The only sure-fire way to kill head lice is to suffocate them. Lice can hold their breath for up to 20 minutes, so you need between 30 minutes and an hour to run through a treatment.
To suffocate lice you need to coat them in some kind of oily substance that sticks around until you choose to wash / wipe it off.
Almost anything oily will work (anything from hair conditioner to mayonnaise!) but the best treatment - and the one recommended by the ministry of health is a 4% Dimethicone-based lotion. Dimethicone is an odourless silicone-like substance that coats the hair (and the lice at the same time) making it impossible for them to breathe.
To kill all the lice
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Starting at the roots and focusing on the scalp (where lice like to live) spread the lotion over the whole hair, making sure you coat all the hair. Give it a good shmoosh around.
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Leave the lotion in the hair for 30 mins. After 30 minutes all the lice should be dead.
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Wash the hair with normal shampoo to get rid of the lotion. You might need to wash it a couple of times to get rid of all the oily residue
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Dry the hair normally and brush the dead lice out. They will fall out naturally over time, so you don’t need to go crazy
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Check you’ve got them all by doing a careful and thorough workover of the head - check carefully behind ears and at the nape of the neck for any lice that didn’t get coated.
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Repeat if necessary
Once you’ve killed all the live lice you need to get rid of the nits (the lice eggs) to avoid getting a re-infection when the eggs hatch in 7 days.