Solution

Add a 1 Amp test source at the terminals of the network. If we can find the voltage across the source, then Req = Vtest/Itest.
In order to find Vtest, let's try node-voltage analysis.

I chose the bottom as the reference node. I labeled three nodes V1, V2, V3. (and Vtest=V1). Here are my KCL equations:
KCL for V1:
-1 + (V1-V2)/2k +(V1-V3)/1k=0
KCL for V2:
V2/1k + (V2-V1)/2k + (V2-V3)/1k=0
KCL for V3:
V3/1k + (V3-V2)/1k + (V3-V1)/1k=0

Solve this set of three equations with three unknowns. First I multiply everything through by 2k to clean up the denominators:

-2k + (V1-V2) + 2*(V1-V3)=0
2V2 + (V2-V1) + 2*(V2-V3)=0
2V3 + 2*(V3-V2) + 2*(V3-V1)=0

Now group like terms together so that the unknowns line up:

3V1 - V2 - 2V3 = 2k
- V1 + 5V2 -2V3 = 0
-2V1 - 2V2 + 6V3=0

I'll eliminate V3 by subtracting the second equation from the first and adding three times the second equation to the third:

4V1 - 6V2 = 2k
-5V1 + 13V2=0

Now eliminate V2 by adding 13/6 times the first equation to the second:

(4*13/6 -5)V1 = (13/6)2k

V1 = (2k*13/6)/(4*13/6 -5) = 1.18kV

Now that we have Vtest,

Req = Vtest/Itest

= 1.18kV/1A

= 1.18Kohms.