate furthest or shortest length between vertex_start and vertex_end using BellmanFord algorithm
        
        Args:
            N (int): number of vertices
            M (int): number of edges
            ABC (list): [(ai, bi, ci) for _ in range(N)] where i-th edge is directed from vertex ai to vertex bi and the length is ci 
            vertex_start (int): start vertex. usually use 0.
            vertex_end (int): end vertex. usually use N-1.
            value_if_inf (int or string as you like, optional): value you want when the furthest (or shortest) distance is infinite (or -infinite). Defaults to -1.
            find_shortest (bool, optional): choose False to find furthest path. Defaults to False.
        
        Returns:
            int or string: normally int (but can be str if you set value_if_inf to str)
            
        Example:
                N, M, P = R()
                ABC = [R() for _ in range(M)]
                ABC = [(a-1, b-1, c-P) for a, b, c in ABC]
                print(BellmanFord(N, M, ABC, 0, N-1, value_if_inf = 'inf'))

        """

        def make_reachable_list(N, M, ABC, vertex_start, vertex_end):
            reachable_to_direct = defaultdict(list)
            reachable_from_direct = defaultdict(list)
            reachable_from_start = [False] * N
            reachable_to_end = [False] * N
            reachable_from_start[vertex_start] = True
            reachable_to_end[vertex_end] = True
            reachable_from_both_sides = [False] * N
            dfs_from_start = []
            dfs_to_end = []
            for a, b, c in ABC:
                reachable_to_direct[a].append(b)
                reachable_from_direct[b].append(a)
                if a == vertex_start:
                    dfs_from_start.append(b)
                    reachable_from_start[b] = True
                if b == vertex_end:
                    dfs_to_end.append(a)
                    reachable_to_end[a] = True
            while dfs_from_start:
                v = dfs_from_start.pop()
                for i in reachable_to_direct[v]:
                    if not reachable_from_start[i]:
                        reachable_from_start[i] = True
                        dfs_from_start.append(i)
            while dfs_to_end:
                v = dfs_to_end.pop()
                for i in reachable_from_direct[v]:
                    if not reachable_to_end[i]:
                        reachable_to_end[i] = True
                        dfs_to_end.append(i)
            for i in range(N):
                reachable_from_both_sides[i] = reachable_from_start[i] and reachable_to_end[i]
            return reachable_from_both_sides
        
        reachable_from_both_sides = make_reachable_list(N, M, ABC, vertex_start, vertex_end)
        
        if find_shortest:
            dist = [INF for i in range(N)]
        else:
            dist = [-INF for i in range(N)]
        
        dist[vertex_start] = 0
        for i in range(N):
            updated = False
            for a, b, c in ABC:
                if not reachable_from_both_sides[a]:
                    continue
                elif find_shortest:
                    update_condition = dist[a] + c < dist[b]
                else:
                    update_condition = dist[a] + c > dist[b]
                if dist[a] != INF and update_condition:
                    dist[b] = dist[a] + c
                    updated = True
                    if i == N-1:
                        return value_if_inf
            if not updated:
                break
        return dist[vertex_end]

    """ initialize variables and set inputs
    
        
        
            
            
    
        
        
        
        
        
        
        
    
        
        
        
        
    
        
        
    """

    
    r = lambda: sys.stdin.readline().strip()
    r_int = lambda: int(r())
    R = lambda: list(map(int, r().split()))
    Rmap = lambda: map(int, r().split())
    Rfloat = lambda: list(map(float, r().split()))
    Rtuple = lambda: tuple(map(int, r().split()))

    """ how to treat input
    
    
    
    
    
    
    
    """

    
    if sample_file:
        sys.stdin = open(sample_file)
        
    
    

    N = r_int()
    dp = [[0] * (14) for _ in range(10**5+1)]
    for i in range(10**5+1):
        dp[i][0] = i
    unit_money = [6**i for i in range(1, 7)] + [9**i for i in range(1,6)]

    for col, unit in enumerate(unit_money, start=1):
        for i in range(10**5+1):
            dp[i][col] = dp[i][col-1]
            if i >= unit:
                dp[i][col] = min(dp[i][col], dp[i-unit][col]+1)
    print(dp[N][11])

    
    
    

    """memo: how to use defaultdict of list
    
    Dic = defaultdict(list)
    
    Dic[x].append(y)
    
    for k, v in Dic.items():
    """
    
    """memo: how to solve binary problems
    
    >>> a = 5
    >>> bin_str_a = format(a, '
    >>> print(bin_str_a)
    0b0101 
    """
    
    """memo: how to solve the problem
    creating simple test/answer
    greed
    simple dp
    graph
    """

if __name__ == '__main__':
    main()

Solution (Python):
from collections import deque
import math

n = int(input())


a1 = [1]
a6 = [6**x for x in range(1, int(math.log(10**6, 6))+1)]
a9 = [9**x for x in range(1, int(math.log(10**6, 9))+1)]
b = a1+a6+a9
b.sort()


dp = [-1] *(n+1)
dp[0]  = 0


d = deque([0])


for _ in range(10**12):
    
    if len(d) == 0:
        break
        
    
    tmp = d.pop()
    
    
    for x in b:
        if tmp+x > n:
            break
        if dp[tmp+x] == -1:
            dp[tmp+x] = dp[tmp]+1
            d.appendleft(tmp+x)

print(dp[n])

Solution (Python):
d = tuple([1] + [6**i for i in range(1, 7)] + [9**i for i in range(1, 6)])
M = 10**6+1
B = [float("inf")]*M
B[0] = 0
for i in range(M):
  for j in d:
    if i+j < M:
      B[i+j] = min(B[i+j], B[i]+1)
n = int(input())
print(B[n])

Solution (Python):

N = int(input())

dp = ['inf']*(N+1)

init = 6 if N > 5 else N+1
for n in range(init):
	dp[n] = n

for n in range(6,N+1):
	cnd = [dp[n-1]+1]
	for i in range(1,7):
		if n > 6**i-1:
			cnd.append(dp[n-6**i]+1)
	for j in range(1,6):
		if n > 9**j-1:
			cnd.append(dp[n-9**j]+1)
	dp[n] = min(cnd)
	
print(dp[N])

Solution (Python):
N = int(input())
dp = [100000] * (N + 1)
dp[0] = 0
dp[1] = 1
for i in range(1, N + 1):
    dp[i] = dp[i - 1] + 1
    
    prev = 6
    while i - prev >= 0:
        dp[i] = min(dp[i], dp[i - prev] + 1)
        prev *= 6
    
    prev = 9
    while i - prev >= 0:
        dp[i] = min(dp[i], dp[i - prev] + 1)
        prev *= 9
print(dp[N])

Solution (Python):
N = int(input())
ans = float('inf')
for i in range(N+1):
    count = 0
    n = i
    while n > 0:
        count += n % 6
        n //= 6
    n = N-i
    while n > 0:
        count += n % 9
        n //= 9
    ans = min(ans, count)

print(ans)

Solution (Python):
from itertools import count

n = int(input())

dp = [0]*(n+1)


def it(n):
  yield 1
  m = 6
  while m <= n:
    yield m
    m *= 6
  m = 9
  while m <= n:
    yield m
    m *= 9


for i in range(1,n+1):
  
  
  dp[i] = min(dp[i-t] for t in it(i))+1
  
print(dp[n])

Solution (Python):
import sys
sys.setrecursionlimit(10**6)
from functools import lru_cache

@lru_cache(maxsize=None)
def solve(val , now):
    if val < 6 or now == 0:
        ret_val = val
    
    elif val < counts[now]:
        ret_val = solve(val , now - 1)
    
    else:
        ret_val = min(solve(val , now - 1) , solve(val - counts[now] , now) + 1)
    return ret_val

N = int(input())
coins = [1,6,9]
counts = [1]
for i in range(1,3):
    for j in range(1,N):
        if coins[i] ** j <= N:
            counts.append(coins[i] ** j)
        else:
            break

print(solve(N , len(counts) - 1))

Solution (Python):
N = int(input())

M1 = [6**i for i in range(7)]
M2 = [9**i for i in range(6)]
M = list(set(M1 + M2))
M.sort()



dp = [0 for i in range(N+1)]
dp[0] = 1

for i in range(1, N+1):
    if i in M:
        dp[i] = 1
    else:
        dp[i] = min(dp[i-1] + 1, dp[i])
    for j in M:
        if i + j <= N:
            if dp[i+j] == 0:
                dp[i+j] = dp[i] + 1
            else:
                dp[i+j] = min(dp[i] + 1, dp[i+j])

            

print(dp[N])

Solution (Python):
INF = float('inf')
N = int(input())

dp = [INF for _ in range(N+1)]
dp[0] = 0

for i in range(1, N+1):
    
    dp[i] = min(dp[i], dp[i-1] + 1)
    
    power = 6
    while power <= i:
        dp[i] = min(dp[i], dp[i-power] + 1)
        power *= 6
    
    power = 9
    while power <= i:
        dp[i] = min(dp[i], dp[i-power] + 1)
        power *= 9

print(dp[N])

Solution (Python):

def examC(inf):
    N = I()
    A = []
    A.append(1)
    for i in range(1, 7):
        A.append(6 ** i)
    for i in range(1, 6):
        A.append(9 ** i)
    A.sort()
    dp = [10 * 5 + 10] * (N + 1)
    dp[0] = 0
    for i in range(1, N + 1):
        for s in range(len(A)):
            if i - A[s] >= 0:
                dp[i] = min(dp[i - A[s]] + 1, dp[i])
    ans = dp[N]
    print(ans)

import sys,copy,bisect,itertools,heapq,math
from heapq import heappop,heappush,heapify
from collections import Counter,defaultdict,deque
def I(): return int(sys.stdin.readline())
def LI(): return list(map(int,sys.stdin.readline().split()))
def LSI(): return list(map(str,sys.stdin.readline().split()))
def LS(): return sys.stdin.readline().split()
def SI(): return sys.stdin.readline().strip()
mod = 10**9 + 7
inf = float('inf')

examC(inf)

Solution (Python):
n = int(input())

ans = float("inf")

for i in range(n+1):
    kari = 0

    a = i
    while a >= 6:
        kari += a % 6
        a //= 6
    kari += a

    b = n - i
    while b >= 9:
        kari += b % 9
        b //= 9
    kari += b

    ans = min(ans, kari)

print(ans)

Solution (Python):
def nbasesum(value,n):
    nbasevalue = 0
    while value > 0:
        nbasevalue += value%n
        value //= n
    return nbasevalue

N = int(input())
ans = nbasesum(N,9)

for i in range(N+1):
    c6 = i
    c9 = N-i
    ans = min(ans, nbasesum(c9,9)+nbasesum(c6,6))

print(ans)

Solution (Python):
def nbasesum(value,n):
    nbasevalue = 0
    while value >= n:
        nbasevalue += value%n
        value //= n
    nbasevalue += value
    return nbasevalue

N = int(input())
ans = nbasesum(N,9)

for i in range(N+1):
    c6 = i
    c9 = N-i
    ans = min(ans, nbasesum(c9,9)+nbasesum(c6,6))

print(ans)

Solution (Python):
from collections import Counter,defaultdict,deque
import sys
import bisect
import math
import itertools
import string
import queue
import copy


from itertools import permutations, combinations
from heapq import heappop, heappush

sys.setrecursionlimit(10**8)
mod = 10**9+7
 
def inp(): 
    return int(input())
def inpm(): 
    return map(int,input().split())
def inpl(): 
    return list(map(int, input().split()))
def inpls(): 
    return list(input().split())
def inplm(n): 
    return list(int(input()) for _ in range(n))
def inpll(n): 
    return [list(map(int, input().split())) for _ in range(n)]

def main():
    n=inp()
    ans=n
    for t in range(min(5,n//46656)+1):
        x1 = n-t*46656
        for s in range(min(5,x1//7776)+1):
            x2=x1-7776*s
            for r in range(min(5,x2//1296)+1):
                x3=x2-1296*r
                for q in range(min(5,x3//216)+1):
                    x4=x3-216*q
                    for p in range(min(5,x4//36)+1):
                        x5=x4-36*p
                        for o in range(min(5,x5//6)+1):
                            x6=x5-6*o
                            n1=x6//59049
                            x6-=59049*n1
                            n2=x6//6561
                            x6-=6561*n2
                            n3=x6//729
                            x6-=729*n3
                            n4=x6//81
                            x6-=81*n4
                            n5=x6//9
                            x6-=9*n5
                            ans=min(ans,o+p+q+r+s+t+n1+n2+n3+n4+n5+x6)
    print(ans)

if __name__ == "__main__":
    main()

Solution (Python):
n = int(input())

max_N = 1000000
memo = [False]*max_N
import sys
import threading
sys.setrecursionlimit(100000)
threading.stack_size(128*1024*1024)



from functools import lru_cache

lru_cache(maxsize=100000)
def f(n):
	if n < 6:
		return n
	if memo[n]:
		
		return memo[n]
	
	ans = n
	
	x = 6
	while x <= n:
		ans = min(ans, f(n-x)+1)
		x *= 6
	
	x = 9
	while x <= n:
		ans = min(ans, f(n-x)+1)
		x *= 9
	
	memo[n] = ans
	return ans

def main():
	print(f(n))
	
threading.Thread(target=main).start()

Solution (Python):
N = int(input())
dp = dict()

dp[0] = 0

for n in range(1, N+1):
  dp[n] = dp[n-1] + 1
  
  
  x = 6
  while x <= n:
    dp[n] = min(dp[n], dp[n-x]+1)
    x *= 6
  
  x = 9
  while x <= n:
    dp[n] = min(dp[n], dp[n-x]+1)
    x *= 9
    
  
  
print(dp[N])

Solution (Python):
N = int(input())
dp = dict()

dp[0] = 0
for n in range(1, N+1):
  ans = dp[n-1] + 1
  
  
  x = 6
  while x <= n:
    ans = min(ans, dp[n-x]+1)
    x *= 6
  
  x = 9
  while x <= n:
    ans = min(ans, dp[n-x]+1)
    x *= 9
    
  dp[n] = ans
  
print(dp[N])

Solution (Python):
n=int(input())

m=n//6+n%6
for i in range(n+1):
  c=0
  t=i
  while t>0:
    c+=t%6
    t//=6
  t=n-i
  while t>0:
    c+=t%9
    t//=9
  m=min(m,c)

print(m)

Solution (Python):
def pow_n(n,k): 
    pow_n = 0
    while k //n != 0:
        pow_n += 1
        k = k//n
    return(pow_n)

n = int(input())
dp= [0]*(n+1)
dp[0] = 0

for i in range(1,n+1):
    pow_6 = pow_n(6,i)
    pow_9 = pow_n(9,i)
    dp[i] = dp[i-1]+1
    for j in range(1,pow_6+1):
        dp[i] = min(dp[i],dp[i-6**j]+1)
    for j in range(1,pow_9+1):
        dp[i] = min(dp[i],dp[i-9**j]+1)

print(dp[n])

Solution (Python):
N=int(input())
dp=[i for i in range(N+1)]
n=1
while n<=N:
  n*=6
  for j in range(N-n+1):
    dp[j+n]=min(dp[j+n],dp[j]+1)
    
n=1
while n<=N:
  n*=9
  for j in range(N-n+1):
    dp[j+n]=min(dp[j+n],dp[j]+1)

print(dp[N])

Solution (Python):
n=int(input())
dp=[10**6]*(n+1)
dp[0]=0
for i in range(n+1):
	j=0
	while i-6**j>=0:
		dp[i]=min(dp[i-6**j]+1,dp[i])
		j+=1
	j=0
	while i-9**j>=0:
		dp[i]=min(dp[i-9**j]+1,dp[i])
		j+=1
print(dp[-1])

Solution (Python):
n = int(input())
ans = float("inf")
for i in range(n):
    if i * 9 > n:
        break
    j = (n-(i*9)) // 6
    k = (n-(i*9)) % 6

    tmp = 0
    tmp += k
    while i:
        tmp += i % 9
        i = i // 9
    while j:
        tmp += j % 6
        j = j // 6
    if tmp < ans:
        ans = tmp

print(ans)

Solution (Python):
import sys
import math
if sys.platform =='ios':
  sys.stdin=open('Untitled.txt')
input = sys.stdin.readline


N = int(input())
dp = [float("INF") for _ in range(N+1)]
dp[0] = 0

MAX_JOU = int(math.ceil(math.log(N, 6)))

a_six = [6**x for x in range(MAX_JOU+1)]
a_nine = [9**x for x in range(MAX_JOU+1)]

for i in range(N+1):
    if (i-1 >= 0):
            dp[i] = min(dp[i], dp[i-1]+1)
    for six in a_six:
        if (i-six >= 0):
            dp[i] = min(dp[i], dp[i-six]+1)
    for nine in a_nine:
        if (i-nine >= 0):
            dp[i] = min(dp[i], dp[i-nine]+1)

print(dp[-1])

Solution (Python):
from sys import stdin
N = int(stdin.readline().rstrip())

dp = [99999999]*(N+1)
dp[0] = 0
for i in range(0, N):
    dp[i+1] = min(dp[i] + 1, dp[i+1])
    k = 6
    while i+k <= N:
        dp[i+k] = min(dp[i]+1, dp[i+k])
        k *= 6
    k = 9
    while i+k <= N:
        dp[i+k] = min(dp[i]+1, dp[i+k])
        k *= 9
print(dp[N])

Solution (Python):
n = int(input())
mn = 1e10

for i in range(n+1):
    ans = 0
    a, b = i, n-i

    while a>=6:
        a, mo = divmod(a, 6)
        ans += mo
    ans += a

    while b>=9:
        b, mo = divmod(b, 9)
        ans += mo
    ans += b

    mn = min(mn, ans)
print(mn)

Solution (Python):
n = int(input())
mn = 1e10

for i in range(n+1):
    ans = 0
    a, b = i, n-i

    di, mo = divmod(a, 6)
    ans += mo
    while di>=6:
        di, mo = divmod(di, 6)
        ans += mo
    ans += di

    di, mo = divmod(b, 9)
    ans += mo
    while di>=9:
        di, mo = divmod(di, 9)
        ans += mo
    ans += di

    mn = min(mn, ans)
print(mn)

Solution (Python):
n = int(input())

r = n

for i in range(n+1):
    cnt = 0
    nine = n - i
    while i > 0:
        cnt += i % 6
        i = i // 6
    while nine > 0:
        cnt += nine % 9
        nine = nine // 9
    if r > cnt: r = cnt

print(r)

Solution (Python):

import numpy as np

N = int(input())

dp = dict()
dp[0] = 0
for n in range(1, N+1):
  ans = dp[n-1] + 1
  
  x = 6
  while x <= n:
    ans = min(ans, dp[n-x]+1)
    x *= 6
  
  x = 9
  while x <= n:
    ans = min(ans, dp[n-x]+1)
    x *= 9
    
  dp[n] = ans
  
print(dp[N])

Solution (Python):
N = int(input())
dp = [10 ** 6] * (N + 1)
dp[0] = 0

for i in range(N+1):
    p = 1
    while p <= i:
        dp[i] = min(dp[i], dp[i-p] + 1)
        p *= 6
    p = 1
    while p <= i:
        dp[i] = min(dp[i], dp[i-p] + 1)
        p *= 9

print(dp[N])

Solution (Python):
N = int(input())
dp = [10 ** 6] * (N + 1)
dp[0] = 0

for i in range(N+1):
    p = 1
    while p <= N:
        dp[i] = min(dp[i], dp[i-p] + 1)
        p *= 6
    p = 1
    while p <= N:
        dp[i] = min(dp[i], dp[i-p] + 1)
        p *= 9

print(dp[N])

Solution (Python):
N = int(input())
res = N

for i in range(N+1):
    cc = 0
    t = i
    while t > 0:
        cc += int(t % 6)
        t //= 6
    t = N - i
    while t > 0:
        cc += int(t % 9)
        t //= 9
    res = min(res, cc)

print(int(res))

Solution (Python):
n=int(input())
dp=[10**5]*(n+1)
dp[0]=0
for i in range(n+1):
    pow=1
    while pow<=n:
        dp[i]=min(dp[i],dp[i-pow]+1)
        pow*=6
    pow=1
    while pow<=n:
        dp[i]=min(dp[i],dp[i-pow]+1)
        pow*=9
print(dp[n])

Solution (Python):
N=int(input())

C=[1]
for i in range(1,100):
    if 6**i<=100000:
        C.append(6**i)
    else:
        break
    
for i in range(1,100):
    if 9**i<=100000:
        C.append(9**i)
    else:
        break

dp=[10**6 for i in range(N+1)]
dp[0]=0
for i in C:
    for j in range(1,len(dp)):
        if j-i>=0:
            dp[j]=min(dp[j],dp[j-i]+1)
print(dp[N])

Solution (Python):
import bisect
N = int(input())
enable = set()

k = 0
while(6**k <= N):
    enable.add(6**k)
    k += 1
k = 0
while(9**k <= N):
    enable.add(9**k)
    k += 1


enable = list(enable)
enable.sort()

ans = 10**9
l = len(enable)
for i in range(l):
    n = N
    tmp = 0
    while(n != 0):
        id = bisect.bisect_right(enable, n) - 1
        if id != 0 and n % enable[id] > 2 and n % enable[id] < 6:
            id -= 1
        n -= enable[id]
        tmp += 1
    enable.pop()

    ans = min(tmp, ans)

print(ans)

Solution (Python):
N = int(input())
X = [0]
for n in range(1,N+1):
    x = X[-1]+1

    r6 = 6
    while n-r6>=0:
        x = min(x, X[n-r6]+1)
        r6*=6

    r9 = 9
    while n-r9>=0:
        x = min(x, X[n-r9]+1)
        r9*=9

    X.append(x)

print(X[N])

Solution (Python):
n=int(input())
num6=6
num9=9
l=[]
while num6<=n:
  l.append(num6)
  num6*=6
while num9<=n:
  l.append(num9)
  num9*=9
count=[0,1,2,3,4,5,1,2,3,1,2,3]
for i in range(12,n+1):
  min=100000
  for j in l:
    if i-j<0:
      pass
    else:
      if min>count[i-j]+1:
        min=count[i-j]+1
  count.append(min)
print(count[n])

Solution (Python):
n=int(input())
l6=[]
l9=[]
num6=6
num9=9
while num6<=n:
  l6.append(num6)
  num6*=6
while num9<=n:
  l9.append(num9)
  num9*=9
count=[0,1,2,3,4,5,1,2,3,1,2,3]
for i in range(12,n+1):
  min=100000
  for j in l6:
    if i-j<0:
      pass
    else:
      if min>count[i-j]+1:
        min=count[i-j]+1
  for j in l9:
    if i-j<0:
      pass
    else:
      if min>count[i-j]+1:
        min=count[i-j]+1
  count.append(min)
print(count[n])

Solution (Python):
N = int(input())

ans = 10 ** 5
for i in range(N+1):
    j = N-i
    count = 0
    while i > 0:
        count += i % 6
        i //= 6
    while j > 0:
        count += j % 9
        j //= 9
    ans = min(ans, count)

print(ans)

Solution (Python):
N = int(input())
import math 
m = math.floor(math.log(N,6)) 
n = math.floor(math.log(N,9)) 
s = [1]
for i in range(1,m+1):
    s.append(6**i)
for j in range(1,n+1):
    s.append(9**j)
s.sort()
l = len(s)
dp = [ i for i in range(N+1)]
for i in range(1,N+1):
    for j in range(1,6):
        m = 9**j
        
        K = min(i//m,8)
        for k in range(K+1):
            dp[i] = min(dp[i],dp[i-m*k]+k)
    for j in range(1,7):
        m = 6**j
        
        K = min(i//m,5)
        for k in range(K+1):
            dp[i] = min(dp[i],dp[i-m*k]+k)
print(dp[N])

Solution (Python):
import sys

N = int(input())
if N < 6:
	print(N)
	sys.exit()
elif N < 9:
	print(1+(N-6))
	sys.exit()

def Base_10_to_n(X, n):
    X_dumy = X
    out = ''
    while X_dumy>0:
        out = str(X_dumy%n)+out
        X_dumy = int(X_dumy/n)
    return out

ans = float("inf")

for i in reversed(range((N//9)+1)):
	t = N - 9*i
	t_6, t_1 = divmod(t, 6)

	time_9 = Base_10_to_n(i, 9)
	time_6 = Base_10_to_n(t_6, 6)

	ans = min(ans, sum(list(map(int, time_9))) + sum(list(map(int, time_6))) + t_1)

print(ans)

Solution (Python):
import sys
N = int(input())

dp = [10 ** 9 + 7] * (N + 1)
dp[0] = 0
for n in range(N+1):
    pow = 1
    while pow <= n:
        dp[n] = min(dp[n], dp[n-pow] + 1)
        pow *= 6
    pow = 1
    while pow <= n:
        dp[n] = min(dp[n], dp[n-pow] + 1)
        pow *= 9

print(dp[-1])

Solution (Python):
def main():
    n=int(input())
    ln=[1,6,9,36,81,216,729,1296,6561,7776,46656,59049]
    dp=[0 for i in range(n+1)]
    dp[1] = 1
    for i in range(2,n+1):
        lnt = [ln[j] for j in range(12) if i >= ln[j]]
        tdp=[]
        for slnt in lnt:
            tdp.append(dp[i-slnt])
        dp[i]=min(tdp)+1
    return dp[-1]

print(main())

Solution (Python):
n = int(input())
dp=[0]*(n+1)
dp[1]=1
j,k=1,1
for i in range(2,n+1):
    if i==6*j:
        j*=6
    if i==9*k:
        k*=9
    dp[i]=min(dp[i-k],dp[i-j],dp[i-1])+1
print(dp[n])

Solution (Python):
n= int(input())
dp=[2000000000]*(n+1)
dp[0]=0

for i in range(n+1):
    k=1
    while k<=i:
        dp[i]=min(dp[i],dp[i-k]+1)
        k*=6
    k=9
    while k<=i:
        dp[i]=min(dp[i],dp[i-k]+1)
        k*=9
print(dp[n])

Solution (Python):
N = int(input())


candidates = []
for base in [6, 9]:
    n = base
    while n <= N:
        candidates.append(n)
        n *= base




counts = [i for i in range(N + 1)]
for c in candidates:
    for i in range(c, N+1):
        counts[i] = min(counts[i], counts[i-c] + 1)

print(counts[N])

Solution (Python):
n = int(input())
dp = [float("inf")] * (n+1)
dp[0] = 0

for i in range(n+1):
    a = 1
    while a <= i:
        dp[i] = min(dp[i], dp[i-a] + 1)
        a *= 6
    a = 9
    while a <= i:
        dp[i] = min(dp[i], dp[i-a] + 1)
        a *= 9

print(dp[n])

Solution (Python):
import bisect
n = int(input())
add_num = 1
draw_money = []
while add_num <= 10 ** 5:
    draw_money.append(add_num)
    add_num *= 6
    
add_num = 1
while add_num <= 10 ** 5:
    draw_money.append(add_num)
    add_num *= 9


draw_money = list(set(draw_money))
ans = 0
draw_money.sort()

dp = [10000000] * (n + 1)
dp[0] = 0
for i in range(1,n + 1):
    for j in draw_money:
        if  i - j >= 0:
            dp[i] = min(dp[i - 1] + 1, dp[i - j] + 1,dp[i])
        else:
            dp[i] = min(dp[i-1] + 1, dp[i])

print(dp[n])