# What numbers fill the blanks to satisfy the equation?

Complete the grid shown below with the digits 1 to 6 to make the sum correct.

Perform each mathematical operation in the order shown, from left to right, e.g. 1 + 2 x 3 is treated as (1 + 2) x 3 = 9.

Note: there is no ÷ 1, and at no point is a decimal or fraction used.

$$[\;\;\;\;] + [\;\;\;\;] - [\;\;\;\;] * [\;\;\;\;] \div [\;\;\;\;] * [\;\;\;\;] = 50$$

The $[\;\;\;\;]$ are empty spaces and must be filled with numbers.

I was just a little slower than Bailey, but to add further detail:

You start by working out that the last number can only be:

5,2 or 1: 5 is more likely as it involves a smaller answer to using the other numbers

This makes:

[ ] + [ ] - [ ] x [ ] ÷ [ ] = 10

You now want a small number to divide by, that isn't 1 - so this should either be 2 or 3 making

[ ] + [ ] - [ ] x [ ] = 20

Or

[ ] + [ ] - [ ] x [ ] = 30

With the 4 numbers and 3 sums remaining, it is impossible to make the former and are forced down the latter route to create:

4 + 2 - 1 x 6 / 3 x 5 = 50

• I went through the exact same thought process as this - I just so happened to try ÷3 first, which gave me a correct answer and therefore no reason to try ÷2. – Bailey M Jun 8 '15 at 16:03

I "cheated" and wrote a Python script that used the Johnson-Trotter algorithm to permute the order of the numbers that are plugged into the equation. My script spits out 6 possible solutions:

3 + 6 - 4 x 5 ÷ 1 x 2
6 + 3 - 4 x 5 ÷ 1 x 2
6 + 3 - 4 x 2 ÷ 1 x 5
3 + 6 - 4 x 2 ÷ 1 x 5
4 + 2 - 1 x 6 ÷ 3 x 5
2 + 4 - 1 x 6 ÷ 3 x 5

Though, given the extra conditions of the puzzle (no ÷1 and never dealing with decimals/fractions), you're left with 2 solutions because addition is commutative.

4 + 2 - 1 x 6 ÷ 3 x 5
2 + 4 - 1 x 6 ÷ 3 x 5

I'm sure the pen-and-paper method accomplished above is probably more satisfying to perform, though!

Python Source

Play with the code here.

This is just a Johnson-Trotter permutation class I wrote. It uses a step function to go to the next permutation. I thought about making an iterator version, but was too lazy =/. It may be a little lengthy, but it can be used to permute any list of objects. I'm sure it could be improved and condensed; I just threw this together for this puzzle =P

jtpermute.py

class JohnsonTrotterPermute(object):
objlist = []   # contains the objects given to the class
nodelist = []  # contains the JTNode objects that wrap around each given object
numnodes = 0   # the current number of nodes added (for numbering)
steps = 1      # the number of iterations generated (number of steps)

def __init__(self, objlist):
self.objlist = objlist
for obj in self.objlist:
self.nodelist.append(JTNode(self.numnodes, obj))
self.numnodes += 1

# returns True if there exists at least one mobile JTNode
def has_mobile(self):
for i in range(len(self.nodelist)):
if self.check_mobility(i):
return True
return False

# returns True if the JTNode at the given index is mobile
def check_mobility(self, index):
if index < 0 or index >= len(self.nodelist):
return False
if self.nodelist[index].direction == -1: # pointing left
if index == 0: # on the left edge of list
return False
if self.nodelist[index].num > self.nodelist[index - 1].num:
return True
if self.nodelist[index].direction == 1:  # pointing right
if index == len(self.nodelist) - 1:  # on the right edge of list
return False
if self.nodelist[index].num > self.nodelist[index + 1].num:
return True
return False

# iterates through the nodes and flips the direction of any node
# whose .num is larger than the given number
def flip_larger_than_num(self, num):
for node in self.nodelist:
if (node.num > num):
node.flip()

# returns the index of the JTNode that has the largest .num value
# and is mobile
def get_largest_mobile_index(self):
index = -1
largest = -1
for i in range(len(self.nodelist)):
if self.nodelist[i].num > largest and self.check_mobility(i):
largest = self.nodelist[i].num
index = i
return index

# swaps the node at the given index with the node that it's pointing
# at. Returns True if the swap is successful, False if the swap fails
# (e.g. the node is pointing left, but it's the left-most node)
def move_node(self, index):
node = self.nodelist[index]
tempnode = None
if node.direction == -1: # pointed left
if index == 0: # on left edge of list
return False
tempnode = self.nodelist[index]
self.nodelist[index] = self.nodelist[index - 1]
self.nodelist[index - 1] = tempnode
self.flip_larger_than_num(tempnode.num)
if node.direction == 1: # pointed right
if index == len(self.nodelist) - 1: # on right edge of list
return False
tempnode = self.nodelist[index]
self.nodelist[index] = self.nodelist[index + 1]
self.nodelist[index + 1] = tempnode
self.flip_larger_than_num(tempnode.num)
return True

# steps the list order to its next iteration
def step(self):
if self.has_mobile():
largest_mobile = self.get_largest_mobile_index()
self.move_node(largest_mobile)
self.steps += 1
return True
return False

# returns the list of items that the user initially provided in the
# order of the current iteration
def get_items(self):
returnlist = []
for node in self.nodelist:
returnlist.append(node.obj)
return returnlist

# a wrapper class for the list items the user will provide
class JTNode(object):
obj = None      # the actual user-provided list item
num = -1        # the number of the node
direction = -1  # pointing direction (-1 -> left, 1 -> right)

def __init__(self, num, obj):
self.obj = obj
self.num = num

# flip the direction the node is pointing
def flip(self):
self.direction *= -1


puzzle.py

from jtpermute import *

# computes the puzzle equation on the given list of integers
def do_maths(intlist):
if len(intlist) < 6:
return 0
num = intlist
num += intlist
num -= intlist
num *= intlist
num /= intlist
num *= intlist
return num

# prints the puzzle's function with the numbers plugged in
def print_maths(intlist):
if len(intlist) < 6:
return 0
# a string builder of sorts
strlist = []
strlist.append(str(intlist))
strlist.append(" + ")
strlist.append(str(intlist))
strlist.append(" - ")
strlist.append(str(intlist))
strlist.append(" * ")
strlist.append(str(intlist))
strlist.append(" / ")
strlist.append(str(intlist))
strlist.append(" * ")
strlist.append(str(intlist))
strlist.append(" = ")
strlist.append(str(do_maths(intlist)))
print "".join(strlist)

jtp = JohnsonTrotterPermute([1, 2, 3, 4, 5, 6])
# the equivalent of a do-while loop in Python. I wanted to put the
# step function in the loop conditional, but it had to be evaluated
# before the first .step call. When the step function returns False,
# it has reached the end of the permutations
while True:
if do_maths(jtp.get_items()) == 50:
print_maths(jtp.get_items())
if jtp.step() == False:
break
print "Number of permutations: %d" % jtp.steps


Output of puzzle.py

3 + 6 - 4 * 5 / 1 * 2 = 50
6 + 3 - 4 * 5 / 1 * 2 = 50
6 + 3 - 4 * 2 / 1 * 5 = 50
3 + 6 - 4 * 2 / 1 * 5 = 50
4 + 2 - 1 * 6 / 3 * 5 = 50
2 + 4 - 1 * 6 / 3 * 5 = 50
Number of permutations: 720

• Could you please show your Python script? – Kritixi Lithos Jun 9 '15 at 4:46

 +  -  x  ÷  x  = 50

Or, a cleaner version:

((4 + 2) - 1) x ((6 / 3) x 5) = 50