For the solenoids shown in the diagram (which are assumed to be close to each other), the resistance of the left-hand circuit is slowly increased. In which direction does the current flow through galvanometer in the right-hand circuit?

Increasing the resistance causes a decrease in the current on the left. This reduces the field strength for the solenoid B=μ0nI. By the right-hand rule, the field through the solenoid is directed to the right. Hence, in the right-hand circuit. Since there will be fewer field lines directed to the right, there is an induced electric field that will produce a current in the wires to "replace" the lost field lines. This means that current will be directed with the same orientation as the current in the left circuit. ∴ The galvanometer will deflect to the left.