Double Cryptogram

From the conditions of the problem, $c+d=a+b$ . We need to find ways of writing integers from $5$ to $15$ as sum of $2$ distinct positive integers in $2$ distinct ways.

To justify the lower bound of the sum as $5$ , note that $5$ is the first integer which can be expressed as sum of $2$ distinct positive integers in $2$ distinct ways. We can write $4$ as $1+3$ or $2+2$ , but the latter is not allowed since the two integers must be distinct. Similarly, the upper bound of the sum is $15$ . There is only one way to express $16$ as sum of two distinct positive integers, that is $9+7$ .

Below are the ways of writing $5, \cdots , 15$ as sum of two distinct positive integers.

$5=1+4=2+3$

$6=1+5=2+4$

$7=1+6=2+5=3+4$

$8=1+7=2+6=3+5$

$9=1+8=2+7=3+6=4+5$

$10=1+9=2+8=3+7=4+6$

$11=2+9=3+8=4+7=5+6$

$12=3+9=4+8=5+7$

$13=4+9=5+8=6+7$

$14=5+9=6+8$

$15=6+9=7+8$

Let $n$ number of possible ways to write a certain integer as sum of $2$ distinct positive integers. To enumerate the number of quadruples $(a,b,c,d)$ , we take $8\displaystyle \sum_{} {n \choose 2}$ .

The factor of $8$ is introduced to account for permutations. Note that $(a,b)$ is interchangeable with $(c,d)$ and the elements in $(a,b)$ and $(c,d)$ themselves are interchangeable.

A quick computation yields the answer.

$8(4. {2 \choose 2} +4. {3\choose 2}+ 3. {4 \choose 2}) = 8(34)= \boxed{272}$

Here is my code in python.

I use this JS code (paste it in Chrome Console)

I used this python 3 code:

ans=0
for a in range(1,10):
    for b in range(1,10):
        for c in range(1,10):
            for d in range(1,10):
                if 1000*a+100*b+10*c+d+1000*b+100*c+10*d+a==1000*d+100*c+10*b+a+1000*c+100*b+10*a+d and a!=b and b!=c and c!=d and a!=d and b!=d and a!=c:
                    ans+=1
                else:
                    pass
print (ans)