I sat down on a park bench and stood up with gum on the back of my jeans. My first instinct was to reach back and pull it off. That was the wrong move. The gum stretched, split, and worked itself deeper into the denim. By the time I stopped pulling I had made the situation significantly worse. I spent the walk home trying to remember if I had any ice at home and whether that was actually a thing that worked or just something people said.
It works. But the reason it works is not what most people think, and understanding the reason is what tells you why everything else you are probably trying is failing. Getting gum out of clothes is not a chemistry problem. It is a physics problem. Once you treat it that way, the fix is simple.

The Short Answer:
To get gum out of clothes: place the garment in a sealed plastic bag with the gum exposed and freeze for one to two hours until the gum is completely solid. Scrape the hardened gum off with a plastic spoon or dull knife. Wash the garment normally to remove any residue.
If you cannot get to a freezer: press ice directly onto the gum for 15 to 20 minutes until it hardens completely, then scrape. Or spray with canned air (computer keyboard duster) to freeze it instantly.
Do not pull at room-temperature gum. Do not put the garment in the washing machine before the gum is removed. Do not put anything with gum in the dryer.

Why Gum Is Different From Every Other Stain
Every other stain in a laundry guide is a chemistry problem. A pigment, a protein, an oil: something that can be dissolved, broken down, or lifted from fabric fibers with the right cleaning agent. Gum is not a stain in that sense. It does not absorb into fabric. It bonds to fabric. And the chemistry of that bond is what makes everything you would normally try completely useless.
Gum base typically contains synthetic polymers: primarily polyisobutylene, styrene-butadiene rubber, and polyvinyl acetate, plus resins, plasticizers, and waxes. These are the same class of materials used in car tires and industrial rubber. They are hydrophobic, meaning they repel water. They are insoluble in water. And critically, hydrophobic polymers attract other hydrophobic surfaces, which is exactly what fabric fibers are.
When you chew gum, the water-soluble components (sugar and flavoring) gradually dissolve and are released. What remains is the pure polymer base: synthetic rubber, now warm and pliable, adhering to whatever surface it contacts. When it contacts fabric, the polymers lock into the fiber structure through hydrophobic attraction. Water cannot reach them because water and hydrophobic polymers repel each other. Soap helps with many oil-based stains but cannot dissolve a cross-linked synthetic rubber polymer. Scrubbing spreads it. Pulling stretches it, and stretched polymers work their way deeper into the weave.
This is why nothing in your usual cleaning toolkit touches gum. You are not dealing with a substance you can dissolve. You need to change its phy 

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