Same Diagonal, Same Area, Different Boxes

From the conditions in the problem, we can set up equations as:

$a^2 + b^2 + c^2 = d^2 + e^2 + f^2$ for the same diagonal constraint.

$2(ab + bc + ca) = 2(de + ef + fd)$ for the same area constraint.

Adding both equations together, we will get:

$a^2 + b^2 + c^2 + 2(ab + bc + cd) = d^2 + e^2 + f^2 + 2(de + ef + fd)$

$(a+ b+ c)^2 = (d + e+f)^2$

Because all variables are natural numbers, we can conclude $a + b + c = d + e + f$ .

With $a + b + c = d + e + f$ , we are looking for the number partition into six different numbers. Taking $a, b, c, d, e, f \in S$ for some space set of natural numbers, if $S = \{1, 2, 3, 4, 5, 6\}$ , whose elements are as least as possible, no equal summation is possible because $1+2+3+4+5+6 = 21$ , which can't be halved.

Hence, $7$ must be involved in $S$ . Then if $S = \{1, 2, 3, 4, 5, 6, 7\}$ , we will have $1+2+3+4+5+6+7 = 28$ . Thus, the summation will be less than $\dfrac{28}{2} = 14$ because only six numbers can be selected, and the excluded one must be even to follow the constraint.

To yield the least summation, we will attempt to examine the trials by excluding the most possible even integer. First, by taking $6$ out, we will reach $11 = 7+3+1 = 5+4+2$ as the only possible partition. However, $7^2 +3^2 +1^2 \neq 5^2+4^2+2^2$ . It's not our desired solution.

Then by taking $4$ out, we will reach $12 = 7+3+2 = 6+5+1$ as the only possible partition, and $7^2 +3^2 +2^2 = 6^2+5^2+1^2 = 62$ , which works perfectly.

As a result, the least possible total surface area equals $2(7\times 3 + 3\times 2 + 2\times 7) = 2(6\times 5 + 5\times 1 + 1\times 6) = \boxed{82}$ .

If we observe properly then there is a very nice pattern in the solutions which is ac=df,b=d+f,e=a+c. Now if ac=df=6, (for least value) a,b,c,d,e,f=2,7,3,1,5,6; respectively so the minimum area is 2(2 7+2 3+7*3)=82...

I wrote my solution in python. At first I had the max set to 5 but after running it for 10 minutes no answers would show up, so I increased it to 20 and I got a whole bunch. So I let it run for a bit and 82 was the smallest number it came up with.

import random

allsets = []
max = 20

lowest = 10000000000

def generateSet():
    a = random.randint(1, max)
    b = a
    while b == a:
        b = random.randint(1, max)
    c = b
    while c == a or c == b:
        c = random.randint(1, max)
    d = c
    while d == a or d == b or d == c:
        d = random.randint(1, max)
    e = d
    while e == a or e == b or e == c or e == d:
        e = random.randint(1, max)
    f = e
    while f == a or f == b or f == c or f == d or f == e:
        e = random.randint(1, max)
    theset = set([a, b, c, d, e, f])
    if theset in allsets:
        return generateSet()
    else:
        allsets.append(theset)
        return [a,b,c, d,e,f]

def calculatearea(a, b, c):
    return 2*(a*b + a*c + b*c)

def calculatediagonalsq(a, b, c):
    return a**2 + b**2 + c**2

def checkarea(a,b,c, d,e,f):
    return calculatearea(a,b,c) == calculatearea(d,e,f)

def checkdiagonal(a,b,c, d,e,f):
    return calculatediagonalsq(a,b,c) == calculatediagonalsq(d,e,f)

while True:
    theset = generateSet()
    a = theset[0]
    b = theset[1]
    c = theset[2]
    d = theset[3]
    e = theset[4]
    f = theset[5]
    if checkdiagonal(a,b,c, d,e,f) and checkarea(a,b,c, d,e,f) and calculatearea(a,b,c) < lowest:
        lowest = calculatearea(a,b,c)
        print(calculatearea(a,b,c))
        print(a,b,c, d,e,f)
    if checkdiagonal(a,b,d, c,e,f) and checkarea(a,b,d, c,e,f) and calculatearea(a,b,d) < lowest:
        lowest = calculatearea(a,b,d)
        print(calculatearea(a,b,d))
        print(a,b,d, c,e,f)
    if checkdiagonal(a,b,e, d,c,f) and checkarea(a,b,e, d,c,f) and calculatearea(a,b,e) < lowest:
        lowest = calculatearea(a,b,e)
        print(calculatearea(a,b,e))
        print(a,b,e, d,c,f)
    if checkdiagonal(a,b,f, d,e,c) and checkarea(a,b,f, d,e,c) and calculatearea(a,b,f) < lowest:
        lowest = calculatearea(a,b,f)
        print(calculatearea(a,b,f))
        print(a,b,f, d,e,c)
    if checkdiagonal(a,c,d, b,e,f) and checkarea(a,c,d, b,e,f) and calculatearea(a,c,d) < lowest:
        lowest = calculatearea(a,c,d)
        print(calculatearea(a,c,d))
        print(a,c,d, b,e,f)
    if checkdiagonal(a,c,e, b,d,f) and checkarea(a,c,e, b,d,f) and calculatearea(a,c,e) < lowest:
        lowest = calculatearea(a,c,e)
        print(calculatearea(a,c,e))
        print(a,c,e, b,d,f)
    if checkdiagonal(a,c,f, b,d,e) and checkarea(a,c,f, b,d,e) and calculatearea(a,c,f) < lowest:
        lowest = calculatearea(a,c,f)
        print(calculatearea(a,c,f))
        print(a,c,f, b,d,e)
    if checkdiagonal(a,d,e, b,c,f) and checkarea(a,d,e, b,c,f) and calculatearea(a,d,e) < lowest:
        lowest = calculatearea(a,d,e)
        print(calculatearea(a,d,e))
        print(a,d,e, b,c,f)
    if checkdiagonal(a,d,f, b,c,e) and checkarea(a,d,f, b,c,e) and calculatearea(a,d,f) < lowest:
        lowest = calculatearea(a,d,f)
        print(calculatearea(a,d,f))
        print(a,d,f, b,c,e)
    if checkdiagonal(a,e,f, b,c,d) and checkarea(a,e,f, b,c,d) and calculatearea(a,e,f) < lowest:
        lowest = calculatearea(a,e,f)
        print(calculatearea(a,e,f))
        print(a,e,f, b,c,d)