I was given the following task: Richard likes to ask his classmates questions like the following:
4 x 4 x 3 = 13
In order to solve this task, his classmates have to fill in the missing arithemtic operators (+, -, *, /) that the created equation is true. For this example the correct solution would be 4 * 4 - 3. Write a piece of code that creates similar questions to the one above. It should comply the following rules:
Create riddles up to 15 arithmetic operands, which follow these rules.
My progress so far:
import random
import time
import sys
add = lambda a, b: a + b
sub = lambda a, b: a - b
mul = lambda a, b: a * b
div = lambda a, b: a / b if a % b == 0 else 0 / 0
operations = [(add, '+'),
(sub, '-'),
(mul, '*'),
(div, '/')]
operations_mul = [(mul, '*'),
(add, '+'),
(sub, '-'),
(div, '/')]
res = []
def ReprStack(stack):
reps = [str(item) if type(item) is int else item[1] for item in stack]
return ''.join(reps)
def Solve(target, numbers):
counter = 0
def Recurse(stack, nums):
global res
nonlocal counter
valid = True
end = False
for n in range(len(nums)):
stack.append(nums[n])
remaining = nums[:n] + nums[n + 1:]
pos = [position for position, char in enumerate(stack) if char == operations[3]]
for j in pos:
if stack[j - 1] % stack[j + 1] != 0:
valid = False
if valid:
if len(remaining) == 0:
# Überprüfung, ob Stack == target
solution_string = str(ReprStack(stack))
if eval(solution_string) == target:
if not check_float_division(solution_string)[0]:
counter += 1
res.append(solution_string)
if counter > 1:
res = []
values(number_ops)
target_new, numbers_list_new = values(number_ops)
Solve(target_new, numbers_list_new)
else:
for op in operations_mul:
stack.append(op)
stack = Recurse(stack, remaining)
stack = stack[:-1]
else:
if len(pos) * 2 + 1 == len(stack):
end = True
if counter == 1 and end:
print(print_solution(target))
sys.exit()
stack = stack[:-1]
return stack
Recurse([], numbers)
def simplify_multiplication(solution_string):
for i in range(len(solution_string)):
pos_mul = [position for position, char in enumerate(solution_string) if char == '*']
if solution_string[i] == '*' and len(pos_mul) > 0:
ersatz = int(solution_string[i - 1]) * int(solution_string[i + 1])
solution_string_new = solution_string[:i - 1] + solution_string[i + 1:]
solution_string_new_list = list(solution_string_new)
solution_string_new_list[i - 1] = str(ersatz)
solution_string = ''.join(str(x) for x in solution_string_new_list)
else:
return solution_string
return solution_string
def check_float_division(solution_string):
pos_div = []
solution_string = simplify_multiplication(solution_string)
if len(solution_string) > 0:
for i in range(len(solution_string)):
pos_div = [position for position, char in enumerate(solution_string) if char == '/']
if len(pos_div) == 0:
return False, pos_div
for j in pos_div:
if int(solution_string[j - 1]) % int(solution_string[j + 1]) != 0:
# Float division
return True, pos_div
else:
# No float division
return False, pos_div
def new_equation(number_ops):
equation = []
operators = ['+', '-', '*', '/']
ops = ""
if number_ops > 1:
for i in range(number_ops):
ops = ''.join(random.choices(operators, weights=(4, 4, 4, 4), k=1))
const = random.randint(1, 9)
equation.append(const)
equation.append(ops)
del equation[-1]
pos = check_float_division(equation)[1]
if check_float_division(equation):
if len(pos) == 0:
return equation
for i in pos:
equation[i] = ops
else:
'''for i in pos:
if equation[i+1] < equation[i-1]:
while equation[i-1] % equation[i+1] != 0:
equation[i+1] += 1'''
new_equation(number_ops)
else:
print("No solution with only one operand")
sys.exit()
return equation
def values(number_ops):
target = 0
equation = ''
while target < 1:
equation = ''.join(str(e) for e in new_equation(number_ops))
target = eval(equation)
numbers_list = list(
map(int, equation.replace('+', ' ').replace('-', ' ').replace('*', ' ').replace('/', ' ').split()))
return target, numbers_list
def print_solution(target):
equation_encrypted_sol = ''.join(res).replace('+', '○').replace('-', '○').replace('*', '○').replace('/', '○')
print("Try to find the correct operators " + str(equation_encrypted_sol) + " die Zahl " + str(
target))
end_time = time.time()
print("Duration: ", end_time - start_time)
input(
"Press random button and after that "ENTER" in order to generate result")
print(''.join(res))
if __name__ == '__main__':
number_ops = int(input("Number of arithmetic operators: "))
# number_ops = 10
target, numbers_list = values(number_ops)
# target = 590
# numbers_list = [9, 3, 5, 3, 5, 2, 6, 3, 4, 7]
start_time = time.time()
Solve(target, numbers_list)
Basically it's all about the "Solve(target, numbers)" method, which returns the correct solution. It uses Brute-Force in order to find that solution. It receives an equation and the corresponding result as an input, which has been generated in the "new_equation(number_ops)" method before. This part is working fine and not a big deal. My main issue is the "Solve(target, numbers)" method, which finds the correct solution using a stack. My aim is to make that program as fast as possible. Currently it takes about two hours until an arithmetic task with 15 operators has been found, which follows the rules above. Is there any way to make it faster or maybe another approach to the problem besides Brute-Force? I would really appreciate your help:)
This is mostly brute force but it only takes a few seconds for a 15 operation formula.
In order to check the result, I first made a solve
function (recursive iterator) that will produce the solutions:
def multiplyDivide(numbers):
if len(numbers) == 1: # only one number, output it directly
yield numbers[0],numbers
return
product,n,*numbers = numbers
if product % n == 0: # can use division.
for value,ops in multiplyDivide([product//n]+numbers):
yield value, [product,"/",n] + ops[1:]
for value,ops in multiplyDivide([product*n]+numbers):
yield value, [product,"*",n] + ops[1:]
def solve(target,numbers,canGroup=True):
*others,last = numbers
if not others: # only one number
if last == target: # output it if it matches target
yield [last]
return
yield from ( sol + ["+",last] for sol in solve(target-last,others)) # additions
yield from ( sol + ["-",last] for sol in solve(target+last,others)) # subtractions
if not canGroup: return
for size in range(2,len(numbers)+1):
for value,ops in multiplyDivide(numbers[-size:]): # multiplicative groups
for sol in solve(target,numbers[:-size]+[value],canGroup=False):
yield sol[:-1] + ops # combined multipicative with rest
The solve
function recurses through the numbers building an addition or subtraction with the last number and recursing to solve a smaller problem with the adjusted target and one less number.
In addition to the additions and subtraction, the solve
function groups the numbers (from the end) into consecutive multiplications/divisions and processes them (recursing into solve
) using the resulting value that will have calculation precedence over additions/subtractions.
The multiplyDivide
function (also a recursive generator) combines the group of numbers it is given with multiplications and divisions performed from left to right. Divisions are only added when the current product divided by the additional number produces an integer intermediate result.
Using the solve
iterator, we can find a first solution and know if there are more by iterating one additional time:
def findOper(S):
expression,target = S.split("=")
target = int(target.strip())
numbers = [ int(n.strip()) for n in expression.split("x") ]
iSolve = solve(target,numbers)
solution = next(iSolve,["no solution"])
more = " and more" * bool(next(iSolve,False))
return " ".join(map(str,solution+ ["=",target])) + more
Output:
print(findOper("10 x 5 = 2"))
# 10 / 5 = 2
print(findOper("10 x 5 x 3 = 6"))
# 10 / 5 * 3 = 6
print(findOper("4 x 4 x 3 = 13"))
# 4 * 4 - 3 = 13
print(findOper("1 x 3 x 4 = 13"))
# 1 + 3 * 4 = 13
print(findOper("3 x 3 x 4 x 4 = 25"))
# 3 * 3 + 4 * 4 = 25
print(findOper("2 x 6 x 2 x 4 x 4 = 40"))
# 2 * 6 * 2 + 4 * 4 = 40 and more
print(findOper("7 x 6 x 2 x 4 = 10"))
# 7 + 6 * 2 / 4 = 10
print(findOper("2 x 2 x 3 x 4 x 5 x 6 x 3 = 129"))
# 2 * 2 * 3 + 4 * 5 * 6 - 3 = 129
print(findOper("1 x 2 x 3 x 4 x 5 x 6 = 44"))
# 1 * 2 + 3 * 4 + 5 * 6 = 44
print(findOper("1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 8 x 7 x 6 x 5 x 4 x 1= 1001"))
# 1 - 2 - 3 + 4 * 5 * 6 * 7 / 8 * 9 + 8 * 7 - 6 + 5 + 4 + 1 = 1001 and more
print(findOper("1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 8 x 7 x 6 x 5 x 4 x 1= 90101"))
# no solution = 90101 (15 seconds, worst case is not finding any solution)
In order to create a single solution riddle, the solve
function can be converted to a result generator ( genResult
) that will produce all possible computation results. This will allow us to find a result that only has one combination of operations for a given list of random numbers. Given that the multiplication of all numbers is very likely to be a unique result, this will converge rapidly without having to go through too many random lists:
import random
def genResult(numbers,canGroup=True):
*others,last = numbers
if not others:
yield last
return
for result in genResult(others):
yield result - last
yield result + last
if not canGroup: return
for size in range(2,len(numbers)+1):
for value,_ in multiplyDivide(numbers[-size:]):
yield from genResult(numbers[:-size]+[value],canGroup=False)
def makeRiddle(size=5):
singleSol = []
while not singleSol:
counts = dict()
numbers = [random.randint(1,9) for _ in range(size)]
for final in genResult(numbers):
if final < 0 : continue
counts[final] = counts.get(final,0) + 1
singleSol = [n for n,c in counts.items() if c==1]
return " x ".join(map(str,numbers)) + " = " + str(random.choice(singleSol))
The reason we have to loop for singleSol
(single solution results) is that there are some cases where even the product of all numbers is not a unique solution. For example: 1 x 2 x 3 = 6
could be the product 1 * 2 * 3 = 6
but could also be the sum 1 + 2 + 3 = 6
. There aren't too many of those cases but it's still a possibility, hence the loop. In a test generating 1000 riddles with 5 operators, this occurred 4 times (eg 4 x 1 x 1 x 4 x 2
has no unique solution). As you increase the size of the riddle, the occurrence of these no-unique-solution patterns becomes more frequent (eg 6 times generating 20 riddles with 15 operators).
output:
S = makeRiddle(15) # 7 seconds
print(S)
# 1 x 5 x 2 x 5 x 8 x 2 x 3 x 4 x 1 x 2 x 8 x 9 x 7 x 3 x 2 = 9715
print(findOper(S)) # confirms that there is only one solution
# 1 * 5 * 2 * 5 * 8 * 2 * 3 * 4 - 1 + 2 + 8 * 9 + 7 * 3 * 2 = 9715
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