Question

Find the number of solutions to the equation
1/a+1/b+1/c+1/d=1
where a, b, c, d are positive integers and a≤b≤c≤d

Answer 1

Starting with the smallest values of a,b,c,d we can see that

1/2 + 1/3 + 1/6 = 1

but that's only 3 fractions. So, if we can split 1/6 into two fractions, we are ok:

1/6
= 1/7 + 1/42
= 1/8 + 1/24
= 1/9 + 1/18
= 1/10 + 1/15
= 1/11 + 5/66 but that's not allowed
= 1/12 + 1/12 and we are done, since there are no other possibilities. That is, any higher divisors won't be unit fractions (except those already listed above)

That gives us 4 solutions starting with 1/2 + 1/3 + * + *

Now, try 1/2+1/4+*+* where *+*=1/4 and so on till you run out of possibilities.