Spoiler for New Scientist Enigma 1748: “Quo Vadis?” (Follow the link to see the puzzle.)
I’ll render the starting position in ASCII, and label rows and columns, like this:
a

b

c

d



1 
3>

<1


2  
3  
4 
We’re looking for a closed loop, with each square pointed to by exactly one other square — I assume that’s what they mean by “circuit” (and there is such a solution). So from c1 we cannot use <1 to b1 (there isn’t one) or <2 to a1 (short circuit, i.e. we’ve formed a loop that doesn’t get to all 12 blank squares) so must use 3v to c4.
From b1 we cannot use <1 to a1 (there isn’t one) or 1> to c1 (non loop, i.e. two squares both point to c1) so must use 1v to b2 or 3v to b4. But 3v is used at c4. Therefore b1 must be 1v. Now the only way to get to b1 is 3^ from b4, and the only way out of b2 is 2> to d2.
To get to a1, we cannot use 3^ from a4 (in use), so must use 2^ from a3.
To get to a3 we must use 1^ from a4 or ❤ from d3. 1^ doesn’t work; in that case a4 cannot be reached from c4 with <2 (short circuit) so must be ❤ from d4, then c4 can only be 1> to d4 (short circuit). So d3 is <3.
Now at a4 we can’t use 1^ (non loop) so it must be 1>. We can only get to a4 with <2 from c4. Then 2v at d2 and 1^ at d4 complete the circuit:
a

b

c

d



1 
3>

1v

3v

<1

2 
2>

2v


3 
2^

❤


4 
1>

3^

<2

1^
