Math riddle: What is the smallest positive possible outcome?

Question

I spent hours trying to solve this riddle, I got $412-385=27$ as the smallest outcome, does anyone have a way to find the smallest positive outcome and prove it? The riddle:

You have the digits $1, 2, 3, 4, 5, 8.$ You need to place them in these squares without repeating the same digit twice so that you will get the smallest positive possible outcome:

The image preview isnt working. the image is in this link: https://i.stack.imgur.com/ip2rq.png — YahavB, Feb 27 '22 at 19:07
It might be fun for you to write a program in python, for example, to solve this. It would be good programming practice and will run very fast. — DOUGLAS BRUNSON, Feb 27 '22 at 19:09
The left most digits of $4$ and $3$ certainly look inevitable. If they are accepted, it is hard to then dispute the positions of the $1$ and $8$. — user2661923, Feb 27 '22 at 19:55

score 2 · Accepted Answer · answered Feb 27 '22 at 20:50

If you want a rigorous method, check this. By inserting the digits $x_1, x_2, x_3$ in the top and $x_4, x_5, x_6$ in the bottom, and taking the difference, the result is $$d = 100 (x_1 - x_4) + 10 (x_2 - x_5) + (x_3 - x_6)$$

Since we want $d >0$, we must have $x_1 - x_4 > 0$. To have it as small as possible, we need $x_1 - x_4 = 1$. We do not need to decide them atm. Now, for $x_2 - x_5$, we have no restrictions, since it is not possible to make $d$ negative if $100(x_1 - x_4) > 0.$ Hence, we need to make $x_2 - x_5$ as small as possible, which clearly is done by setting $x_2 = 1$ and $x_5 = 8$. Similarly for $x_3 - x_6$, but we can not choose 1 or 8, so the best option is $x_3 = 2$ and $x_6=5$. Now we could choose $x_1=4$ and $x_4 = 3$ and we get $$d = 412 - 385 = 27$$

In case of your python code, its much simpler to use the formula above for $d$. Something like this (pseudo):

for p in Permutations([1,2,3,4,5,8], size=6):
d = 100(p[0] - p[1]) + .....
if (d&gt;0)....

score 1 · Answer 2 · answered Feb 27 '22 at 19:51

Hi I solved the question using python, Here is the code:

from itertools import permutations
combinations1 = list(permutations('123458', 3))
lst = []
min1 = 731
flag = True
for i in range(len(combinations1)):
    for j in range(len(combinations1)):
        for k in combinations1[i]:
            if k in combinations1[j]:
                break;
        else:
            res = int(''.join(combinations1[i])) - int(''.join(combinations1[j]))
            if res > 0 and res < min1:
                min1 = res
print(min1)

score 1 · Answer 3 · answered Feb 27 '22 at 20:15

1

Step 1:In the hundreds position of both numbers, the difference should be 1, since it is the smallest. So you put 2 (above) and 1 (below), or 3 and 2, or 4 and 3, or 5 and 4, depending on the next step. Step 2:The next two positions in both numbers (tens and units) should be: above the smallest number , below the largest number. The smallest is 12 and the largest is 85, so above we have 12 and below 85. We have 4 and 3 still available, so above 4 and below 3. The smallest difference is indeed 412-385=27

answered Feb 27 '22 at 20:15

Alexandru Ghiveciu

13

As it’s currently written, your answer is unclear. Please [edit] to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. – Community Feb 27 '22 at 20:20

Math riddle: What is the smallest positive possible outcome?

3 Answers3