This circuit will drive a super-bright white LED from a 1.5v cell. The 60 turn inductor is wound on a small ferrite slug 2.6mm dia and 6mm long with 0.25mm wire.
The main difference between this circuit and the two circuits above is the use of a single winding and the feedback to produce oscillation comes from a 1n capacitor driving a high gain amplifier made up of two transistors.
The feedback is actually positive feedback via the 1n and this turns on the two transistors more and more until finally they are fully turned on and no more feedback signal is passed though the 1n. At this point they start to turn off and the signal through the 1n turns them off more and more until they are fully turned off. The 33k turns on the BC557 to start the cycle again.
If you do not have a ferrite slug, the inductor can be made from a machine screw 10mm long and about 3-4mm dia. Wind 150 turns of 0.25mm wire. Or you can use a brass ferrule 20mm long x 5mm. Wind 150 turns.
RESULTS for the same brightness:
Brass Spacer: 18mA
Machine screw: 14mA