Over here we use Special Effects. I don't know if you live in a large or small town, but often this can be purchased at a "punk rock barber shop" type place. You shouldn't pay much more than ten dollars for it. A place like the punk mall store chain "Hot Topic" will absolutely gouge you for 15 dollars or more, which is okay if you just want one bottle one time. I do not recommend the "classic" brands Punky Color or Manic Panic, because in my experience Special Effects has at least four times the staying power, and this is with a normal shampoo regimen.
If you're in a small town, then I recommend this website if you're going to make a habit of it. Click on the name of the color to see it on people's heads. These are REALLY nice people out of Indiana and I have dealt with them several times. It's more worthwhile to me to pay for shipping because I buy in bulk. I actually end up spending less.
I don't know know what color your kiddo is naturally, but with brown/dark hair you're going to get a "wash" or "tint" effect that can still be quite dramatic. With naturally blonde hair you're going to get more tinting. My kiddo has sun-bleached hair and it's pretty porous and visible. My personal opinion is that while you can get a dramatic effect with bleaching, that's a pretty risky way to do it with kids. My big kid was subjected to bleach at her dad's house for a wedding to make her "presentable" and all she remembers two years later is the bleach burns. Sad panda.
Words to the wise for new hair-dyers, especially of little kids:
Special Effects is VERY stain-y and transfers to almost anything. Set yourself up in a "safe" place that can get trashed, or can be cleaned easily. Old clothes and dark/trash towels are a must! The dye can transfer onto combs, pillowcases, shirt collars, anything, especially when wet. Think of it as a gooier version of food dye. You know how that stuff can spread. By the way, dyeing outside in the summer can be fun, and a smart idea.
During dyeing (I recommend a half-hour), keep a close watch, especially if you have a wiggle-worm. Once I was doing my big kid's hair when she was about four and my phone rang. I turned around to grab it, and, BAM, there she was rolling on the floor, spreading a pink stain all over the apartment carpet. EEP! I recommend wrapping the colored hair in a bun or ponytail, depending on how much you're going to do. Sometimes I will wrap hair in foil to contain it.
Use gloves! Use gloves! They can be had at the hardware, beauty supply, or drugstore.
Get creative! I have done my kid's whole head, but I have also done her tips and streaks, which is also fun, still dramatic and less messy.
I don't know how independent your kiddo is, but I still help mine the first time she gets in the shower, by directing her how to rinse effectively. I remind her to hold her head back and to give it a good rinse before she digs her hands in. If your kid can put her head upside down in a sink, I recommend this the first time. You may consider wearing gloves to rinse her head, to keep your hands less stained. I also find it handy to jump into the shower with her to assist her hands-on.
To have a really bright color, you want to do less rinsing and shampooing the first time out. If you are okay with a more pastel color, and want to minimize transferring the color around your house, shampoo a couple of times and try to get the water to run clear(ish) before containing it in a towel.
4. Damage control
The price of colored hair, especially if it's past the ears, is that the color will spread a bit. But there's some things you can do to help.
If it gets on cloth and won't come out, trying exposing it to sun. The sun saved me with my apartment carpet disaster…after a while the sun "ate" the color until there was no trace. Repeated machine washings for most clothes will remove the color eventually.
Something bleach-related usually works with the tub, sink, or other parts of the bathroom. Try Soft Scrub or Comet and if it's stubborn and the surface can take it, make a paste and let it sit for a while under a wet paper towel or something.
On the skin, the best thing to do is prevention, such as wearing gloves. If her neck or face ends up with stains, I have had good luck with scrubbing gently with a washcloth and then following up with drugstore face wipes like these.
Makeup remover can help. too. For my own body I use exfoliating gloves, but this would probably be too harsh for little ones. If you shower the stains can linger in unexpected places as it runs off your body. I have hopped out of the shower, dried off, and then discovered orange streaks on the back of my leg. Grand. Sometimes I wrap my hair in a towel when it's clean and rinsed, and then rinse my body one more time to make sure.
One more tip to keep in mind is that the darker the dye, the more it will spread around. I'm not sure about dark skin, but in my experience as a light-skinned person, when I pick shades that are in the same color family as my skin (pinks or warm reds) it follows that they show up less as stains on my face and neck. Recently I did my kid's hair midnight blue, and streaks on her neck were a lot more obvious.
Don't worry, after a few shampoos the rest of the color just "clings" and you will hardly notice any leeching anymore.
Here are some good light colors:
Cupcake pink (very long-lived as a light pink)
Joyride (light purple which fades to pink)
Burgundies, blues, and deep reds will be more challenging. Blue velvet is what we used recently, which was tricky, but has faded to a VERY pretty cornflower.
I know this sounds complicated, but I wanted to share the benefit of ten-plus years of experience with you. Have fun experimenting!
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